TRANSICIONES IRREVERSIBLES EN REDES COMPLEJAS · en redes complejas (Pikovsky et al., 2002; Osipov et al., 2007). Este proce-so, inherente a sistemas con un gran numero de elementos

CENTRO DE TECNOLOGIA BIOMEDICA

ESCUELA TECNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACIONUNIVERSIDAD POLITECNICA DE MADRID

TESIS DOCTORAL:

TRANSICIONES IRREVERSIBLESEN REDES COMPLEJAS

Adrian Navas Santo-TomasMCs en Fısica Fundamental

SUPERVISORES:

Inmaculada Leyva Callejas

Alexander Pisarchik

2015

Dedicado amis padres, por todo.

I

AGRADECIMIENTOS

Gracias a...

Luis Garay, por los intermedios hablando de cine, por sus papeles este-lares, por preguntarme en el pasillo de la Facultad “¿Pero tu que quiereshacer?” y meterme el gusanillo de la investigacion cuando nunca me lo habıaplanteado.

David Gomez-Ullate, por proponerme una tesina cuando le pedı un tra-bajo y por abrirme la puerta.

Inmaculada Leyva, por acoger y tenerle una paciencia infinita a un estu-diante de Master que buscaba cerebros y encontro osciladores.

Stefano Boccaletti, por contar conmigo sin conocerme y por la bisteccafiorentina.

Francisco del Pozo y al personal del CTB/UPM por darme la posibilidadde realizar esta tesis.

Jose Antonio Villacorta por compartir conmigo el mejor ultimo ano demi beca.

Y a todos los que habeis estado ahı, compartiendo vuestro camino yvuestro conocimiento conmigo sin pedir nada a cambio, en esa red invisibledel dıa a dıa.

III

INDICE GENERAL

Abstract VII

Resumen IX

1. Introduccion 1

2. Sistemas complejos 7

2.1. Teorıa de redes complejas . . . . . . . . . . . . . . . . . . . . 8

2.1.1. Propiedades generales . . . . . . . . . . . . . . . . . . 8

2.1.2. Clasificacion de redes . . . . . . . . . . . . . . . . . . . 9

2.1.3. Centralidad en redes . . . . . . . . . . . . . . . . . . . 11

2.2. Sistemas dinamicos no lineales . . . . . . . . . . . . . . . . . . 14

2.2.1. Soluciones de equilibrio y estabilidad . . . . . . . . . . 15

2.2.2. Soluciones periodicas . . . . . . . . . . . . . . . . . . . 19

2.2.3. Bifurcaciones . . . . . . . . . . . . . . . . . . . . . . . 21

2.3. Sincronizacion de sistemas dinamicos en redes complejas . . . 30

2.3.1. El modelo de Kuramoto original . . . . . . . . . . . . . 30

V

2.3.2. El modelo de Kuramoto en redes complejas: la ruta ala sincronizacion . . . . . . . . . . . . . . . . . . . . . 37

2.3.3. El modelo de Kuramoto con inercia . . . . . . . . . . . 44

2.3.4. Introduccion a la sincronizacion explosiva . . . . . . . . 48

3. Transiciones de fase irreversibles 57

4. Conclusiones 97

A. Variedades invariantes 103

VI

ABSTRACT

In this thesis we address the emergence of synchronization in systems ofcoupled oscillators in complex networks. We focus our attention on a par-ticular kind of discontinuous transitions, named explosive synchronization,where the system changes abruptly from an incoherent state to a synchronousstate. This emergent phenomena is analogous to those first order transitionstypically associated with changes among the aggregate states of matter, andit is important in many different fields, such as spontaneous synchronizationof neurons or spontaneous desynchronization in power grids.

To analyze it, we introduce some methods of increasing generality inorder to induce such a discontinuous transition by acting over the topologyand the natural frequencies in several different ways. Likewise, we addressthe study of a more complex model in order to acquire deeper knowledgeon the properties of this kind of transitions, where a hysteretic behavior isspecially relevant.

Finally, we propose a new quantitative approach in order to find the im-portance of each node in the route to synchronization, aiming to provide acharacterization of the effects over the network’s units of the different met-hods able to induce an explosive transition. This approach allows us to showthe inner mechanisms behind such explosive behavior in networks of coupledoscillators, being rooted by a frustration of the local synchronization processprevious to the emergence of global coherence.

VII

RESUMEN

En esta tesis se aborda la emergencia de sincronizacion en sistemas deosciladores acoplados. En particular, nos centraremos en la emergencia de untipo de transicion discontinua entre el estado incoherente y el estado sıncrono,llamada transicion explosiva. Este fenomeno es analogo al de las transicionesde fase de primer orden asociadas a los cambios de agregacion de la materia,cuya importancia abarca diversos campos, desde la sincronizacion espontaneade redes neuronales al riesgo de desincronizacion subita entre los osciladoresque componen la red de suministro de potencia electrica.

Para analizar el problema, se introducen varios metodos de creciente ge-neralidad cuyo efecto es inducir una transicion explosiva al imponer una seriecondiciones sobre la topologıa y las frecuencias naturales de cada oscilador.Ası mismo, se aborda el estudio de un modelo algo mas complejo con carac-terısticas similares para entender en mayor profundidad las caracterısticasasociadas a este tipo de transiciones, siendo la histeresis una de las masdestacadas.

Finalmente, se propone un metodo cuantitativo para describir la impor-tancia de cada nodo en el proceso de sincronizacion con el objetivo de estudiary caracterizar el efecto sobre los nodos del sistema de los diversos metodosque inducen una transicion explosiva. Este nuevo enfoque permite descubrirun proceso de frustracion de la sincronizacion local en redes de osciladoresacoplados, siendo el responsable de la emergencia de la sincronizacion explo-siva.

IX

CAPITULO 1

INTRODUCCION

Al contrario de lo que sucedio durante el siglo XX, la Ciencia del si-glo XXI se caracteriza por disponer de numerosos metodos experimentalesque le permiten obtener ingentes cantidades de datos. Hoy dıa, el volumende estos datos puede ser tal que ya solo su gestion es un desafıo, cuantomas su analisis. Por tanto, el desarrollo de metodos que permitan explorarlas propiedades de sistemas colectivos con un gran numero de elementos sevuelve imprescindible. La gran revolucion tecnologica en los campos de lastelecomunicaciones y el VLSI (very large scale integration) ha tenido unaparticipacion fundamental en la espectacular mejorıa que la generacion y al-macenamiento de datos ha experimentado, conformando no solo un problemade interes economico sino un campo de experimentacion practicamente ilimi-tado. La accesibilidad de estos datos ha propiciado un acercamiento de lasciencias menos formales a los metodos matematicos de las ciencias naturales.Esta eclosion en diferentes areas ha promocionado a su vez el desarrollo ygeneralizacion de metodos originalmente adscritos a campos de investigacionmas especializados, como el estudio de sistemas caoticos, formacion de patro-nes, analisis de senales, machine learning o teorıa de la informacion. Ahoraes comun encontrar analisis economicos basados en modelos estocasticos, ca-racterizar procesos epidemicos mediante automatas celulares o cuantificar laspropiedades de un idioma en terminos de teorıa de grafos.

La cuestion fundamental que esta situacion plantea es como tratar con unnumero tan elevado de elementos a la hora de intentar predecir la evolucionen el tiempo de las dinamicas subyacentes. En este contexto, hay que destacarlos trabajos pioneros de Ludwig Boltzmann en uno de los hitos de la fısicamoderna, la Mecanica Estadıstica, en la cual los fenomenos macroscopicos

1

CAPITULO 1. INTRODUCCION

emergentes de la materia se derivan de la mecanica clasica mediante metodosprobabilısticos. La posterior revolucion conceptual introducida por la FısicaCuantica y, mas tarde, por la Teorıa del Caos, ha consolidado el papel dela probabilidad en contra del concepto clasico de trayectoria, reforzando elcaracter impredecible de los eventos reales, donde conocer las condicionesiniciales que caracterizan un sistema (si es que es posible) es insuficientepara conocer su estado final.

Todos estos factores han hecho especialmente relevante el estudio de losllamados sistemas complejos, donde el esfuerzo combinado de muchas disci-plinas permiten analizar aquellos sistemas con un gran numero de elementosen interaccion que necesitan de un enfoque holıstico para su comprension. Enparticular, tres pilares fundamentales sostienen la formulacion de los sistemascomplejos. En primer lugar, la Teorıa de Redes (Berge, 1958; Boccaletti et al.,2006; Newman, 2010) establece el contexto formal necesario para estudiar laspropiedades estructurales de sistemas con un gran numero de elementos inter-conectados. Los logros obtenidos por sı misma son suficientes para justificarsu auge en los ultimos anos, aplicandose, por ejemplo, en la identificacionde genes y proteınas relevantes en multiples procesos, en el analisis de losflujos de informacion en redes sociales o en estudios sobre la robustez deredes tecnologicas reales, e. g. las redes de energıa electrica. Por otro lado,la Teorıa de Sistemas no Lineales (Jordan and Smith, 1987; Strogatz, 2001;Wiggins, 2003) permite trabajar con sistemas donde, ademas de la estructuratopologica, cada elemento tiene una cierta dinamica natural propia. De estaforma, los metodos de analisis de ecuaciones diferenciales no lineales permi-ten investigar tanto el comportamiento propio de los elementos del sistemacomo los cambios en la dinamica colectiva mediante tecnicas que permitenesquivar o vencer las limitaciones derivadas de la no analiticidad de los siste-mas reales. Por ultimo, la Mecanica Estadıstica (Reichl and Prigogine, 1980;Stanley, 1987) es el primer hito cientıfico en el estudio de la conexion entrelas partes de un sistema a escala microscopica y los fenomenos emergentes aescala macroscopica. Sus herramientas permiten describir, apoyandose en labase matematica de la Teorıa de Sistemas no Lineales, procesos tales comotransiciones de fase, sistemas estocasticos o sistemas fuera del equilibrio.

Dentro del amplio espectro de fenomenos emergentes en sistemas comple-jos, esta tesis se encuadra exclusivamente en el contexto de la sincronizacionen redes complejas (Pikovsky et al., 2002; Osipov et al., 2007). Este proce-so, inherente a sistemas con un gran numero de elementos conectados entresı a traves de una cierta estructura relacional, se manifiesta en numerosossistemas naturales y biologicos donde las unidades tienden a interactuar de

2


forma coordinada bajo determinadas circunstancias. Como ejemplos, bastenombrar el funcionamiento de las celulas neuronales, las funciones metaboli-cas, las epocas de apareamiento, el uso del ritmo en los ritos tribales, lasfases del sueno, las redes de telecomunicaciones o las redes de distribucion deenergıa electrica. Ası pues, el grado de sincronizacion depende de cuan coordi-nados esten los elementos de cada sistema. Matematicamente, dicho grado secuantifica a traves de un cierto parametro de orden, cuya evolucion es tıpica-mente funcion de la fuerza de interaccion entre sus elementos. De esta forma,si la interaccion es debil, el sistema se mantiene en un estado incoherente oasıncrono. Segun se incrementa dicha interaccion, la dinamica del sistema sereorganiza microscopicamente hasta alcanzar un cierto acoplamiento crıticotal que sus elementos pueden sincronizar entre sı. Si la interaccion es fuerte,el sistema es capaz de sincronizar completamente. Esta transicion del esta-do incoherente al estado sıncrono puede a su vez clasificarse en dos clases.Por un lado, si el valor del parametro de orden es una funcion suave de laintensidad de la interaccion, hablamos de transiciones de fase continuas, enanalogıa con las transiciones de fase de segundo orden, e. g. la orientacionde los dipolos magneticos en presencia de un campo magnetico externo. Noobstante, bajo ciertas condiciones el sistema evoluciona subitamente de unestado totalmente asıncrono a un estado totalmente coherente, siendo habi-tual la irreversibilidad del proceso a traves de un fenomeno de histeresis, yen tal caso hablamos de transiciones de fase discontinuas, en analogıa con lastransiciones de fase de primer orden, e. g. los cambios de estado clasicos dela materia.

Las condiciones que determinan si la transicion sera continua o discon-tinua y/o irreversible son el eje central de esta tesis, y dependen tanto delas propiedades dinamicas de cada elemento como de su estructura relacio-nal. Los primeros resultados al respecto (Pazo, 2005) demostraron que alimponer una distribucion uniforme de frecuencias naturales en un sistemade osciladores de Kuramoto (Kuramoto, 1975), los osciladores se mantenıandesincronizados hasta alcanzar un cierto acoplamiento crıtico, a partir delcual el sistema sincronizaba completamente con tal de esperar un tiemposuficiente, generando una discontinuidad en torno al punto crıtico1. Al pro-poner su modelo, Kuramoto ya hizo notar como una distribucion bimodalde frecuencias naturales forzaba la aparicion de dos subpoblaciones local-

1Cabe destacar que la discontinuidad aparece como un cambio abrupto en el parametrode orden al incrementar el parametro de control, si bien este resultado asume que estamosevaluando el parametro de orden para tiempos asintoticos. Naturalmente, dado un ciertovalor del parametro de control, el sistema siempre evolucionara en el tiempo de formacontinua hacia su estado final.

3


mente sıncronas en torno a los maximos de la distribucion, alterando laspropiedades de la transicion. Este fenomeno fue explorado posteriormenteencontrando transiciones discontinuas y con histeresis (Bonilla et al., 1992;Crawford, 1994; Martens et al., 2009). No obstante, fue la publicacion de unartıculo en el campo de percolacion (Achlioptas et al., 2009) el que atrajo laatencion sobre la existencia de una inesperada transicion abrupta, bautizadacomo transicion explosiva. En el contexto clasico de percolacion, la transi-cion se produce al aumentar el numero de enlaces de un sistema inicial con nnodos aislados. Si el numero de enlaces anadidos esta por debajo de un cier-to umbral, el tamano de las componentes conexas (subconjuntos de nodosconectados entre sı) es de orden O(log(n)), coexistiendo multiples compo-nentes desconectadas entre sı. No obstante, para un cierto numero crıtico deenlaces anadidos emerge una componente cuyo tamano es de orden O(n).Este comportamiento es un ejemplo clasico de transicion de segundo orden,pues una vez pasado el umbral crıtico, el tamano de la componente de ordenO(n) se incrementa de forma continua. Sobre esta base, el caso propuestopor Achlioptas et al. consistıa en aumentar el numero de enlaces siguiendo loque los autores bautizaron como la regla del producto, dando lugar ası a unatransicion discontinua. Este fenomeno fue rapidamente exportado al contex-to de sistemas dinamicos, proliferando una serie de trabajos que describıantransiciones explosivas en redes de osciladores de Kuramoto para topologıasheterogeneas (Gomez-Gardenes et al., 2011), ası como en sistemas caoticos(Leyva et al., 2012), siempre y cuando la frecuencia natural de cada nodo ifuera igual a su grado o numero de vecinos, ωi = ki. A este tipo de transicionabrupta se le llamo sincronizacion explosiva (ES por su nombre en ingles).

Estos tres artıculos (Achlioptas et al., 2009; Gomez-Gardenes et al., 2011;Leyva et al., 2012) marcan el inicio de la investigacion expuesta en estatesis, que a lo largo de cinco publicaciones pretende alcanzar tres objetivosfundamentales:

O.1 Estudiar diversos mecanismos que inducen sincronizacion explosiva.

O.2 Entender y unificar los fundamentos de las transiciones explosivas.

O.3 Describir los principios subyacentes a la histeresis que tıpicamente exhi-ben las transiciones irreversibles.

Para ello, esta tesis se ha estructurado en cuatro capıtulos. Tras el presentecapıtulo de introduccion, el Capıtulo 2 se dedica, en primer lugar, a pre-sentar la teorıa de redes complejas (2.1) y la teorıa de sistemas dinamicos

4


(2.2). La combinacion de ambos factores (topologıa y dinamica) determinalas propiedades de la transicion entre el estado asıncrono y el estado coheren-te (2.3) y establece las bases de la transicion explosiva (2.4). Posteriormente,el Capıtulo 3 contiene las publicaciones derivadas del estudio de estas tran-siciones irreversibles, cuyos resultados se resumen a continuacion:

1. Hysteretic transitions in the Kuramoto model with inertia (Olmi et al.,2014)

Si bien este artıculo no se engloba dentro del fenomeno de la sincro-nizacion explosiva, describe un modelo de Kuramoto con un terminode inercia (Tanaka et al., 1997a,b), donde aparecen los dos fenomenosfundamentales de las transiciones de primer orden: una discontinuidaden el parametro de orden y un fenomeno de histeresis. Este artıculocompleta el estudio inicial generalizandolo al caso de redes complejas,con aplicacion a la red de energıa electrica italiana, y sirve como mo-tivacion de la ubicuidad e importancia de las transiciones irreversiblesen sistemas reales.

2. Explosive transitions to synchronization in networks of phase oscillators(Leyva et al., 2013a)

Tras los primeros resultados de explosividad en redes heterogeneas, seplantea la cuestion de si es posible inducir una transicion explosiva enel caso de redes homogeneas. La primera propuesta al respecto consisteen generar las redes de forma aleatoria con la condicion extra de que unpar de nodos i y j solo se podran conectar si sus frecuencias naturalescumplen la condicion |ωi − ωj| > γ, donde γ es un parametro. Deesta forma, la transicion resulta tanto mas explosiva cuanto mayor seael valor de γ, siempre y cuando este por encima de un cierto umbralmınimo γc.

3. Explosive synchronization in weighted complex networks (Leyva et al.,2013b)

El siguiente paso es considerar la extension natural del modelo anteriora cualquier tipo de red. Para ello, se considera la matriz que define laconectividad de una red dada a priori y se ponderan los enlaces de talforma que aquellos con distancias |ωi − ωj| pequenas sean muy pocorelevantes. Este metodo es valido tanto para topologıas homogeneascomo heterogeneas con tal de elegir el peso adecuado para los enlaces,i. e. Ωij = |ωi − ωj| y Ωij = |ωi − ωj|lij/

∑j lij respectivamente, siendo

5


lij una particular medida de la importancia del enlace correspondiente(betweenness).

4. Effects of degree correlations on the explosive synchronization of scale-free networks (Sendina-Nadal et al., 2015)

Establecido en los trabajos anteriores un metodo general para indu-cir explosividad en una red cualquiera, exploramos los lımites de lacondicion de correlacion ωi = ki en redes heterogeneas como inductorde transiciones explosivas. De esta forma, encontramos que el metodode construir la red heterogenea es determinante a la hora de inducirexplosividad, y lo relacionamos con las correlaciones existentes en ladistribucion conjunta de grado P (k, k′).

5. Effective centrality and explosive synchronization in complex networks(Navas et al., 2015)

Para concluir el estudio de la explosividad en redes complejas, propone-mos un nuevo enfoque para el analisis cuantitativo de la evolucion delsistema en su ruta a la sincronizacion. De esta forma, somos capacesde analizar y unificar los diferentes metodos propuestos en la literatu-ra, encontrando que la sincronizacion explosiva es el resultado de unafrustracion en la emergencia de la sincronizacion.

Finalmente, el Capıtulo 4 reexamina el fenomeno de las transiciones irre-versibles a partir de los resultados expuestos en los artıculos precedentes,concluyendo cuales son los aspectos principales involucrados en dichas tran-siciones y proponiendo algunas lıneas de trabajo futuro.

6

CAPITULO 2

SISTEMAS COMPLEJOS

Una primera aproximacion al concepto de sistema complejo consiste enestablecer como tal a un conjunto de elementos que interaccionan a traves deuna estructura relacional no trivial y de unas reglas que describen la dinami-ca local de cada elemento. No obstante, esta definicion es insuficiente paraque un sistema sea categorizado como complejo, pues para ello es necesarioincluir la impredictabilidad de ciertos sucesos. Es precisamente en este con-texto que el concepto de complejidad adquiere sentido: los sistemas complejosexhiben, a escalas macroscopicas, fenomenos emergentes cuyas propiedadesno se pueden deducir unicamente a partir de los subsistemas microscopicosque lo componen (Gell-Mann, 1995; Fuchs, 2014). Ası pues, en el estudio dela complejidad se hacen necesarios dos enfoques microscopicos: uno estructu-ral, para poder describir correctamente las conexiones entre los elementos delsistema, y otro dinamico, donde establecer el comportamiento fundamentalde cada elemento. Del estudio conjunto de la topologıa del sistema, de ladinamica de sus elementos y de sus interacciones se espera entender aque-llos fenomenos macroscopicos caracterısticos de los sistemas complejos, entrelos cuales el fenomeno de sincronizacion ha logrado, como veremos, un lu-gar predominante en las ultimas decadas. Para ello, estudiaremos en primerlugar los aspectos relacionados con el analisis estructural del sistema e in-troduciremos las herramientas necesarias para describir tanto la dinamica decada elemento como los cambios en la dinamica global, aplicando finalmenteambas teorıas al estudio de los modelos estudiados en la presente tesis.

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CAPITULO 2. SISTEMAS COMPLEJOS

2.1. Teorıa de redes complejas

A la hora de describir la estructura de un sistema complejo, es necesariodesarrollar una serie de herramientas que nos permitan abordar el estudio desistemas cuyo tamano y complejidad exceden con mucho enfoques analıticoso descriptivos. Para ello, el uso de la Teorıa de la Probabilidad y de losmetodos propios de la Mecanica Estadıstica, ası como su aplicacion mediantemetodos numericos, se vuelven indispensables. La Teorıa de Redes, basadaen la Teorıa de Grafos (Erdos and Renyi, 1959; Bollobas, 1998b,a; Westand otros, 2001; Bondy and Murty, 2008), se ha perfilado en los ultimosanos como un excelente candidato en la busqueda de regularidades en redesreales, ası como en la elaboracion de algoritmos de analisis y modelos quecuantifiquen y expliquen las propiedades de dichas redes (para una revisioncompleta de la teorıa de redes complejas, se puede consultar (Newman, 2003;Boccaletti et al., 2006)).

Se entiende por red un conjunto de nodos y conexiones, de tal forma queel numero de nodos determina el tamano de la red, mientras que el numero deconexiones y su distribucion determinan sus propiedades topologicas. Formal-mente, una red se define mediante una 2-tupla (N ,L), dondeN es el conjuntoformado por los nodos o vertices de la red y L es un subconjunto de paresde N , indicando ası que existe una conexion entre los elementos de dichospares. En particular, si los enlaces determinan pares no ordenados (el enlacea → b es indistinguible del b → a) hablaremos de redes bidireccionales o nodirigidas. Si el orden de los pares es determinante, entonces la red sera unidi-reccional o dirigida. En base a esto, podemos representar una red medianteuna matriz de adyacencia A, tal que Aij = 1 si el par (i, j) ⊂ L, y Aij = 0en el caso contrario. Por tanto, que una red sea bidireccional (unidireccional)se traduce en que su matriz de adyacencia sea simetrica (asimetrica). Final-mente, aunque sea un aspecto asociado a la dinamica, es comun encontrarsituaciones en las que a cada enlace se le hace corresponder un determinadopeso (Barrat et al., 2004). En tal caso, la matriz de adyacencia toma valorescontinuos, tıpicamente normalizados, de forma que Ai,j → Wi,j ∈ (0, 1).

2.1.1. Propiedades generales

La primera propiedad que caracteriza estadısticamente una red es su dis-tribucion de grado. Definimos el grado de un nodo i como el numero deenlaces que inciden en dicho nodo, es decir, ki =

∑Nj=1Aij. Por tanto, la

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distribucion de grado P (k) determina la probabilidad de encontrar un nodocon un cierto grado k. Ası mismo, la probabilidad de que un nodo de grado keste conectado con un nodo de grado k′ esta determinada por la distribucionconjunta P (k, k′). Esta distribucion permite caracterizar las redes en funcionde la existencia de correlaciones de grado, definiendo una red asortativa (di-sasortativa) como aquella con una correlacion positiva (negativa) entre losgrados de los nodos vecinos.

Respecto a las posibles particiones de la red, se dice que una red es conexasi todos sus nodos estan directa o indirectamente conectados entre sı. Enel caso de redes no conexas, se define una componente conexa como aquelsubconjunto de nodos de la red que cumplen la condicion previa. El conceptode componente conexa es especialmente util en el contexto de percolacion,donde se puede demostrar que en el regimen subcrıtico tienen un tamano deorden O(log(n)). En particular, se define la componente gigante como aquellacomponente conexa cuyo tamano es de orden O(N).

Finalmente, una red presenta una estructura modular si existen parti-ciones de la red donde la razon entre los enlaces dentro de cada particion(intra-enlaces) y los enlaces entre particiones (inter-enlaces) es alta y parti-cularmente mayor que la esperada en el caso de redes aleatorias.

2.1.2. Clasificacion de redes

Las propiedades estadısticas de P (k) determinan el grado medio 〈k〉 ylas fluctuaciones de grado 〈k2〉 como el primer y segundo momento de ladistribucion respectivamente. En funcion de dichas propiedades, podemoscaracterizar dos tipos de redes muy importantes por su extensa aplicacion enel contexto de redes complejas.

Redes homogeneas

Erdos y Renyi inauguraron en 1959 lo que hoy conocemos como teorıade grafos a traves del estudio de redes aleatorias (o ER) (Erdos and Renyi,1959), es decir, aquellas en las cuales la probabilidad p de que un par denodos esten conectados es la misma para todos los pares de la red. Si biense puede construir dicho tipo de red fijando el numero de nodos y enlaces ydistribuyendolos al azar, el proceso mas extendido es el de establecer un grado

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medio 〈k〉 objetivo e ir conectando pares de nodos aleatorios con probabilidadp hasta alcanzar dicho grado medio. Por tanto, en este caso la construccionde una topologıa aleatoria es un proceso binomial, donde la probabilidad deque el nodo i tenga un grado ki = k viene dada por la expresion

P (ki = k) =

(N − 1k

)pk(1− p)N−1−k. (2.1)

Esta probabilidad converge a la distribucion de Poisson si N es suficiente-mente grande (lımite termodinamico) y 〈k〉 es constante:

P (k) = e−〈k〉〈k〉kk!

. (2.2)

Por tanto, la distribucion de grado de las redes de tipo ER esta centradaen torno a un valor medio 〈k〉 y tiene unas fluctuaciones 〈k2〉 finitas, dandolugar a que la mayor parte de los nodos tengan grados semejantes. De estaforma, las redes Erdos-Renyi son el paradigma de redes homogeneas, que sibien constituyen los inicios de la teorıa de grafos, difıcilmente se encuentranen la naturaleza, teniendo que recurrir a otros modelos para poder describirla(Newman et al., 2001; Katifori et al., 2010; Newman and Girvan, 2004; Latoraand Marchiori, 2001).

Redes heterogeneas

Una de las propuestas de modelos alternativos mas relevantes por suadaptacion a multitud de sistemas reales es la de redes libres de escala (oSF, del ingles scale-free), cuyo nombre remite al hecho de que el grado sedistribuye a todas las escalas, sin estar acotado en ningun intervalo concreto1.Estas redes estan generalmente asociadas a distribuciones muy sesgadas quepresentan las llamadas “heavy tails”, es decir, aquellos casos en los que la colade la distribucion se aleja significativamente del valor medio. Como resultado,encontramos topologıas en las cuales la probabilidad de encontrar un nodocon muchas conexiones es pequena, mientras que la probabilidad de encontrarnodos con pocas conexiones es muy grande. Esta propiedad, paradigma delas redes heterogeneas, se manifiesta matematicamente a traves de una ley depotencias en la distribucion de grado p(k) ∼ kγ, introducida por primera vezen el contexto de redes complejas por D. Price al analizar la red de citas entre

1Matematicamente, la ley libre de escala tiene una repercusion extra: si p = xγ , bastaconocer el valor de la funcion para una determinada escala del sistema, p0 = xγ0 , paraconocer el valor de la funcion a cualquier escala, pues p = xγ = (αx0)γ = αγxγ0 = αγp0.

10


artıculos cientıficos (de Solla Price, 1965). Para una excelente introduccion alas leyes de potencia, se recomienda (Newman, 2005).

Computacionalmente, hay dos formas estandar de construir redes libresde escala. La primera de ellas hace uso del modelo de configuracion (CM)(Bender and Canfield, 1978), valido en general para construir redes dadauna determinada secuencia de grados D = k1, k2, ..., kN . En este caso, lasecuencia se obtiene a partir de una distribucion de grado libre de escala,asociando al primer nodo k1 radicales, al segundo nodo k2, y ası sucesiva-mente. Finalmente, los radicales libres se conectan entre sı aleatoriamente,excluyendo autoconexiones, hasta que todos quedan emparejados. La segun-da forma de construir redes SF consiste en utilizar el modelo de crecimientopropuesto por Barabasi-Albert (BA) (Barabasi and Albert, 1999). Partiendode m0 ≥ 2 nodos conectados todos con todos (para asegurarnos de que la redsea conexa), los nuevos nodos se conectan de uno en uno a los nodos de la redseminal, con una probabilidad proporcional al grado de cada nodo objetivo(elegido aleatoriamente de entre los nodos ya existentes). La probabilidadde conexion es por tanto pi = ki/

∑ki, configurando un proceso de creci-

miento selectivo con una seleccion preferencial del enlace. De esta forma, losnodos de grado alto (hubs) tienden a acumular rapidamente mas enlaces queel resto de la red. Finalmente, convergiendo asintoticamente (N → ∞) a ladistribucion teorica de una red SF con γ = 3, 〈k〉 = cte, 〈k2〉 =∞.

2.1.3. Centralidad en redes

Otra propiedad fundamental de las redes complejas es la llamada centra-lidad topologica (Borgatti, 2005), que da cuenta de la importancia de cadanodo en la red. Si bien existen diferentes criterios para determinar la cen-tralidad de un nodo, nos ceniremos a las tres medidas que se usaran en elCapıtulo 3.

Centralidad de grado

En primer lugar, la centralidad de un nodo se puede caracterizar direc-tamente por su grado, ki =

∑Nj=1Aij o, en el caso de redes pesadas, por su

strength si =∑N

j=1Wij. La importancia se mide en funcion del numero devecinos o del peso total que estos adquieren, y es por tanto una medida decentralidad puramente local.

11


Centralidad de autovector

Alternativamente, podemos definir la importancia de un nodo en fun-cion de la importancia de sus vecinos. Esta medida recursiva da lugar a unacentralidad global, llamada centralidad de autovector, que se define de lasiguiente forma:

Sea Xi la centralidad del nodo i. Supondremos que la centralidad del no-do i depende de la centralidad del resto de la red de forma lineal, es decir,Xi ∝

∑j∈Γi

Xj, donde Γi es el conjunto de vecinos de dicho nodo. Usando la

definicion de la matriz de adyacencia, Xi ∝∑N

j=1AijXj, donde ahora pode-mos sumar sobre todos los nodos de la red. En notacion vectorial, podemosreexpresar esta ecuacion como AX = λX, siendo λ la inversa de la constantede proporcionalidad. Esta ecuacion es precisamente la ecuacion de autovalo-res asociada al operador A, de forma que a cada autovalor λ(i) le correspondeun autovector X(i), cuyas componentes son los valores de la centralidad decada nodo. Finalmente, para resolver el problema de la degeneracion de cen-tralidades (hay tantos autovectores como numero de nodos), el teorema dePerron-Frobenius garantiza que podemos ordenar los autovalores de tal formaque λ1 ≥ λ2 ≥ ... ≥ λn, donde solo el autovector asociado al mayor autovalortiene todas sus componentes positivas. Dado que la centralidad solo esta de-finida para valores positivos, se concluye que la centralidad de un nodo i esla componente i -esima del autovector X1 asociado al autovalor λ1.

Centralidad de betweenness

La ultima medida de centralidad que usaremos es la llamada betweennesscentrality, una centralidad global que puede ser aplicada tanto a nodos comoa enlaces. Para definirla, debemos introducir previamente el concepto degeodesica o shortest path como aquella secuencia de enlaces que une dosnodos de la red con el mınimo numero de pasos (Wasserman, 1994; Latoraand Marchiori, 2003). Por tanto, una medida de centralidad natural en elcontexto de flujos de informacion es aquella que tiene en cuenta el numerode geodesicas que pasan por un cierto nodo o enlace uniendo cada par denodos de la red. De esta forma,

Bi =∑

j,k∈N , j 6=k

njk(i)

njk, (2.3)

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donde njk(i) es el numero de geodesicas que unen el par de nodos (j, k) pa-sando por el nodo i y nij es el numero total de geodesicas entre (i, j) (pues, engeneral, puede ser mayor que 1). Por tanto, la betweenness centrality permitedetectar cuellos de botella, nodos o enlaces a traves de los cuales la red secomunica entre sı, y que pueden influir fuertemente en la estimulacion o frus-tracion de procesos que tengan que ver con la propagacion de interacciones,como veremos en el caso de la sincronizacion explosiva en redes SF.

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2.2. Sistemas dinamicos no lineales

Si la Teorıa de Redes es la base en la descripcion y analisis de la estructu-ra de los sistemas complejos, la Teorıa de Sistemas no Lineales (Jordan andSmith, 1987; Strogatz, 2001; Wiggins, 2003) permite establecer un segundonivel de complejidad, otorgando a los nodos de la red una dinamica parti-cular. Ası mismo, gracias a la Teorıa de Bifurcaciones podemos analizar loscambios en el comportamiento de la dinamica colectiva del sistema, siendopor tanto la teorıa fundamental en el estudio de las transiciones de fase. Elcomportamiento dinamico de cada elemento del sistema viene descrito, engeneral, por una ecuacion diferencial o un mapa:

x = f (x, t;µ) , (2.4)

x 7→ g (x;µ) , (2.5)

donde x ∈ U ⊂ Rn, t ∈ R1 y µ ∈ V ⊂ Rp, siendo U y V entornos abiertosen Rn y Rp respectivamente. A partir de ahora, nos restringiremos al caso desistemas dinamicos continuos y autonomos, es decir, aquellos que cumplen laecuacion

x = f (x;µ) , (2.6)

donde x ≡ dxdt

es la derivada parcial con respecto a la variable independientet, identificada en este contexto con el tiempo, y µ es un conjunto de parame-tros caracterısticos del sistema. De esta forma, la ecuacion (2.6) describe uncampo vectorial cuyas soluciones estan determinadas por un mapa de ciertointervalo I ⊂ R1 a Rn, esto es:

x : I → Rn,

t 7→ x(t),

cuyo codominio Rn determina el espacio de fases de las posibles soluciones,siendo estas tangentes al campo vectorial definido en (2.6). Dado que, por logeneral, es difıcil encontrar soluciones analıticas para problemas no lineales,conocer las propiedades geometricas de las posibles soluciones es, en muchoscasos, mas util que conocer la solucion en sı misma. Por tanto, de ahora enadelante fijaremos nuestra atencion en describir las propiedades geometricasde las soluciones en el espacio de fases, ası como su comportamiento asintoticopara tiempos t→∞.

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2.2.1. Soluciones de equilibrio y estabilidad

Para obtener informacion sobre la geometrıa del espacio de fases sin co-nocimiento a priori de las soluciones del sistema, una primera aproximacionconsiste en buscar aquellas soluciones que no dependen del tiempo y, a con-tinuacion, estudiar su estabilidad, es decir, el sentido de los flujos generadospor el campo vectorial alrededor de dichas soluciones. De esta forma, se puedecaracterizar el comportamiento cualitativo de las trayectorias en el espaciode fases desde un punto de vista puramente geometrico.

Un punto fijo o solucion estacionaria de equilibrio es un punto x ∈ Rn talque

f(x) = 0. (2.7)

En el caso de sistemas unidimensionales, los flujos del espacio de fases se re-ducen a trayectorias cuyo movimiento es monotono o constante. En sistemascon mayor dimensionalidad, el incremento de los grados de libertad otorga alas trayectorias una variedad de comportamientos mayor, como veremos masadelante. No obstante, antes de analizar la estabilidad de los puntos fijosdebemos introducir previamente el concepto de estabilidad de una solucioncualquiera x(t) de la ecuacion (2.6):

Definicion (Estabilidad de Lyapunov.) x(t) se dice que es estable si,dado ε > 0, existe un δ = δ(ε) > 0 tal que, para cualquier otra soluciony(t) de (2.6) que satisface |x(t0)− y(t0)| < δ, entonces |x(t)− y(t)| < ε parat > t0, t0 ∈ R.

Definicion (Estabilidad Asintotica.) x(t) se dice que es asintoticamenteestable si es estable y, para cualquier otra solucion y(t) de (2.6), existe unaconstante b > 0 tal que, si |x(t0)− y(t0)| < b, entonces lım

t→∞|x(t)− y(t)| = 0.

En el caso de que la trayectoria de interes sea un punto fijo del sistema,x(t) = x(t), la estabilidad en el sentido de Lyapunov implica que cualquiertrayectoria suficientemente cerca del punto fijo permanece cerca para todotiempo t > t0, mientras que la estabilidad asintotica, como su propio nombreindica, establece que las trayectorias suficientemente cercanas al punto fijose aproximan a x(t) segun t→∞ (Fig. 2.1).

Si bien estas definiciones nos permiten entender el concepto de estabili-dad, no nos dan un metodo practico para determinar si un punto fijo es es-table o no. Para estudiar el comportamiento de las soluciones en un entorno

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Figura 2.1: (a) Estabilidad de Lyapunov. (b) Estabilidad asintotica.

del punto fijo, basta con introducir una pequena perturbacion x = x(t) + yen (2.6) y estudiar la dinamica asintotica de la perturbacion y linealizandola ecuacion en torno al punto fijo. De esta forma, la expansion en serie deTaylor resulta:

x = ˙x(t) + y = f (x(t)) +Df (x(t)) y +O(|y|2), (2.8)

donde Df es la matriz Jacobiana de f y | · | es la norma en Rn. Usando laidentidad (2.6) en la ecuacion (2.9), obtenemos finalmente

y = Df (x(t)) y +O(|y|2), (2.9)

que es la ecuacion diferencial que rige la dinamica de una pequena pertur-bacion y en torno al punto fijo x(t). Puesto que la solucion al problemade valores iniciales dado por la ecuacion (2.9) linealizada toma la formay(t) = e

∫Df(x(t′))dt′y0, la estabilidad de la solucion de equilibrio depende del

comportamiento de la integral∫Df (x(t′)) dt′. Afortunadamente, las solucio-

nes de equilibrio son, por definicion, independientes del tiempo, y por tantoDf (x(t)) = Df (x) es una matriz n-dimensional de coeficientes constantes,por lo que la solucion analıtica al problema linealizado existe, y evaluar la es-tabilidad de la aproximacion lineal se reduce a diagonalizar la matriz Df (x)y calcular el signo de sus autovalores2.

2Desde un punto de vista geometrico, diagonalizar Df (x) es equivalente a desacoplar el

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Figura 2.2: Tabla de estabilidad de puntos fijos.

Los casos mas importantes que podemos encontrar en el estudio de laestabilidad del punto fijo en la aproximacion lineal se resumen en la Fig. 2.2,donde nos hemos restringido por simplicidad a sistemas bidimensionales deltipo (2.6): son los nodos, los saddles, las espirales y los centros. El analisis dela estabilidad depende del signo de los autovalores λ1,2 = 1

2(τ±√τ 2 − 4∆) de

la matriz Jacobiana, donde τ = λ1 +λ2 es la traza y ∆ = λ1λ2 es el determi-nante de dicha matriz. Los nodos tienen ambos autovalores del mismo signo(∆ > 0), siendo puntos fijos que atraen (estables) o repelen (inestables) todaslas trayectorias de su entorno. Los puntos de tipo silla de montar (saddles)

sistema linealizado, y por tanto cada autovalor corresponde a una combinacion lineal de lasvariables originales que da lugar a una nueva variable desacoplada del resto, cuya soluciones del tipo u(t) = eλtv, siendo v el autovector correspondiente. Ası pues, λ representa latasa de variacion exponencial de las soluciones en el nuevo sistema de referencia, y portanto la estabilidad del sistema se reduce a estudiar la convergencia (estable) o divergencia(inestable) de dichas soluciones a la solucion u = 0 dada por el signo del correspondienteautovalor.

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tienen un autovalor positivo y otro negativo, siendo por tanto inestables. Loscentros espirales son puntos fijos cuyos autovalores toman valores complejos(λ, λ) y su estabilidad depende del signo de la parte real de λ. De esta forma,si Re(λ) < 0 (Re(λ) > 0), las trayectorias se acercan (alejan) al punto fijo enorbitas espirales, siendo el centro asintoticamente estable (inestable). Final-mente, existen casos degenerados que viven en las fronteras de los sectores dela figura, de los cuales los centros son sin duda los mas importantes, estandoasociados a la zona lımite que separa las espirales estables de las inestables.Por tanto, alrededor de los puntos fijos con Re(λ) = 0 surgen orbitas cerradasneutralmente estables, donde una pequena perturbacion lleva de una orbitacerrada a otra tan proxima como pequena sea la perturbacion.

Si bien podemos estudiar la estabilidad de los puntos fijos en la aproxima-cion lineal, no podemos estar seguros de que esta se mantenga al incluir losterminos no lineales de la ecuacion (2.9), esto es, no hay garantıas de que laestructura de las orbitas de un entorno del punto fijo del campo vectorial enla aproximacion lineal sea esencialmente la misma que en el caso del campovectorial no lineal. No obstante, bajo ciertas condiciones podemos garantizarque la aproximacion lineal es representativa de la dinamica del sistema nolineal:

Definicion (Puntos fijos hiperbolicos.) Sea x = x un punto fijo dela ecuacion x = f (x), x ∈ R. Se dice que x es un punto fijo hiperbolico sitodos los autovalores de Df(x) tienen parte real distinta de cero.

Siempre y cuando Df(x) no tenga autovalores con parte real cero, pode-mos decir que la estabilidad del sistema linealizado sigue siendo valida en elcaso del sistema no lineal original (teorema de Hartman-Grobe). Por contra,la presencia de centros asociados a orbitas cerradas en la aproximacion linealrara vez permite deducir la estabilidad del punto fijo del sistema no linealoriginal, y por tanto sera necesario desarrollar otros metodos para el estudiode los casos que implican autovalores con parte real cero (ver Apendice A).Como consecuencia directa del teorema de Hartman-Grobe, podemos formu-lar el siguiente teorema:

Teorema. Sea x un punto fijo del sistema no lineal x = f (x), x ∈ R.Entonces, x es asintoticamente estable si y solo si todos los autovalores deDf(x) tienen parte real negativa.

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2.2.2. Soluciones periodicas

Ademas de las soluciones de equilibrio, existe otro tipo de soluciones fun-damentales en la teorıa de sistemas no lineales:

Definicion (Soluciones Periodicas.) Una solucion x(t) de la ecuacion(2.6) se dice que es periodica y de periodo T si existe un T > 0 tal quex(t) = x(t+ T ) para todo t ∈ R.

Estas soluciones dan lugar a orbitas cerradas presentes en el espacio de fasesque, al igual que los puntos fijos, existen para todo tiempo t y se puedenclasificar en funcion de los criterios de estabilidad. No obstante, para po-der aplicar dichos criterios debemos redefinir la estabilidad en este contexto,introduciendo previamente los siguientes conceptos. Definimos la orbita po-sitiva que pasa por x0 para t ≥ t0 como:

O+(x0, t0) = x ∈ Rn|x = x(t), t ≥ t0, x(t0) = x0 . (2.10)

Ası mismo, la distancia entre un punto y un conjunto se define a continuacion.Sea S ∈ R un conjunto arbitrario y p ∈ Rn un punto arbitrario. Entonces ladistancia entre el punto p y el conjunto S es:

d(p, S) = ınfx∈S|p− x|.

Definicion (Estabilidad Orbital.) x(t) se dice que es una orbita establesi, dado ε > 0, existe un δ = δ(ε) > 0 tal que, para cualquier otra soluciony(t) de (2.6) que satisface |x(t0)− y(t0)| < δ, entonces d (y(t), O+(x0, t0)) < εpara t > t0.

Definicion (Estabilidad Orbital Asintotica.) x(t) se dice que es unaorbita asintoticamente estable si es estable y, para cualquier otra soluciony(t) de (2.6), existe una constante b > 0 tal que, si |x(t0) − y(t0)| < b, en-tonces lım

t→∞d (y(t), O+(x0, t0)) = 0.

El ejemplo mas simple de una solucion periodica es el de aquellas orbitascerradas que emergen alrededor de un centro (Fig. 2.3). Como se ha dichopreviamente, los centros dados por la aproximacion lineal generalmente sonmuy sensibles a las contribuciones no lineales. No obstante, existe un casoen el cual estos centros son especialmente robustos: los sistemas conservati-vos. Se dice que un sistema no lineal x = f(x) es conservativo si existe una

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Figura 2.3: Mapa de fases de un pendulo simple.

funcion continua E(x) que es constante sobre las soluciones del sistema, esdecir,

E(x) = ∇E(x) · x = ∇E(x) · f(x) = 0. (2.11)

Por tanto, el gradiente de las superficies equipotenciales es siempre perpendi-cular al flujo vectorial, siendo posible foliar el espacio de fases en superficiesequipotenciales, de forma que toda solucion queda confinada en un entornoque le impide degenerar en orbitas espirales.

No obstante, este tipo de orbita periodica, caracterıstica de los sistemaslineales, no es estable frente a perturbaciones, y por tanto nos vemos obli-gados a buscar otro tipo de solucion periodica para explicar la mayorıa desistemas oscilatorios autorregulados presentes en la naturaleza. Ası pues, unciclo lımite es una solucion periodica aislada (a diferencia de las orbitas entorno a un centro), que puede ser tanto estable como inestable en funcion desi las trayectorias de su entorno convergen o divergen del ciclo lımite, y queesta estrictamente asociada a sistemas no lineales3.

3Esta relacion intrınseca con sistemas no lineales se puede entender facilmente en termi-nos de Mecanica Clasica. Considerando el sistema bidimensional x = y, y = f(x, y), dondef(x, y) = −g(x)− h(x, y), el sistema combinado x+ g(x) = −h(x, y) se puede interpretarcomo un sistema conservativo x+ g(x) = 0 sometido a una cierta fuerza externa −h(x, x).La energıa total del sistema conservativo es E = T + V = x2/2 +

∫dxg(x), y la tasa de

variacion de la energıa cuando se tiene en cuenta la fuerza externa es dE/dt = −xh(x, x),donde se ha sustituido la ecuacion x = −g(x)− h(x, x). Por tanto, la condicion de orbitaaislada implica que debe existir una trayectoria cerrada de energıa constante dE/dt = 0que separe una region interior con dE/dt > (<)0 de una region exterior con dE/dt < (>)0,

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Figura 2.4: Formacion de un ciclo lımite entre las regiones dE/dt > 0 (amor-tiguacion negativa) y dE/dt < 0 (amortiguacion positiva).

2.2.3. Bifurcaciones

Una caracterıstica fundamental de los sistemas no lineales es la gran va-riedad de respuestas que presentan frente a condiciones iniciales y cambiosen los parametros propios. Ası pues, un sistema inicialmente en reposo puedecomenzar a oscilar con tan solo cambiar el valor de un parametro, transi-cion que se describe en terminos matematicos mediante el paso de un puntofijo estable a un ciclo lımite para un cierto valor crıtico del parametro encuestion. Por tanto, el estudio de las transiciones entre comportamientosdinamicos distintos o bifurcaciones es fundamental para entender la emer-gencia de fenomenos macroscopicos. Atendiendo a la estructura del espaciode fases cuando la bifurcacion tiene lugar, estas se pueden clasificar en dosgrupos, locales y globales.

de tal forma que las soluciones convergen (divergen) a la solucion periodica, como se ve enla Fig. (2.4). De esta forma, dE/dt = −yh(x, y) ≡ F (x, y) = 0 es la ecuacion de una curvaen el espacio de fases, siendo una curva cerrada si y solo si F (x, y) es de orden n ≥ 2 ensus variables. Dado que F (x, y) = −yh(x, y), la unica forma de satisfacer esta condiciones que el sistema contenga terminos no lineales a traves de h(x, y).

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Bifurcaciones locales

En el primer caso, la bifurcacion hace referencia al cambio de estabili-dad en el entorno de una solucion de equilibrio debido al cambio en algunparametro del sistema. En concreto, una bifurcacion local tiene lugar cuan-do, como resultado del cambio en dicho parametro, la parte real de uno ovarios autovalores del Jacobiano del sistema cambia de signo. Esta definicionaparentemente arbitraria es bastante natural si atendemos a la estructuratopologica del espacio de fases. Como se dijo anteriormente, un punto fi-jo hiperbolico es estructuralmente estable, es decir, su estabilidad no se vealterada por una pequena perturbacion de los parametros del sistema. Portanto, resulta natural que un cambio cualitativo en la estructura topologicadel espacio de fases solo pueda tener lugar cuando la parte real de algunautovalor se hace nula como resultado de la variacion de algun parametro,rompiendo ası la hiperbolicidad del punto fijo. El resultado de esta transicionpuede acarrear fenomenos dinamicos de los mas variado, como por ejemplo, laaparicion o desaparicion de nuevos puntos fijos o de nuevos comportamientostemporales, como periodicidad, cuasiperiodicidad o caos. A su vez, el numerode autovalores con Re(λ) = 0 determinara cuan exotico puede llegar a ser elnuevo comportamiento del sistema.

Comenzaremos el estudio de la ruptura de hiperbolicidad considerandoel caso mas sencillo: cuando la variacion de un unico parametro µ hace queun solo autovalor cambie de signo. La complejidad de este estudio puedeincrementarse en dos vertientes. Por un lado, podemos considerar el numerode parametros necesarios para dar lugar a una bifurcacion, lo que determinala codimension de dicha bifurcacion. En nuestro caso, nos restringiremos alestudio de bifurcaciones de codimension 1. Por otro lado, podemos considerarque haya mas de un autovalor con parte real cero. En esta introduccion tansolo estudiaremos aquellos casos con un maximo de dos autovalores nulos,lo que nos llevara a definir una de las bifurcaciones mas importantes ensistemas dinamicos: la bifurcacion de Poincare-Andronov-Hopf. En amboscasos, el estudio consistira en explorar la ruptura de la hiperbolicidad delpunto fijo en un entorno del parametro crıtico (de ahı el termino bifurcacionlocal), es decir, segun µ se acerca al valor crıtico µc.

Consideremos el siguiente sistema parametrico no lineal:

x = f(x, µ), x ∈ Rn, µ ∈ Rp, (2.12)

donde f es una funcion Cr definida en un un abierto de Rn × Rp. Supon-

22


dremos que el sistema tiene un punto fijo en el origen4 y que dicho puntofijo es no hiperbolico. En el caso mas sencillo, el Jacobiano del sistema tieneun unico autovalor nulo y hay un solo parametro libre. De esta forma, ladinamica del sistema se divide en dos escalas temporales: una rapida, aso-ciada a los autovalores con parte real no nula cuya dinamica es exponencial,y otra lenta, asociada al autovalor con parte real nula, donde tienen lugarlos cambios estructurales del espacio de fases. Por tanto, podemos restringirel estudio del campo vectorial (2.12) al subespacio definido por los autovec-tores correspondientes a la dinamica lenta (la llamada variedad central, verApendice A), de tal forma que la ecuacion (2.12) sobre la variedad central sereduce a

u = f(u, µ), u ∈ R1, µ ∈ R1, (2.13)

tal que

f(0, 0) = 0, (2.14)

∂f

∂u(0, 0) = 0. (2.15)

La ecuacion (2.14) expresa la condicion de punto fijo, mientras que la ecua-cion (2.15) es la condicion de no hiperbolicidad de la proyeccion del campovectorial original sobre la variedad central.

En estas condiciones, existen cuatro tipos de bifurcaciones determinadaspor la correspondiente forma normal, esto es, el campo vectorial asociado adicho comportamiento en su forma mas simple:

a) Bifurcacion de tipo saddle-node5: u = µ+ u2

La variacion del parametro µ crea o destruye un par de puntos fijos, unoestable y otro inestable. El punto fijo correspondiente es u1,2 = ±√−µ.Por tanto, la ecuacion u = 0 tiene dos soluciones reales si µ < 0 y nin-guna si µ > 0. En el primer caso, f ′(u1) = 2

√−µ > 0 (inestable) yf ′(u2) = −2

√−µ < 0 (estable). Cuando µ = 0, el nodo y el punto desilla convergen, siendo f ′(u1,2) = 0, y por tanto µc = 0 es el punto debifurcacion.

4De no ser ası, siempre podemos hacer un cambio del sistema de referencia tal que(x′, µ′) = (x− x, µ− µc), donde (x, µc) satisface f(x, µc) = 0. Por simplicidad, nos referi-remos a las variables en el nuevo sistema de coordenadas sin tildes.

5Aunque literalmente saddle-node significa silla de montar-nodo, la traduccion resultaun tanto incomoda, y por tanto el termino se conserva en ingles.

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b) Bifurcacion transcrıtica: u = µu− u2

Dos puntos fijos, uno estable y otro inestable, intercambian su estabili-dad como consecuencia de la variacion de un parametro. Los puntos fijoscorrespondientes son u1 = 0 y u2 = µ. Por tanto, la ecuacion u = 0 tienedos soluciones reales cuya estabilidad depende del signo de µ. En el casoµ < 0, u1 es estable (f ′(u1) = µ < 0) y u2 es inestable (f ′(u2) = −µ > 0).En el caso µ > 0, u1 es inestable (f ′(u1) = µ > 0) y u2 es estable(f ′(u2) = −µ < 0). Cuando µ = 0, el nodo y el punto de silla convergen,siendo f ′(u1,2) = 0, y por tanto µc = 0 es el punto de bifurcacion.

c) Bifurcacion de pitchfork supercrıtica6: u = µu− u3

La variacion del parametro µ permite transitar de una estructura mono-estable a una estructura biestable y viceversa. Los puntos fijos correspon-dientes son u1 = 0 y u2,3 = ±√µ. Por tanto, la ecuacion u = 0 tieneuna solucion real si µ < 0 y tres si µ > 0. En el primer caso, u1 = 0es estable (f ′(u1) = µ < 0). En el segundo caso, la solucion u1 = 0 sevuelve inestable (f ′(u1) = µ > 0) y aparecen dos nuevas soluciones esta-bles (f ′(u2,3) = −µ < 0) con sus correspondientes cuencas de atraccion(los semiplanos positivo y negativo respectivamente). Cuando µ = 0, lostres puntos fijos convergen, siendo f ′(u1,2,3) = 0, y por tanto µc = 0 es elpunto de bifurcacion.

d) Bifurcacion de pitchfork subcrıtica: u = µu+ u3

Esta bifurcacion es analoga al caso supercrıtico, cuyos correspondientespuntos fijos son u1 = 0 y u2,3 = ±√−µ. El cambio de signo dentro dela raız cuadrada provoca una inversion en la estabilidad de las ramas deldiagrama de bifurcacion, donde los puntos fijos no nulos existen solo siµ < 0. En el intervalo µ > 0, el unico punto fijo (la solucion trivial)es inestable. De esta forma, cualquier solucion con u(0) 6= 0 diverge entiempo finito. Fısicamente este tipo de comportamiento “explosivo” seevita debido a contribuciones de orden superior7. En este caso, podemosintroducir un termino de orden cinco que respeta la simetrıa del problemay compensa la inestabilidad debida al termino cubico. Ası, la ecuacionu = µu+ u3 − u5 da lugar al siguiente diagrama de bifurcaciones:

6Al igual que en el caso de la bifurcacion de tipo saddle-node, conservaremos el terminopitchfork en lugar de horca.

7Recordemos que las formas normales son los terminos principales de un desarrollo deTaylor de la ecuacion dinamica en un sistema de coordenadas adecuado

24


Figura 2.5: Histeresis y discontinuidad.

Como se puede ver, en un entorno de u pequeno el diagrama es equivalen-te al de una bifurcacion de pitchfork subcrıtica, mientras que para valoresmayores de u aparecen dos ramas que atrapan las soluciones divergentes.Este sistema tiene ademas una caracterıstica muy importante: la histere-sis. Al incrementar progresivamente el parametro µ, la solucion trivial esestable hasta el punto crıtico µ = 0, donde la solucion evoluciona haciala rama estable superior, apareciendo una discontuinuidad en sus valoresasintoticos. Ası mismo, al disminuir µ la rama superior mantiene su es-tabilidad hasta alcanzar µ = µc, donde se vuelve inestable, forzando a lasolucion una vez mas a regresar al punto fijo u = 0. Este es el mecanismomatematico para describir las transiciones discontinuas y con histeresisque estudiaremos a lo largo de esta tesis.

Finalmente, consideremos el caso de un solo parametro libre y dos autovalorescon parte real cero, por ser de especial interes en la mayorıa de oscilacionespresentes en sistemas biologicos:

e) Bifurcacion de Poincare-Andronov-Hopf

Esta bifurcacion (tambien llamada bifurcacion de Hopf) es propotipo desistemas en los cuales el cambio en un parametro permite la transicion en-tre una dinamica de reposo asociada a un punto fijo estable y una dinamicaoscilatoria asociada a un ciclo lımite, siendo fundamental en el estudio desistemas excitables (e. g. modelos neuronales). La expresion general de uncampo vectorial no lineal sobre la variedad central bidimensional asociada

25


a dos autovalores complejos con parte real nula viene dada por(uv

)=

(Reλ(µ) −Imλ(µ)Imλ(µ) Reλ(µ)

)(uv

)+

(f 1(u, v, µ)f 2(u, v, µ)

), (2.16)

donde el primer termino corresponde al campo vectorial linealizado8 y elsegundo termino corresponde a la parte no lineal. Por tanto, los auto-valores de la matriz Jacobiana son λ(µ) = α(µ) + iω(µ) y su complejoconjugado λ(µ) = α(µ) − iω(µ), de forma que α(0) = 0 es la condicionde punto fijo no hiperbolico. Se puede demostrar que existe un cambio debase tal que la parte no lineal del sistema (2.23) se reduce a su forma massimple. En coordenadas polares, dicha forma normal toma la expresion

r = αr + ar3 +O(r5),

θ = ω + br2 +O(r4).(2.17)

Si desarrollamos en serie de Taylor los coeficientes α(µ), ω(µ), a(µ) y b(µ)en torno a µ = 0 (es decir, en un entorno de µc donde se produce laruptura de hiperbolicidad), se obtiene

r = α′(0)µr + a(0)r3 +O(µ2r, µr3, r5),

θ = ω(µ) + b(µ)r2 +O(r4),(2.18)

donde ′ implica la derivada con respecto a µ. Despreciando los terminosde orden superior en (2.18), la forma normal (2.17) queda reducida a

r = dµr + ar3,

θ = ω + cµ+ br2,(2.19)

siendo α′(0) ≡ d, a(0) ≡ a, ω(0) ≡ ω, ω′(0) ≡ c y b(0) ≡ b.

De esta forma, resulta inmediato comprobar que la ecuacion radial tienedos puntos fijos,

r1 = 0 y r2 =√−dµ/a, (2.20)

siendo r2 el radio asociado a una orbita periodica, cuya amplitud es pro-porcional a

√|µ|. Tambien parece inmediato que la estabilidad de dicha

orbita depende del signo de d, a y µ, pues la derivada de la ecuacion radialrespecto al radio evaluada en la orbita periodica es

∂rf(r2) = dµ+ 3|dµ| a|a| . (2.21)

8Se asume que el sistema de referencia es tal que la matriz Jacobiana adquiere estaestructura. De no ser ası, siempre se puede encontrar la transformacion lineal necesariapara ello.

26


No obstante, un estudio mas cuidadoso revela que la estabilidad de la orbi-ta periodica solo depende del signo de a debido a la restriccion implıcitaen (2.20). Ası pues, a > 0 implica que d y µ han de tener signos opuestospara garantizar que r2 sea real, y por tanto la orbita periodica es inestable(∂rf(r2) = 2|dµ| > 0). Ası mismo, a < 0 implica que d y µ han de tenerel mismo signo, y por tanto la orbita periodica es asintoticamente estable(∂rf(r2) = −2dµ < 0).

Por simplicidad, en lo que queda de apartado nos restringiremos al casomas simple (d = 1, a = −1, c = 0):

r = µr − r3,

θ = ω + br2.(2.22)

Cuando µ < 0, el origen es el unico punto fijo, siendo asintoticamenteestable, y ası todas las soluciones convergen sobre el formando espirales.No obstante, una vez µ cruza el punto crıtico, el origen se vuelve inestabley un ciclo lımite asintoticamente estable de radio

√µ emerge segun se

incrementa el valor de µ, ilustrando ası la transicion entre una dinamicaestacionaria y una oscilatoria. Una vez mas, podemos considerar el casosubcrıtico (a > 0) y, en analogıa con la bifurcacion de Pitchfork subcrıtica,anadir un termino −r5 a la ecuacion para compensar el caracter desesta-bilizador del termino cubico:

r = µr + r3 − r5,

θ = ω + br2.(2.23)

De esta forma, el resultado de introducir tales cambios en el sistema semanifiesta a traves de un proceso de histeresis, como se vera justificado enel siguiente apartado. En particular, los puntos fijos de la ecuacion radialson los mismos que en el caso de la bifurcacion de Pitchfork subcrıtica,con la salvedad de que ahora excluimos los radios negativos:

r1 = 0, r2,3 =

√(1∓

√1 + 4µ

)/2 (2.24)

Por tanto, para valores de µ ligeramente negativos, un punto fijo estableen el origen y un ciclo lımite estable coexisten, separados por un ciclolımite inestable9. Para µ = 0 el radio de la orbita inestable se reduce

9Cabe destacar la necesidad de la existencia de dicho ciclo lımite inestable, pues porcontinuidad de las trayectorias, un ciclo lımite estable no puede contener en su interior unpunto fijo estable sin que exista algun tipo de separatriz entre ambas variedades invariantes

27


al origen (r2 = 0), mientras que r3 = 1 permanece. Ası pues, cuandoµ > 0 el ciclo lımite inestable desaparece (r2 ∈ C) y el origen se vuelveinestable, forzando a las soluciones a “saltar” al unico atractor estable,el ciclo lımite asociado a r3. En otras palabras, el sistema sufre de formabrusca oscilaciones de gran amplitud que permanecen incluso para valoresde µ < 0, y solo desapareceran cuando los ciclos lımite estable e inestablecolisionen.

Bifurcaciones globales

Como su nombre indica, las bifurcaciones globales no pueden detectarseexplorando un entorno pequeno de un punto fijo, dado que tienen lugar enregiones amplias del espacio de fase. De esta forma, aparecen nuevos meca-nismos de creacion o destruccion de ciclos lımite.

a) Bifurcacion de tipo saddle-node en ciclos lımite

Un ejemplo de bifurcacion global es precisamente la colision entre lasorbitas estable e inestable de las que hablabamos en el caso µ < 0 dela bifurcacion de Hopf subrıtica (2.23). Si bien en el apartado anteriornos hemos preocupado por la bifurcacion en µ = 0, para valores de µligeramente menores que cero las dos orbitas periodicas r2 y r3 coexistensiempre y cuando el discriminante en (2.24) sea positivo. Cuando µ =−1/4, el discriminante se hace cero y por tanto las dos orbitas periodicasse fusionan en una sola orbita de radio r =

√1/2, desapareciendo (r2,3 ∈

C) para valores µ < −1/4. Por tanto, queda justificado el fenomeno dehisteresis del caso subcrıtico, pues las oscilaciones que emergen cuando elparametro cruza el punto crıtico µ = 0 no desaparecen hasta llegar a labifurcacion global en µ = −1/4.

b) Bifurcacion de perıodo infinito

Este tipo de bifurcacion esta caracterizado por la aparicion de dos puntosfijos sobre un ciclo lımite. Como ejemplo, veamos la siguiente combinacionde dos sistemas unidimensionales particularmente simples:

r = r(1− r2),

θ = µ− sinθ.(2.25)

En la componente radial, el origen es un punto fijo inestable y todas lastrayectorias se aproximan al ciclo lımite estable con r = 1. En la compo-nente angular, el sistema es analogo a un oscilador con frecuencia natural

28


µ y un termino perturbativo. Ası mismo, tiene un punto de bifurcacionen µ = 1 asociado a una bifurcacion de tipo saddle-node. Por tanto, siµ > 1 (altas frecuencias), la ecuacion angular no tiene puntos fijos y elsistema presenta oscilaciones con sentido antihorario. Segun µ → 1+, es-tas oscilaciones comienzan a ralentizarse en torno a θ = π/2, hasta que enµ = 1 aparece un punto fijo en θ = 2π y el tiempo necesario para realizaruna oscilacion completa diverge a infinito. Para valores de µ < 1 (bajasfrecuencias), el punto fijo se divide en dos, uno estable y otro inestable,determinados por θ1,2 = sin−1µ, desapareciendo ası la orbita periodica.

c) Bifurcacion homoclınica

Una orbita homoclınica es una orbita cerrada que conecta un punto fijo detipo saddle consigo mismo. Por tanto, este tipo de bifurcacion tiene lugarcuando la variacion de un parametro hace colisionar un ciclo lımite conun saddle, formando una orbita homoclınica para un cierto valor crıtico.El estudio de este tipo de bifurcacion se lleva a cabo mediante un enfoqueperturbativo llamado metodo de Melnikov (Guckenheimer and Holmes,1983).

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2.3. Sincronizacion de sistemas dinamicos en

redes complejas

Una vez descritas las propiedades estructurales de una red y la teorıa ge-neral de sistemas dinamicos, procedemos al estudio de los fenomenos emer-gentes que resultan de la combinacion de ambos factores y que requieren, portanto, de una aproximacion holıstica. En nuestro caso, el fenomeno emergentefundamental es la sincronizacion de sistemas de osciladores acoplados de for-ma no lineal, cuyo descubrimiento se adjudica historicamente durante el sigloXVII a C. Huygens (Huygens and Oscillatorium, 1932). En particular, Huy-gens advirtio como dos relojes colgados en la misma pared de su habitacionoscilaban al unısono, hecho que por experiencia sabıa que no podıa deberseal azar. De esta forma, dedujo que ambos relojes debıan sentir una interac-cion mutua a traves de la pared. A continuacion, analizaremos la emergenciade sincronizacion dentro del marco conjunto de la Teorıa de Bifurcaciones(dinamica) y de la Teorıa de Redes (topologıa) en el caso de los dos modelosestudiados en esta tesis, el modelo de Kuramoto y su extension inercial. Parauna revision completa de la sincronizacion en el contexto de redes complejasse puede consultar (Arenas et al., 2008).

2.3.1. El modelo de Kuramoto original

Este modelo fue propuesto por Yoshiki Kuramoto (Kuramoto, 1975, 2003)como paradigma en el estudio de la sincronizacion de sistemas biologicos,motivado particularmente por el trabajo de Winfree sobre la sincronizacion enpoblaciones de luciernagas (Winfree, 1967). No obstante, este sencillo modeloresoluble analıticamente ha resultado ser extremadamente util, sirviendo en lamodelizacion de problemas en campos tan diversos como la Fısica (Wiesenfeldet al., 1998), la Quımica (Kim et al., 2004) o la Neurociencia (Cumin andUnsworth, 2007). En la literatura pueden encontrarse numerosas aportacionesdebidas a su importancia como modelo para la comprension y descripcion dela sincronizacion en sistemas reales. Para una excelente aproximacion generalal problema, ası como una revision de la literatura al respecto, cabe destacar(Strogatz, 2000; Acebron et al., 2005). Respecto al estudio de la estabilidadde las soluciones destacamos (Strogatz and Mirollo, 1991; Strogatz et al.,1992; Mirollo and Strogatz, 2005, 2007), estudio extendido y corregido enterminos de la variedad central (Crawford, 1994). Ası mismo, la distribucionde frecuencias naturales puede dar lugar a transiciones con histeresis tanto en

30


el modelo de Kuramoto (Martens et al., 2009) como en la version estocastica(Bonilla et al., 1992). Por ultimo, un interesante enfoque en terminos defunciones de Lyapunov se puede encontrar en (Van Hemmen and Wreszinski,1993).

El modelo de Kuramoto consiste en un conjunto de osciladores con unafrecuencia natural wi propia (aquella que determina su fase individualmente)y un termino de acoplamiento no lineal que depende de la diferencia de lasfases de cada par de osciladores conectados. En el modelo original, todos lososciladores estan acoplados entre sı y la ecuacion que rige la dinamica delsistema es:

θi = wi +σ

N

N∑

j=1

sin(θj − θi), i = 1, ..., N, (2.26)

donde σ es la constante de acoplamiento y N el numero de osciladores delsistema. Es inmediato advertir que el modelo de Kuramoto es precisamen-te la formulacion moderna y general del fenomeno observado por Huygens,donde los osciladores acoplados intercambian energıa gracias a una fuerzade interaccion. Como resultado, las oscilaciones se van ajustando progresiva-mente hasta alcanzar el estado de sincronizacion θ1 = θ2 = ... = θN , siemprey cuando la constante de acoplamiento sea suficientemente grande.

Antes de proseguir con el tratamiento general, consideremos el modelo deKuramoto en su formulacion mas sencilla. Si atendemos a un solo elementoaislado, un oscilador de Kuramoto es simplemente un flujo en la circunferen-cia caracterizado por la ecuacion

θ = ω. (2.27)

Por tanto, las soluciones son oscilaciones regulares del tipo θ = ωt + θ0. Laparticularidad del modelo de Kuramoto esta en considerar un acoplamientono lineal. En el caso mas simple, la ecuacion de Kuramoto para dos osciladoreses

θ1 = ω1 + σ sin(θ2 − θ1),

θ2 = ω2 + σ sin(θ1 − θ2).(2.28)

Este sistema de dos ecuaciones se puede reducir a un sistema unidimensionalmediante el cambio de variables φ = θ2 − θ1, de forma que

φ = ∆− 2σ sin(φ), (2.29)

donde ∆ = ω2 − ω1, σ regula la intensidad de la interaccion y φ = 0 es lacondicion de sincronizacion en frecuencias. Redefiniendo la escala temporal

31


t′ = 2µt y el parametro µ = ∆/2σ, recuperamos la ecuacion angular delejemplo de la bifurcacion de perıodo infinito (ver seccion 2.2.3), caracterizadapor una bifurcacion de tipo saddle-node en µ = 1 (σ = ∆/2). Para valoresde acoplamiento σ < ∆/2, el sistema no tiene puntos fijos y los osciladoresno pueden sincronizarse. Cuando σ > ∆/2 aparecen dos puntos fijos, unoestable y otro inestable, siendo el punto fijo estable el atractor responsablede que ambos osciladores sincronicen. El punto crıtico σc = ∆/2 que separa elestado sıncrono del estado incoherente resulta ser proporcional a la diferenciaen frecuencias naturales. De esta forma, cuanto mas diferentes sean ω1 y ω2,tanto mas se debera incrementar σ para lograr que el sistema sincronice. Unavez alcanzado el estado de sincronizacion asociado a φ = 0, los osciladoresgiran con una diferencia de fase φ constante a la misma frecuencia instantaneade sincronizacion ωs = 〈ω〉. Si incrementamos aun mas σ, la diferencia defases se reduce progresivamente, y en el lımite σ → ∞ el sistema acabasincronizando tanto en frecuencia como en fase.

Una vez entendido el modelo en su configuracion mas sencilla, podemosestudiar el caso original de un sistema con N osciladores acoplados todos contodos. Introduciendo notacion compleja, podemos reexpresar (2.26) como

θi = wi + σrsin(ψ − θi), i = 1, ..., N, (2.30)

con tal de introducir el cambio de variables

r(t)eiψ(t) =1

N

N∑

j=1

eiθj(t). (2.31)

De esta forma, reiψ es un numero complejo cuyo argumento ψ es la fasepromedio del sistema, y cuyo modulo r da cuenta de la sincronizacion ins-tantanea de los osciladores. Considerando cada oscilador como un vector demodulo unidad y angulo θ en el plano complejo, si el sistema esta en el estadoincoherente la suma de todos los vectores en (2.31) dara como resultado unvector de modulo r ' 0, mientras que si el estado es sıncrono, todos los vec-tores rotaran al unısono y el parametro de orden sera r ' 1 (cumpliendose laigualdad estricta solo cuando t→∞ en el lımite termodinamico). La transi-cion de un estado al otro determina un cierto valor crıtico de la constante deacoplo σc tal que, para σ > σc, el sistema evoluciona del estado incoherenteal estado (parcialmente) sıncrono.

Para poder definir el grado de sincronizacion global del sistema debemosintroducir una serie de consideraciones previas. En primer lugar, el atractordel sistema (ver Apendice A) es una variedad diferenciable definida por la

32


condicion de sincronizacion θ1 = θ2 = ... = θN . Esta variedad invarianteexiste siempre, pero su estabilidad como atractor depende del valor de σ,siendo inestable para valores de σ < σc. Segun se incrementa el valor de σpor encima de dicho punto crıtico, las soluciones convergen progresivamentey de forma asintotica sobre la variedad. Por tanto, el grado de sincroniza-cion global ha de dar cuenta del numero de osciladores que asintoticamenteconvergen sobre la variedad invariante. Partiendo del grado de sincroniza-cion instantaneo r(t), esta condicion de comportamiento asintotico equivalea descartar el intervalo de tiempo correspondiente al estado transitorio delsistema, garantizando ası que el grado de sincronizacion es representativo desu dinamica asintotica. Ası mismo, para evitar fluctuaciones debidas a efec-tos de tamano finito, tomaremos un promedio temporal 〈.〉T del parametrode orden (una vez descartado el transitorio) sobre un intervalo de tiempo Tsuficientemente grande. De esta forma, definimos la sincronizacion global deun sistema con N osciladores como

S =1

N〈|

N∑

j=1

eiθj |〉T . (2.32)

Una vez definida la sincronizacion global del sistema, procedemos al estudioanalıtico de la sincronizacion del modelo de Kuramoto en el lımite termo-dinamico mediante el paso al continuo de las ecuaciones (2.30) y (2.31)10.Este procedimiento implica la introduccion de una funcion densidad de pro-babilidad ρ(θ, ω, t) que debe satisfacer la condicion de normalizacion y laecuacion de continuidad: ∫ ∞

−∞

∫ 2π

0

dθdωρ(θ, ω, t) = 1, (2.33)

∂tρ(θ, ω, t) + ∂θ(ρ(θ, ω, t)θ) + ∂ω(ωρ(θ, ω, t)ω) = 0. (2.34)

Por tanto, esta formulacion se interpreta como un continuo de osciladoressobre la circunferencia unidad, de tal forma que ρ(θ, ω, t)dθdω es la fraccionde osciladores con fases y frecuencias naturales comprendidas en los intervalos(θ, θ+dθ) y (ω, ω+dω) respectivamente. Si consideramos ası mismo la funcionde distribucion de las frecuencias naturales, g(ω), como la probabilidad deque un oscilador i tenga ωi = ω, entonces

g(ω) =

∫ 2π

0

dθρ(θ, ω, t), (2.35)

10Si bien Kuramoto hizo uso del concepto de densidades, no utilizo la formulacion conti-nua para obtener soluciones analıticas. En su lugar, a partir de la condicion de sincroniza-cion propuso dos subpoblaciones de osciladores (aquellos que la verificaban y aquellos queno) y una serie de condiciones sobre las densidades que los describıan. Para un tratamientodetallado se puede consultar [Strogatz, 2000].

33


y por tanto, la ecuacion de continuidad se reduce a

∂tρ(θ, ω, t) + ∂θ(vρ(θ, ω, t)) = 0, (2.36)

donde hemos tenido en cuenta que las frecuencias naturales no dependen deltiempo (ω = 0) y que la densidad de probabilidad se puede descomponercomo

ρ(θ, ω, t) = ρ(θ, ω, t)g(ω), (2.37)∫ 2π

0

dθρ(θ, ω, t) = 1. (2.38)

Antes de sustituir la expresion (2.30) para la velocidad v = θ en la ecua-cion de continuidad, queremos llamar la atencion sobre un par de aspectosimportantes. En primer lugar, la condicion de punto fijo en la ecuacion decontinuidad corresponde a la condicion de estacionariedad de la densidad dedistribucion ρ(θ, ω, t). Esta condicion es equivalente, en el caso de la ecuacionde Kuramoto, a exigir soluciones estacionarias del parametro de orden r(t),garantizando que los osciladores sincronizan a la frecuencia de sincronizacionΩs mientras su fase se aproxima a la fase promedio ψ. En el lımite σ → ∞,todos los osciladores tienen la misma frecuencia de sincronizacion y la mismafase ψ(t) = Ωst. Esta ultima ecuacion permite elegir como sistema de referen-cia aquel que gira con frecuencia Ωs mediante la transformacion θ′ = θ+Ωst,y por tanto, sustituyendo ambas ecuaciones en (2.30) y omitiendo el cambiode notacion, la ecuacion de evolucion de un oscilador con fase θ y frecuencianatural ω queda reducida a

θ = w − Ωs − σrsinθ. (2.39)

En segundo lugar, esta ecuacion determina dos subpoblaciones, ρL (locked)y ρD (drift), caracterizadas por diferentes comportamientos a largo plazoderivados de la condicion de sincronizacion, ω − Ωs = σrsinθ. Por un la-do, los osciladores pertenecientes a ρL estan caracterizados por la condicion|ω − Ωs| ≤ σr, que garantiza la existencia de un punto fijo estable corres-pondiente a θ = sin−1(ω/σr), siendo |θ| ≤ π/2. Por tanto, los primerososciladores en converger sobre el atractor son aquellos con frecuencia naturalproxima a Ωs, y su numero se va incrementando segun aumenta el valor deσ−σc. Cuanto mas extremal sea la frecuencia natural de un oscilador, tantomayor tendra que ser el acoplamiento necesario para conseguir que cumplala condicion de sincronizacion. Por otro lado, los osciladores que cumplen|ω − Ωs| > σr oscilan de forma no uniforme, lo que podrıa dar lugar a unaviolacion de la condicion de punto fijo de la formulacion discreta del modelo.

34


Por ello, Kuramoto exigio “ad hoc” que la distribucion de osciladores a laderiva ρD fuera una distribucion estacionaria. Esta exigencia intuitiva se vuel-ve natural en la formulacion continua, donde la condicion de estacionariedad∂tρ = 0 es de por sı la condicion de punto fijo. A esta ventaja conceptual sele une el hecho de que la formulacion continua es susceptible de un analisisclasico de estabilidad (ver referencias al principio de este apartado).

Una vez llegados a este punto, la formulacion continua de las ecuaciones(2.30) y (2.31) es

∂tρ(θ, ω, t) + ∂θ ([ω + σrsin(ψ − θ)]ρ(θ, ω, t)) = 0 (2.40)

reiψ =

∫ ∞

−∞

∫ 2π

0

dθdωg(ω)ρ(θ, ω, t)eiθ. (2.41)

Por tanto, las soluciones estacionarias de (2.40) son las soluciones de la den-sidad de probabilidad asociadas al estado incoherente, al estado parcialmentesıncrono (donde coexisten ρL y ρD) y al estado completamente sıncrono. Enel primer caso, la solucion incoherente es equivalente a que la probabilidadde encontrar θ en cualquier punto entre 0 y 2π sea constante, y por tantoρ = 1/2π queda determinada por la condicion de normalizacion (2.38). Sus-tituyendo ρ = 1/2π en (2.40) y (2.41), la solucion trivial verifica la ecuacionde continuidad y la condicion de incoherencia r = 0. En el caso opuesto, lacondicion de sincronizacion completa es equivalente a exigir que todos lososciladores tengan la misma fase ψ, y por tanto ρ = δ(θ − ψ) satisface laecuacion de continuidad y la condicion de sincronizacion r = 1.

En el estado de sincronizacion parcial (r > 0) la distribucion estacio-naria debe ser un caso intermedio, donde los osciladores pertenecientes alcentro de la distribucion corresponden con la subpoblacion sincronizada, cu-ya fase esta determinada por el punto fijo del modelo de Kuramoto (2.30),mientras que las colas de la distribucion corresponden con la subpoblaciona la deriva. Por ello, buscaremos aquellas distribuciones representativas delcomportamiento asintotico ρ∞(θ, ω) ≡ ρ(θ, ω, t → ∞) tales que r > 0 (Fig.2.6). La condicion de estacionariedad ∂tρ∞ = 0 implica que vρ∞ = C(ω) esindependiente de la fase θ, y por tanto

ρ∞(θ, ω) =C(ω)

ω + σr∞sin(ψ − θ) . (2.42)

Esta distribucion se divide en dos subpoblaciones en funcion del signodel denominador: si |ω| < σr∞, el denominador cambia de signo de forma

35


Figura 2.6: Evolucion del parametro de orden en funcion del tiempo paraacoplamientos super y subcrıticos.

periodica, y dado que ρ∞ ≥ 0, necesariamente C(ω) = 0. Si |ω| > σr∞, lacondicion anterior implica que C(ω) es positiva (negativa) cuando ω > 0(ω < 0), quedando determinada por la condicion de normalizacion (2.38).Por tanto,

ρ∞(θ, ω) =

δ(θ − ψ − sin−1( ω

σr∞)), |ω| < σr∞

sign(ω)√ω2−σ2r2/2π

ω+σr∞sin(ψ−θ) , |ω| > σr∞(2.43)

donde |ω| < σr∞ corresponde al conjunto de osciladores cuya fase −π/2 ≤θ ≤ π/2 es igual a la del punto fijo θ = ψ+sin−1( ω

σr∞), mientras que |ω| > σr∞

corresponde a la distribucion estacionaria de osciladores a la deriva. Usandoel sistema de referencia corrotante y sustituyendo (2.43) en la ecuacion (2.41),el parametro de orden resulta

r∞ =

∫ σr∞

−σr∞

∫ 2π

0

dθdωg(ω)δ(θ − θ)eiθ (2.44)

Las integrales correspondientes a los intervalos (−∞,−σr∞) y (σr∞,∞, ) secancelan entre sı, pues el integrando es impar bajo la transformacion (θ, ω)→(θ+π,−ω), ya que ρ(θ+π,−ω) = ρ(θ, ω), g(−ω) = g(ω) y ei(θ+π) = −eiθ. Portanto, tan solo los osciladores pertenecientes a ρL contribuyen al parametrode orden, dando como resultado

r∞ =σr∞

2

∫ 2π

0

dθcos2θg(σr∞sinθ). (2.45)

Eliminando la solucion trivial, r∞ queda determinada implıcitamente por laecuacion

1 =σ

2

∫ 2π

0

dθcos2θg(σr∞sinθ), (2.46)

36


Figura 2.7: Evolucion del parametro de orden en funcion del acoplamientoen los casos (a) supercrıtico y (b) subcrıtico. En ambos casos, la region r = 0se vuelve inestable para σ > σc. En el caso subcrıtico, la lınea discontinuarepresenta una region inestable de la solucion sıncrona.

cuyas soluciones existen solo para σ > σc. Tomando el lımite r∞ → 0+

correspondiente a σ = σc e integrando se obtiene 1 = (σc/2)πg(0). Por tanto,

σc =2

πg(0). (2.47)

Ası mismo, desarrollando en serie de Taylor la funcion de distribucion delas frecuencias naturales, se obtiene que el parametro de orden escala conel parametro crıtico como r∞ ∝

√µ/− g′′(0), donde µ = (σ − σc)/σc. Por

tanto, y en analogıa con la expresion r2 =√d√−µ/a del punto fijo de la

ecuacion radial de la bifurcacion de Hopf (2.20), el signo de g′′(0) determinasi la bifurcacion en el modelo de Kuramoto es supercrıtica, g′′(0) < 0 (Fig.2.7(a)), o subcrıtica, g′′(0) > 0 (Fig. 2.7(b)), y por tanto sujeta a histeresis.

2.3.2. El modelo de Kuramoto en redes complejas: laruta a la sincronizacion

El modelo de Kuramoto original es un modelo de campo medio dondetodos los osciladores estan sometidos al mismo campo de interaccion σr de-bido a que todos los nodos estan conectados entre sı. No obstante, podemosgeneralizar este modelo a un sistema topologicamente no trivial sin mas queintroducir la matriz de acoplamiento σij y la matriz de adyacencia Aij, dondeAij = 1 si los nodos i y j estan conectados y Aij = 0 en caso contrario. Laecuacion queda por tanto como sigue:

37


θi = wi +1

N

N∑

j=1

σijAijsin(θj − θi), i = 1, ..., N (2.48)

En este contexto, la matriz de adyacencia11 define una particular topo-logıa en el sistema, cuya eleccion concreta puede ser determinante en el com-portamiento dinamico y especialmente en las caracterısticas de la transicionentre los estados sıncrono y asıncrono (Moreno and Pacheco, 2004; Ichinomi-ya, 2004; Restrepo et al., 2005).

El hecho de introducir una topologıa especıfica abre nuevas posibilidadesen el analisis de la sincronizacion. En ausencia de una aproximacion de cam-po medio, ¿es adecuado mantener el parametro de orden global del modelode Kuramoto? Este parametro de orden da cuenta del comportamiento me-dio del sistema. No obstante, perdemos toda informacion sobre los procesosinternos de sincronizacion que estan teniendo lugar entre los elementos dela red. Debido a la complejidad del problema, la mayor parte de los estu-dios realizados para tratar de responder a la pregunta son numericos, si bienexisten algunos tratamientos analıticos bajo ciertas restricciones sobre la to-pologıa de la red. Comenzaremos por este caso, y en particular, seguiremosel analisis propuesto por (Restrepo et al., 2005).

Para paliar las deficiencias de un parametro de orden global, se puedeintroducir un parametro de orden local ri mediante la identidad rie

iφi =∑Nj=1Aij〈eiθj〉t, redefiniendo el parametro de orden global como

r =

∑Ni=1 ri∑Ni=1 ki

. (2.49)

De esta forma, y para el caso de un acoplamiento de campo medio, i. e.σij = σ, la ecuacion original del modelo de Kuramoto (2.26) se transformaen

θi = wi − σrisin(θj − θi)− σhi(t), (2.50)

donde hi(t) = Ime−iθi∑Nj=1Aij(〈eiθj〉t−e

iθj) es despreciable fuera de un en-

torno del punto crıtico y siempre y cuando la conectividad de la red sea lo

11Si bien es fısicamente mas significativo usar la notacion expuesta en la ecuacion (2.48),es comun expresar el caso de un acoplamiento no homogeneo a traves de una matriz deadyacencia pesada, i. e. Ωij ≡ σijAij . Este caso es especialmente util cuando la interaccionse varıa de forma global, asociando a un parametro de acoplamiento homogeneo σ unavariacion no homogenea de la interaccion de cada enlace, i. e. σΩ.

38


suficientemente grande, es decir, que la red sea densa. A continuacion, seprocede aplicando un tratamiento analogo al del apartado anterior con laasuncion extra de que el grado de los primeros vecinos es alto, lo que permitedespreciar la contribucion debida a los osciladores a la deriva. El parametrode orden local queda por tanto reducido a la expresion

ri =∑

|ωj |≤σrj

Aij

√1−

(ωjσrj

)2

. (2.51)

Las limitaciones de esta aproximacion radican tanto en las asunciones sobrelas propiedades de la red como en el hecho de que se deben conocer exacta-mente todas las frecuencias del sistema para poder resolver la ecuacion (2.51).Por tanto, en base a la condicion impuesta de redes densas, podemos asumirque el conjunto de frecuencias naturales de los vecinos de un cierto nodo sonrepresentativas de la distribucion g(ω). De esta forma, podemos ponderarcada contribucion ωi con su correspondiente probabilidad, resultando

ri =∑

j

Aij

∫ σrj

−σrjg(ω)

√1−

(ω

σrj

)2

dω. (2.52)

Si introducimos el cambio de variables x = ω/σrj y desarrollamos g(ω) enserie de Taylor en torno a rj = 0, la expresion del parametro de orden locala primer orden cuando r → 0+ se reduce a

r0i =

σ

Kc∑

j

Aijr0j , (2.53)

donde Kc = 2/πg(0) es el acoplamiento crıtico del modelo de Kuramoto origi-nal. En notacion vectorial, (2.53) es equivalente a la ecuacion de autovaloresAv = (Kc/σ)v, y por tanto Kc/σ = λ1, ..., λN. El acoplamiento crıticocorresponde con el valor mınimo de σ que satisface (2.53), es decir,

σc =Kcλmax

, (2.54)

siendo λmax el maximo autovalor de la matriz de adyacencia. De esta ecua-cion se deduce que el acoplamiento crıtico es mayor en redes homogeneas queen redes heterogeneas, pues λSFmax > λERmax. No obstante, predice erroneamenteuna ausencia de punto crıtico para redes heterogeneas en el lımite termo-dinamico12, cuya existencia se ha comprobado aplicando finite size scalingmediante simulaciones numericas (Moreno and Pacheco, 2004).

12Esto es debido a la relacion del acoplamiento crıtico con los dos primeros momentosde la distribucion de grado (Eq. 2.59), pues la desviacion tıpica de la distribucion de gradoen redes SF diverge para N →∞.

39


Finalmente, podemos establecer la conexion entre este modelo y el modelode campo medio si consideramos la aproximacion ri ∼ ki, es decir, que el valorde la sincronizacion local es proporcional al numero de vecinos. En particular,ri = rki, y por tanto

r =1

ki

∣∣∣∣∣N∑

j=1

Aij〈riθj〉t∣∣∣∣∣ . (2.55)

Sumando sobre N la ecuacion (2.52) y sustituyendo x = ω/σrj y rj = rkj,se obtiene la expresion

N∑

j

kj = σN∑

j

k2j

∫ 1

−1

g(xσrkj)√

1− x2dx, (2.56)

que en el lımite continuo toma la forma

∫kP (k)dk = σ

∫k2P (k)dk

∫ 1

−1

g(xσrkj)√

1− x2dx. (2.57)

El valor del acoplamiento crıtico corresponde al lımite r → 0+, y por tanto

∫kP (k)dk = σc

∫k2P (k)dk

∫ 1

−1

g(0)√

1− x2dx =σcg(0)π

2

∫k2P (k)dk,

(2.58)es decir,

σc = Kc〈k〉〈k2〉 . (2.59)

De esta forma, la condicion de sincronizacion σ > σc (r > 0) se traduce enla inecuacion

σg(0)π

2

∫k2P (k)dk >

∫kP (k)dk. (2.60)

La otra vıa de aproximacion consiste en exploraciones numericas del puntocrıtico de la transicion a la sincronizacion (Watts, 2001; Hong et al., 2002;Moreno and Pacheco, 2004; Vega et al., 2004), si bien el resultado mas rele-vante en este campo es el debido a (Gomez-Gardenes et al., 2007a,b), dondese introducen una serie de medidas locales para describir la ruta a la sincro-nizacion, es decir, los particulares mecanismos microscopicos que llevan a laemergencia de la sincronizacion global.

En primer lugar, ademas del parametro de sincronizacion global r, de-bemos preguntarnos que informacion local es relevante en el estudio de la

40


Figura 2.8: Evolucion de los parametros de orden r (a) y rlink (b) en funcionde λ para topologıas aleatorias (ER) y con ley de potencias (SF) (Gomez-Gardenes et al., 2007a).

transicion de fase. Para ello consideramos de nuevo la ecuacion (2.48) conσij = σ y definimos un nuevo parametro de orden:

rlink =1

2Nl

∑

i

∑

j∈Γi

∣∣∣∣ lım4t→∞

1

4t

∫ tr+4t

tr

ei(θi(t)−θj(t))dt

∣∣∣∣ (2.61)

donde tr es el tiempo del regimen transitorio y Γi el conjunto de vecinos delnodo i. Por tanto, rlink da cuenta de la sincronizacion promedio a primerosvecinos, permitiendo ver como evoluciona localmente la sincronizacion y comoafecta a la sincronizacion global.

La diferencia entre la evolucion de r y rlink se muestra en la Fig. 2.8. Enparticular, la Fig. 2.8(a) muestra la evolucion del parametro global de ordenr en funcion de la constante de acoplamiento σ para redes ER y SF. Comose puede ver, el valor de acoplamiento crıtico es menor para la topologıa SFque en el caso de una red aleatoria, esto es, el sistema sincroniza con ma-yor facilidad cuando la topologıa cumple una ley de potencias. No obstante,aunque el umbral crıtico en el caso aleatorio es mayor, una vez sobrepasado,r aumenta rapidamente situandose por encima del caso SF y manteniendouna sincronizacion algo superior a partir de ese momento.

Este resultado global nos lleva a pensar que debe existir algun compor-tamiento caracterıstico interno que de cuenta de dos factores:

1. Mayor suavidad en la transicion para redes SF frente a una transicionmas abrupta en el caso aleatorio.

41


Figura 2.9: Tamano de la componente conexa gigiante (GCC) y numero dede componentes conexas de sincronizacion (Nc) en funcion de σ (Gomez-Gardenes et al., 2007a).

2. Mayor capacidad de sincronizacion para acoplamiento alto en el casoaleatorio frente al caso SF.

En la Fig. 2.8(b) esta representada la evolucion del parametro local deorden rlink en funcion de la constante de acoplamiento, tambien para lasdos topologıas propuestas. En este caso, lo primero que llama la atencionson los valores no nulos de rlink en una region donde r es practicamentenulo (estado incoherente). Esto implica necesariamente la emergencia localde pequenas agrupaciones de nodos sincronizados en el regimen subcrıtico.Independientemente de la topologıa, la tasa de crecimiento de rlink cambiauna vez se llega al valor de acoplamiento crıtico correspondiente, siguiendoun comportamiento analogo al observado en el caso del parametro global r.

Este comportamiento esta en concordancia con las conclusiones obteni-das a partir de la Fig. 2.8(a), pero introduce una pista definitiva a la hora deentender el proceso interno que esta teniendo lugar. La aparicion de compo-nentes sıncronas es un proceso clave en la ruta hacia el estado coherente.

Para terminar de arrojar luz sobre este punto, debemos primero introducirun nuevo parametro de orden local, que consiste en una particularizacion porpares del parametro de orden global y se define de la siguiente manera:

42


Figura 2.10: Evolucion de las componentes conexas de sincronizacion en fun-cion de σ (Gomez-Gardenes et al., 2007a).

Sij = Aij

∣∣∣∣ lım4t→∞

1

4t

∫ tr+4t

tr

ei(θi(t)−θj(t))dt

∣∣∣∣ (2.62)

Por tanto, Sij contiene la informacion de cuan sincronizado esta cada parde nodos de la red. Dado que es una matriz pesada entre 0 y 1, debemosestablecer un criterio para convertirla en una matriz de adyacencia. Para ellose elige un umbral tal que la fraccion de pares de nodos sincronizados seaigual a rlink (que representaba la fraccion de todos los posibles enlaces queestan sincronizados). De esta forma, obtenemos una matriz donde Sij = 1implica que los nodos i, j estan sincronizados, y por tanto conectados porun enlace funcional. Este proceso de filtrado nos permite obtener informa-cion topologica sobre la red de sincronizacion, en particular el numero decomponentes cuyos nodos estan sincronizados.

En la Fig. 2.9 se representa el numero de componentes sıncronas ası comoel tamano de la componente gigante. Como se puede ver, para una topologıaSF el numero de componentes aumenta inicialmente para decrecer a con-tinuacion con una tasa constante hasta llegar a una sola componente deltamano de todo el sistema. Sin embargo, en el caso aleatorio, el numero decomponentes crece aproximadamente un 50 % mas y decrece rapidamente,convergiendo a una componente gigante de gran tamano antes que en el casoSF. Cabe destacar que en el lımite del estado incoherente, ambas topologıaspresentan una componente gigante con el 50 % de los nodos del sistema y elmismo numero de componentes menores (puntos de corte de ambas curvas).

43


Finalmente, en la Fig. 2.10 se muestra el proceso de agregacion que tienelugar segun aumenta la constante de acoplamiento: en las redes aleatorias, nu-merosos componentes de sincronizacion emergen hasta ser capaces de formaruna componente gigante de gran tamano, proceso que, una vez se desestabi-liza el estado incoherente, converge muy rapidamente. En el caso SF, tras laprimera emergencia de componentes, se forma una componente gigante cuyotamano es el 50 % del tamano de la red, y a partir de ahı se impone un ritmoconstante de agregacion de los pequenos componentes sıncronas del sistema.

En conclusion, las topologıas homogeneas presentan una gran capacidadpara sincronizar localmente, lo que conlleva como contrapartida una disminu-cion de la sincronizacion global debido a la deslocalizacion de los componentessıncronas. Por contra, en el caso de redes heterogeneas unos pocos nodos (loshubs) influyen fuertemente sobre la dinamica del resto de la red, disminuyen-do la capacidad local de sincronizacion a favor de la sincronizacion global.Este comportamiento sera determinante a la hora de entender el fenomenode explosividad, como se vera en el Capıtulo 3.

2.3.3. El modelo de Kuramoto con inercia

El modelo inercial o modelo de Kuramoto de segundo orden se introdujo,por un lado, para intentar describir la sincronizacion de un sistema cuandoexistıa un cierto grado de adaptacion en las frecuencias naturales de cadaelemento (Ermentrout, 1991; Hanson, 1982). Ası mismo, esta extension es-tuvo motivada por el descubrimiento de su relacion con series de unionesJosephson (Wiesenfeld et al., 1996, 1998; Daniels et al., 2003), dando lugara una generalizacion en redes todos con todos, donde el problema admite untratamiento analıtico bajo ciertas aproximaciones (Tanaka et al., 1997a,b).Estos resultados han fomentando el interes por las aplicaciones de este mode-lo, como en el caso de las redes de suministros de energıa electrica (Filatrellaet al., 2008; Dorfler and Bullo, 2012; Motter et al., 2013; Dorfler et al., 2013).

Al tener en cuenta un termino de inercia (mθi), la ecuacion del modelode Kuramoto inercial resulta

θi = ωi −mθi −σ

N

N∑

j=1

sin(θj − θi), i = 1, ..., N. (2.63)

En analogıa con el problema no inercial, podemos reexpresar esta ecuacionen funcion del numero complejo reiψ en el sistema de referencia corrotante

44


(ψ = 0):

θi = ωi −mθi − σrsin(θi), i = 1, ..., N, (2.64)

En el caso de un oscilador con fase θ y frecuencia natural ω, (2.64) es laconocida ecuacion de un pendulo amortiguado sometido a una fuerza externacuyo torque viene dado por el valor de ω (Strogatz, 2001). De esta forma, siel torque es pequeno ω < ωD (bajas frecuencias), el sistema no tiene energıasuficiente para oscilar y todas las soluciones son atraıdas al punto fijo estableθ = sin−1(ω/σr). En el punto crıtico ωD = σr, los dos puntos fijos coinciden(θ = π/2), dando paso a una bifurcacion de tipo saddle-node. Esta situacioncorresponde al caso en el cual, al caer desde arriba, el pendulo realiza unaoscilacion completa hasta volver a la posicion de partida, tomando para elloun tiempo infinito. Para valores ω > ωD, la energıa es suficiente para queel pendulo comience a oscilar. Esto se traduce en la no existencia de puntosfijos, forzando a las soluciones a converger sobre el ciclo limite.

Una consecuencia de la introduccion de inercia en el sistema es la apa-ricion de un fenomeno de histeresis. Si partimos de condiciones inicialesperiodicas y disminuimos el valor de ω, las soluciones que inicialmente es-taban sobre el ciclo lımite permanecen en el hasta alcanzar el valor crıticoωP = π−1

√2σr/m, donde tiene lugar una bifurcacion homoclınica y cuyo va-

lor se obtiene mediante el metodo de Melnikov. En el intervalo ωP < ω < ωDel sistema es biestable, coexistiendo el ciclo lımite y los dos puntos fijos (es-table e inestable). Al llegar al punto crıtico de la bifurcacion global, el ciclolımite colisiona con el punto fijo inestable, creandose una orbita homoclınica,que se transforma para valores de ω < ωP en espirales convergentes sobre elpunto fijo estable. De esta forma, la ruta de ida y vuelta sobre el parametroω crea y destruye las soluciones periodicas en valores crıticos distintos y portanto, en la region de biestabilidad, el caracter de las soluciones depende delas condiciones iniciales, dando lugar a la caracterıstica curva de histeresis.

Como veremos, la biestabilidad del oscilador inercial se hereda en ladinamica colectiva de una red de osciladores inerciales. Siguiendo el tra-tamiento expuesto en (Tanaka et al., 1997a,b), el estudio analıtico de laecuacion (2.64) requiere tener en cuenta explıcitamente las condiciones ini-ciales, distinguiendo (I) el incremento de σ partiendo del estado incoherentey (II) el decremento de σ partiendo desde el estado sıncrono. En concordan-cia, aparecen dos puntos crıticos asociados a una bifurcacion de tipo saddle-node ωD = σr y a una bifurcacion homoclınica ωP = π−1

√2σr/m, siendo

ωP < ωD. Para valores ω > ωD, no hay punto fijo y el sistema oscila a sufrecuencia natural. Para valores ω < ωP , tan solo existe una solucion estable

45


asociada a un punto fijo, y en el intervalo ωP < ω < ωD coexisten tanto elpunto fijo como el ciclo lımite. Si consideramos un sistema compuesto porN osciladores, la frecuencia natural de cada oscilador determinara su estadodinamico en funcion del valor de ωP,D (pues ambos puntos crıticos dependende r y, por ende, de σ). En el caso (I), σ < σc implica r = 0 y ambos puntoscrıticos coinciden ωP = ωD = 0. Por tanto, todos los osciladores cumpleninicialmente la condicion ω > ωD = 0, viendose forzados a oscilar de formaindependiente. Pasado el punto crıtico, ωP y ωD crecen con r y los osciladorescon frecuencias naturales pequenas satisfacen progresivamente la condicionω < ωP asociada a la ausencia de ciclo lımite, incrementando el valor de lasincronizacion global hasta llegar a r = 1. En el caso (II), al disminuir σ losvalores de ωP y ωD disminuyen tambien, y los osciladores mas rapidos satis-facen progresivamente la condicion ω > ωD asociada a la ausencia de puntofijo, desincronizandose progresivamente. De esta forma, la curva de sincro-nizacion presenta dos puntos crıticos, σ

(1)c y σ

(2)c , asociados a la emergencia

y perdida de sincronizacion respectivamente, dando lugar ası a un fenomenode histeresis.

Con estas consideraciones, podemos calcular el parametro de orden co-rrespondiente a las subpoblaciones ρL y ρD analogamente a como se hizo enel modelo de Kuramoto, con la diferencia de que esta vez ambas poblacionescontribuyen al parametro de orden. En el caso (I), la condicion de sincroniza-cion es ω < ωP , y por tanto, con los lımites de integracion correspondientes,

rIL =

∫

|ω|<ωP

∫ 2π

0

dθdωg(ω)δ(θ − θ)eiθ = σr

∫ θP

−θPdθcos2g(σrsinθ), (2.65)

donde θP = sin−1[ωP/(σr)]. Analogamente, la condicion de sincronizacion enel caso (II) es ω < ωD, y por tanto

rIIL =

∫

|ω|<ωD

∫ 2π

0

dθdωg(ω)δ(θ − θ)eiθ = σr

∫ π/2

−π/2dθcos2g(σrsinθ), (2.66)

con la unica diferencia de que, debido a ωD = σr, los lımites de la integralson θD = ±π/2.

El calculo de las contribuciones debidas a ρD es algo mas complejo. Ladefinicion formal corresponde al intervalo de integracion |ω| > ωP (I) y |ω| >ωD (II). Por tanto,

rI,IID =

∫

|ω|>ωP,D

∫ 2π

0

dθdωg(ω)ρD(θ, ω)eiθ. (2.67)

46


Dada la condicion de estacionariedad ρv = C y la condicion de normalizacion,podemos obtener la constante de proporcionalidad en funcion del periodo Tdel centro de la distribucion:

∫ 2π

0

dθρ(θ, ω, t) =

∫ T

0

dtθρ(θ, ω, t) = 1. (2.68)

Sustituyendo ρv = ρ ˙|θ| = C en la ecuacion anterior:

∫ T

0

dtθ ˙|θ|−1C = CT = 1→ C = T−1, (2.69)

y por tanto ρD = T−1 ˙|θ|−1= (Ω/2π) ˙|θ|−1

, siendo Ω = ˙〈θ〉. Sustituyendo laexpresion de la densidad de osciladores a la deriva en la ecuacion (2.67), seobtiene

rI,IID =1

2π

−ωP,D∫

−∞

∫ T

0

dtdω|Ω|g(ω)eiθ +1

2π

∞∫

ωP,D

∫ T

0

dtdω|Ω|g(ω)eiθ =

=1

2π

−∞∫

−ωP,D

∫ T

0

dtd(−ω)|Ω|g(ω)eiθ +1

2π

∞∫

ωP,D

∫ T

0

dtdω|Ω|g(ω)eiθ =

=1

2π

∞∫

ωP,D

∫ T

0

dtdω|Ω|g(ω)e−iθ +1

2π

∞∫

ωP,D

∫ T

0

dtdω|Ω|g(ω)eiθ,

(2.70)

donde se han descompuesto los lımites de integracion apropiadamente y seha utilizado la propiedad de reflexion trivial θ(t,−ω) = −θ(t, ω) y la paridadde la distribucion g(ω). Finalmente, teniendo en cuenta que z+z∗ = 2Re(z),se obtiene la contribucion rD de la subpoblacion a la deriva:

rI,IID =1

π

∞∫

ωP,D

∫ T

0

dtdωcosθ|Ω|g(ω). (2.71)

Por tanto, la expresion del parametro de orden es la suma de las contribu-ciones correspondientes a las dos subpoblaciones (2.65), (2.66) y (2.71):

rI,II = rI,IIL + rI,IID . (2.72)

47


La expresion analıtica de (2.72) depende de la funcion de distribucion defrecuencias naturales. Ası mismo, para resolver la integral asociada a la con-tribucion de osciladores a la deriva es necesario recurrir al metodo de Poin-care-Lindstead (Jordan and Smith (1987)). Reescalando apropiadamente laecuacion (2.64), obtenemos

θ + δθ + δ2rsinθ = δω/σ (2.73)

en funcion del parametro perturbativo δ = (σm)−1 y del cambio de varia-ble r = δr. Descartando los terminos de orden superior y escogiendo comodistribucion de frecuencias naturales una lorentziana g(ω) = d

π(ω2 + d2), el

parametro de orden queda finalmente:

rI =σr

m2

(1

πd3ln

√d2 + ω2

P

ωP− 1

2πdω2P

)+

+2

πσr

[√d2 + σ2r2tan−1

(√d2 + σ2r2tanθP

)− dθP

],

(2.74)

rII =σr

m2

(1

πd3ln

√d2 + σ2r2

σr− 1

2πσ2r2

)+

1

σr

(√d2 + σ2r2 − d

). (2.75)

Esta expresion analıtica del parametro de orden asociado al modelo inercialde campo medio describe con bastante precision los resultados de las simu-laciones numericas (Fig. 2.11), siendo la base teorica del primer artıculo delCapıtulo 3, donde se generalizara a topologıas complejas y se aplicara al casoparticular de la red de energıa electrica italiana.

2.3.4. Introduccion a la sincronizacion explosiva

A lo largo de la seccion actual hemos estudiado la emergencia de la sincro-nizacion en el modelo de Kuramoto, obteniendo como resultado una transi-cion continua si la distribucion de frecuencias naturales g(ω) es unimodal. Noobstante, cuando esta condicion se incumple es posible encontrar otro tipode transiciones donde el parametro de orden tiene un comportamiento dis-continuo y/o con histeresis. A continuacion, abordamos en primer lugar dosartıculos que estudian el modelo de Kuramoto cuando la distribucion g(ω)es bimodal y uniforme, introduciendo posteriormente la publicacion que diolugar al termino “transicion explosiva” ası como el primer artıculo que indujoeste tipo de transicion en el contexto de sistemas dinamicos.

48


Figura 2.11: Evolucion del parametro de orden en funcion del acoplamientopara (a) m=0.95 y (b) m=2.0. Las lıneas continuas y discontinuas corres-ponden a la prediccion teorica, mientras que los puntos son simulacionesnumericas.

Distribucion bimodal de frecuencias naturales (Bonilla et al., 1992)

Este artıculo expone el estudio de la sincronizacion en el modelo deKuramoto cuando la distribucion de frecuencias naturales g(ω) es bimo-dal. En particular, g(ω) se define como una suma de deltas de Dirac, i.e. g(ω) = 1

2[δ(ω + ω0) + δ(ω − ω0)], permitiendo ası un tratamiento analıti-

co de la estabilidad de la solucion incoherente. Para controlar la forma dela distribucion g(ω), se anade un ruido gaussiano cuyo efecto es ensancharg(ω) en torno a las deltas de Dirac a traves de la intensidad del ruido D. Deesta manera, la forma de g(ω) queda parametrizada por la distancia entremaximos 2ω0 y el grado de dispersion D.

Los resultados principales muestran como, para σ > σc, la solucion in-coherente se desestabiliza dando lugar a dos posibles soluciones: una solucionestacionaria parcialmente sıncrona y una solucion periodica. En el primer ca-so, si la separacion 2ω0 es suficientemente pequena el sistema pasa del estadoincoherente al parcialmente sıncrono de forma continua. No obstante, cuandoω0 ∈ (D/

√2, D) la solucion parcialmente sıncrona se bifurca subrıticamen-

te de la solucion incoherente, dando lugar a una transicion discontinua ycon histeresis (Fig. 2.7(b)). En el segundo caso, cuando ω0 > D la separa-cion entre los maximos de g(ω) da lugar a un parametro de orden periodico,representando ası la competencia entre dos subpoblaciones incapaces de sin-cronizar globalmente.

49


Esta ultima solucion ya fue predicha en los trabajos seminales de Kura-moto, donde se especula con la formacion de dos subpoblaciones localmentesıncronas en torno a los maximos de g(ω) y por tanto rotando en sentidocontrario. Esta situacion fue confirmada por Crawford (Crawford, 1994) me-diante un estudio de la variedad central en el modelo de Kuramoto con ruidogaussiano.

Posteriormente (Martens et al., 2009), se demostro que los estudios pre-vios eran incompletos gracias al uso de un metodo de reduccion dimensional(Ott and Antonsen, 2008). Este metodo es aplicable cuando la distribucionesta formada por la suma de dos distribuciones lorentzianas desplazadas,simplificando enormemente el tratamiento del problema y completando lascuestiones abiertas por dificultades formales. De esta forma, el estudio delos puntos fijos del problema reducido da lugar a las dos soluciones clasi-cas del modelo de Kuramoto, el estado incoherente y el estado parcialmentesıncrono, y a una nueva solucion denominada standing wave state. Las transi-ciones entre los tres atractores se llevan a cabo mediante un amplio espectrode bifurcaciones (transcrıtica, Hopf, saddle-node, homoclınica, SNIPER), se-paradas por curvas en el espacio de fases (Fig. 2.12).

La perdida de la estabilidad del estado incoherente en el problema redu-cido se lleva a cabo mediante una bifurcacion transcrıtica (TC) o medianteuna bifurcacion de Hopf (HB). El primer caso da paso al estado parcial-mente sıncrono, mientras que el segundo da lugar a la emergencia de dossubpoblaciones de osciladores localmente sıncronos o standing wave state.

Que suceda una u otra transicion (TC o HB) queda determinado una vezmas por la distancia entre los maximos de g(ω). Si la distancia entre maximoses suficientemente pequena, el estado incoherente da lugar al clasico estadoparcialmente sıncrono, pudiendo tener histeresis, mientras que si es dema-siado grande la solucion es un parametro de orden periodico que representala competencia entre dos subpoblaciones incapaces de sincronizar. Entre me-dias aparecen dos subpoblaciones que corresponden al standing wave sate,sincronizables para acoplamientos suficientemente altos.

Por otro lado, el atractor asociado al estado parcialmente sıncrono tam-bien puede aparecer a traves de una bifurcacion de tipo saddle-node (SN),coexistiendo con la solucion incoherente y con la standing wave. De esta ma-nera, aparecen dos regiones de biestabilidad (ABC y ACD) que justifican lahisteresis asociada a las distribuciones bimodales de frecuencias naturales.

50


Figura 2.12: Diagrama de las regiones del espacio de fases y de las bifurca-ciones que tienen lugar en las fronteras, donde p1 y p2 son funciones de losparametros del sistema. La region asociada a la solucion parcialmente sıncro-na estable esta comprendida en el semicırculo TC. Ası mismo, la region aso-ciada a solucion incoherente estable es el cuadrante superior delimitado porTC y HB. La region asociada al standing wave state es el cuadrante inferiordelimitado por HB, HC y SNIPER (saddle-node infinite period). Las regionesde biestabilidad son las areas comprendidas por los puntos ABC (region 1)y ACD (region 2).

Gracias a estos resultados, queda patente la importancia de la distribucionde frecuencias a la hora de determinar las propiedades de la transicion a lasincronizacion en el modelo de Kuramoto, pues si bien no son transicionesexplosivas especıficamente, sı que introducen ya las propiedades generalesque las definen.

Distribucion unimodal de frecuencias naturales (Pazo, 2005)

Este artıculo estudia el proceso de sincronizacion en el modelo de Kura-moto cuando la distribucion de frecuencias naturales g(ω) es uniforme. Elresultado principal es la demostracion de que esta particular distribucion de

51


frecuencias naturales da lugar a una transicion de fase discontinua en la topo-logıa trivial, que en este caso no presenta histeresis. Para ello, se introduce elparametro de orden en el lımite termodinamico, que en el sistema corrotante(ψ = 0) queda reducido a

r =

∫ γ

−γcos θ(ω)g(ω)dω =

1

2

√1− γ2

σ2r2+σr

2γarcsin

( γσr

), (2.76)

donde se ha definido g(ω) como

g(ω) =

1

2γsi , |γ| ≤ 0

0 si |γ| > 0.(2.77)

La solucion existe si σr ≥ γ, satisfaciendo la igualdad σcrc = γ en el puntocrıtico. De esta forma, el parametro de orden en el punto crıtico resultarc = π/4, y por tanto la transicion es discontinua.

El estudio de la transicion se extiende al caso finito, donde es necesarioescoger una discretizacion de las frecuencias naturales adecuada. No obstante,el aspecto mas interesante de este resultado es el hecho de que, a pesar deobtener una transicion discontinua, esta sucede de manera independientedel proceso de histeresis. A estas alturas, tan solo podemos decir que, en elmodelo de Kuramoto, la histeresis parece estar asociada a una distribucionbimodal de frecuencias naturales.

Percolacion explosiva (Achlioptas et al., 2009)

En este artıculo se presenta, por primera vez, el fenomeno bautizado comopercolacion explosiva. La repercusion que tuvo en el campo de la percolacionse debe, fundamentalmente, al hecho de haber introducido un fenomeno ines-perado en un campo ampliamente estudiado13. Achlioptas et al. proponen unmetodo especıfico a la hora de seleccionar si un enlace nuevo se crea duranteel proceso de percolacion. En particular, en lugar de elegir dos nodos al azar,se escogen aleatoriamente dos enlaces, calculando el producto del tamano(PR, de product rule) de las componentes conexas del par de nodos corres-pondiente. De esta forma, el enlace con un valor mayor de PR es desechado,y el resultado final es una transicion discontinua o explosiva.

13Es mas, la identificacion de la percolacion explosiva con una transicion de primer ordenfue duramente criticada (Grassberger et al., 2011; Riordan and Warnke, 2011).

52


Intuitivamente, es facil ver como este metodo favorece la conexion entrecomponentes pequenas o con tamanos inversamente proporcionales, evitandoque componentes grandes se unan entre sı. Matematicamente hablando, lasfluctuaciones en el tamano medio de las componentes disminuyen progresi-vamente, homogeneizando el tamano de todas ellas. Dado que los enlaces seanaden como una fraccion del numero total de nodos, existe un valor crıticopara el cual se conectan simultaneamente un subconjunto de componentesconexas que percolan la red masivamente, en lugar de comenzar a agregar-se poco a poco en torno a la componente de mayor tamano, suprimida porconstruccion.

La importancia de este fenomeno en el contexto de la sincronizacion esevidente cuando se tiene en cuenta el paralelismo entre sincronizacion y perco-lacion. Durante el proceso de percolacion, el numero de enlaces se incrementahasta llegar a un punto crıtico en el cual aparece una componente gigante,cuyo tamano crece de forma continua con el parametro de control. En el casodinamico, podemos representar el grado de sincronizacion local entre cadapar de nodos Sij como un enlace en una red de sincronizacion imponiendoun cierto umbral de binarizacion, tal como se hizo en el apartado (2.3.2). Deesta forma, la sincronizacion local se va incrementando segun aumenta el aco-plamiento, lo que se traduce en un aumento del numero de enlaces de la redde sincronizacion. Al igual que sucede en el caso de la percolacion, al llegaral acoplamiento crıtico el sistema empieza a sincronizar globalmente, emer-giendo ası una componente gigante, cuyo tamano crece de forma continua sig(ω) es unimodal. Por tanto, la existencia de una transicion explosiva en elcontexto de percolacion establece las bases de la sincronizacion explosiva.

Sincronizacion explosiva (Gomez-Gardenes et al., 2011)

Este es el primer resultado que demuestra la existencia de transicionesexplosivas con histeresis en redes complejas de osciladores acoplados. Enparticular, Gomez-Gardenes et al. estudian la aparicion de una transiciondiscontinua e irreversible en redes SF14 siempre y cuando las frecuenciasnaturales de los osciladores sean iguales al numero de vecinos, i. e. ωi = ki.Bajo estas condiciones, los autores demuestran analıticamente el caracterdiscontinuo de la transicion en el caso de la red SF mas sencilla. Para ello,

14Como se vera en el Capıtulo 3 (Sendina-Nadal et al., 2015), el modo de construir lared SF determina el grado de explosividad que se puede inducir mediante la correlaciongrado-frecuencia.

53


Figura 2.13: Evolucion del parametro de orden en funcion del acoplamiento.La area de histeresis esta comprendido entre las simulaciones con acopla-miento creciente (forward) y decreciente (backward).

consideran una topologıa de tipo estrella compuesta por un hub de gradokh = K conectado a K nodos perifericos o radicales de grado ki = 1. Deesta forma, la correlacion grado-frecuencia impone dos frecuencias, ωh = Ky ωi = ω = 1. En el sistema de referencia corrotante las ecuaciones demovimiento se reducen a

φh = (ωh − Ω) + σK+1∑

s=1

sin(φs − φh),

φj = (ω − Ω) + σ sin(φh − φj), j = 1, ..., K

(2.78)

donde φs = φ1, ..., φK , φh, y Ω = (Kω + ωh)/(K + 1) es la frecuenciadel sistema de referencia corrotante. Introduciendo el cambio de variablesreiψ = (K + 1)−1

∑K+1s=1 eiφs(t) con ψ = 0 (sistema corrotante) en la ecuacion

del hub, obtenemos

φh = (ωh − Ω) + σ(K + 1)r sinφh. (2.79)

Analogamente al tratamiento analıtico del modelo de Kuramoto, aplicando lacondicion de sincronizacion (φs = 0 en el sistema inercial) sobre las ecuacionesde movimiento se obtiene la condicion de sincronizacion en funcion de los

54


parametros del sistema. En particular,

sinφh =ωh − Ω

σ(K + 1)r, (2.80)

cosφj =(Ω− ω) sinφh ±

√[1− sin2 φh][σ2 − (Ω− ω)2]

σ. (2.81)

De la segunda ecuacion se deduce que la condicion de sincronizacion tienesolucion real si y solo si Ω − ω ≤ σ, donde la igualdad determina el acopla-miento crıtico σc = Ω − ω. Ası mismo, la expresion del parametro de ordenen el sistema corrotante se reduce a

r =1

K + 1

K+1∑

s=1

cosφs =K cosφj + cosφh

K + 1. (2.82)

Por tanto, sustituyendo (2.80), (2.81), ωh = K y ωj = 1 en la ecuacion (2.82)evaluada en σ = σc, el parametro de orden en el punto crıtico resulta serrc = K

K+1, mostrando ası el caracter discontinuo de la transicion (Fig. 2.13).

Como conclusion principal, esta es la primera vez que una transiciondiscontinua e irreversible no esta asociada a la forma particular de g(ω), sinoa la especıfica relacion entre la estructura topologica y la dinamica natural.De esta forma, imponer una distribucion g(ω) de tipo scale-free es insuficientepara inducir una transicion explosiva si no se cumple ademas la condicion decorrelacion grado-frecuencia. A lo largo de la tesis trataremos de generalizareste resultado sin restricciones sobre la topologıa o la forma de la distribuciong(ω).

55

CAPITULO 3

TRANSICIONES DE FASEIRREVERSIBLES

57

PHYSICAL REVIEW E 90, 042905 (2014)

Hysteretic transitions in the Kuramoto model with inertia

Simona Olmi,1,2 Adrian Navas,3 Stefano Boccaletti,1,2,3 and Alessandro Torcini1,2,*

1CNR (Consiglio Nazionale delle Ricerche), Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy2INFN Sez. Firenze, via Sansone, 1, I-50019 Sesto Fiorentino, Italy

3Centre for Biomedical Technology (UPM), 28922 Pozuelo de Alarcon, Madrid, Spain(Received 14 June 2014; revised manuscript received 11 August 2014; published 6 October 2014)

We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia forfully coupled and diluted systems. In particular, we examine, for a gaussian distribution of the frequencies, thetransition from incoherence to coherence for increasingly large system size and inertia. For sufficiently largeinertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizesand different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka,Lichtenberg, and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associatedto each of these clusters. We have also investigated numerically the limits of existence of the coherent and ofthe incoherent solutions. The minimal coupling required to observe the coherent state is largely independent ofthe system size, and it saturates to a constant value already for moderately large inertia values. The incoherentstate is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes,while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertiathe transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirlingoscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the orderparameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a levelof connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltagepower grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due tothe simultaneous presence of many competing whirling clusters.

DOI: 10.1103/PhysRevE.90.042905 PACS number(s): 05.45.Xt, 64.60.aq, 89.75.−k

I. INTRODUCTION

Synchronization phenomena in phase oscillator networksare usually addressed by considering the paradagmaticKuramoto model [1–4]. This model has been applied in manycontexts ranging from crowd synchrony [5] to synchroniza-tion, learning, and multistability in neuronal systems [6–8].Furthermore, the model has been considered with differenttopologies ranging from homogeneous fully coupled networksto scale-free inhomogeneous systems [9]. Recently it hasbeen employed as a prototypical example to analyze low-dimensional behavior in a single large population of phaseoscillators with a global sinusoidal coupling [10,11], as well asin many hierarchically coupled subpopulations [12]. The studyof the Kuramoto model for nonlocally coupled arrays [13,14]and for two populations of symmetrically globally coupledoscillators [15] led to the discovery of the so-called Chimerastates, whose existence also has been revealed experimentallyin recent years [16–19].

In this paper we will examine the dynamics and syn-chronization properties of a generalized Kuramoto model forphase oscillators with an inertial term both for fully coupledand for diluted systems. The modification of the Kuramotomodel with an additional inertial term was firstly reportedin Refs. [20,21] by Tanaka, Lichtenberg, and Oishi (TLO).These authors have been inspired in their modelization by aprevious phase model developed by Ermentrout to mimic thesynchronization mechanisms observed in the firefly Pteroptixmalaccae [22]. These fireflies synchronize their flashing

*Corresponding author: [email protected]

activity by entraining to the forcing frequency with almostzero phase lag, even for stimulating frequencies different fromtheir own intrinsic flashing frequency. The main ingredientto allow for the adaptation of the flashing frequency to theforcing one is to include an inertial term in a standard phasemodel for synchronization. Furthermore, networks of phase-coupled oscillators with inertia have been recently employedto investigate the self-synchronization in power grids [23–25],as well as in disordered arrays of underdamped Josephsonjunctions [26]. Explosive synchronization has been reportedfor a complex system made of phase oscillators with inertia,where the natural frequency of each oscillator is assumed tobe proportional to the degree of its node [27]. In particular,the authors have shown that the TLO mean-field approachreproduces very well the numerical results for their system.

Our aim is to describe from a dynamical point of view thehysteretic transition observed in the TLO model for finite-sizesystems and for various values of the inertia; we will devotea particular emphasis to the description and characterizationof coexisting clusters. Furthermore, the analysis is extendedto random networks for different levels of dilution and to arealistic case, represented by the high-voltage power grid inItaly. In particular, in Sec. II we will introduce the model, andwe will describe our simulation protocols as well as the orderparameter employed to characterize the level of coherencein the system. The mean-field theory developed by TLO ispresented in Sec. III together with a generalization able tocapture the emergence of clusters of locked oscillators ofany size induced by the presence of the inertial term. Thetheoretical mean-field results are compared with finite-sizesimulations of fully coupled systems in Sec. IV; in thesame section the stability limits of the coherent and incoherent

1539-3755/2014/90(4)/042905(16) 042905-1 ©2014 American Physical Society

CAPITULO 3. TRANSICIONES DE FASE IRREVERSIBLES

58

OLMI, NAVAS, BOCCALETTI, AND TORCINI PHYSICAL REVIEW E 90, 042905 (2014)

0 2 4 6 8 10K

0

0.5

1

1.5

2

2.5

3

3.5

ΩM

ΩM

ΩP(I)

ΩD(II)

r

(a)

KFC

0 5 10 15 20K

0

0.5

1

1.5

2

2.5

3

3.5(b)

ΩD(II)

ΩP(I)

ΩM

ΩM

KFC

r

FIG. 1. (Color online) Average order parameter r (black circles) and maximal locking frequency M (blue triangles) as a function of thecoupling K for two series of simulations performed following protocol (I) (filled symbols) and protocol (II) (empty symbols). The data referto mass: m = 2 (a) and m = 6 (b). For m = 2 (m = 6) we set K = 0.2 (K = 0.5) and KM = 10 (KM = 20), in both cases N = 500,

TR = 5000, and TW = 200. The (magenta) diamonds indicate (I )P = 4

π

√Kr

mfor protocol (I), the (green) squares

(II )D = Kr for protocol (II),

and the (black) dashed vertical line the coupling constant KGFC , whose expression is reported in Eq. (14).

phase are numerically investigated for various simulationprotocols as a function of the mass value and of the systemsize. A last subsection is devoted to the emergence of clustersof drifting oscillators and to their influence on the collectivelevel of coherence. The hysteretic transition for random dilutednetworks is examined in the Sec. V. As a last point thebehavior of the model is analyzed for a network architecturecorresponding to the Italian high-voltage power grid in Sec. VI.Finally, the reported results are briefly summarized anddiscussed in Sec. VII.

II. SIMULATION PROTOCOLSAND COHERENCE INDICATORS

Following Refs. [20,21], we study the following version ofthe Kuramoto model with inertia:

mθi + θi = i + K

Ni

∑j

Ci,j sin(θj − θi), (1)

where θi and i are, respectively, the instantaneous phase andthe natural frequency of the ith oscillator, K is the coupling, thematrix element Ci,j takes the value one (zero) depending if thelink between oscillator i and j is present (absent), and Ni is thein-degree of the ith oscillator. For a fully connected networkCi,j ≡ 1 and Ni = N ; for the diluted case we have consideredundirected random graphs with a constant in-degree Ni = Nc;therefore each node has exactly Nc random connections andCi,j = Cj,i . In the following we will mainly consider naturalfrequencies i randomly distributed according to a Gaussiandistribution g() = 1√

2πσ 2e−2/2σ 2

with zero average and anunitary standard deviation σ .

To measure the level of coherence between the oscillators,we employ the complex order parameter [28]

r(t)eiφ(t) = 1

N

∑j

eiθj , (2)

where r(t) ∈ [0:1] is the modulus and φ(t) the phase of the con-sidered indicator. An asynchronous state, in a finite network,

is characterized by r 1√N

, while for r ≡ 1 the oscillatorsare fully synchronized and intermediate r values correspondto partial synchronization. Another relevant indicator for thestate of the network is the number of locked oscillators NL,characterized by the same (vanishingly) small average phasevelocity 〈θi〉, and the maximal locking frequency M , whichcorresponds to the maximal natural frequency |i | of thelocked oscillators.

In general we will perform sequences of simulations byvarying adiabatically the coupling parameter K with twodifferent protocols. Namely, for the first protocol (I) theseries of simulations is initialized for the decoupled systemby considering random initial conditions for θi and θi.Afterwards the coupling is increased in steps K until amaximal coupling KM is reached. For each value of K ,apart the very first one, the simulations is initialized byemploying the last configuration of the previous simulationin the sequence. For the second protocol (II), starting fromthe final coupling KM achieved by employing the protocol (I)simulation, the coupling is reduced in steps K until K = 0 isrecovered. At each step the system is simulated for a transienttime TR followed by a period TW during which the averagevalue of the order parameter r and of the velocities 〈θ〉, aswell as M , are estimated.

An example of the outcome obtained by performing thesequence of simulations of protocol (I) followed by protocol(II) is reported in Fig. 1 for not negligible inertia, namely,m = 2 and m = 6. During the first series of simulations (I)the system remains desynchronized up to a threshold K =Kc

1 2; above this value r shows a jump to a finite valueand then increases with K , saturating to r 1 at sufficientlylarge coupling.1 By decreasing K one observes that the value

1Please notice that in the data shown in Fig. 1 the final state does notcorrespond to the 100% of synchronized oscillators, but to 99.6% form = 2 and 97.8% for m = 6. However, the reported considerationsare not modified by this minor discrepancy.

042905-2


59

HYSTERETIC TRANSITIONS IN THE KURAMOTO MODEL . . . PHYSICAL REVIEW E 90, 042905 (2014)

of r assumes larger values than during protocol (I), while thesystem desynchronizes at a smaller coupling, namely, Kc

2 <

Kc1 . Therefore, the limit of stability of the asynchronous state

is given by Kc1 , while the partially synchronized state can exist

down to Kc2 ; thus asynchronous and partially synchronous

states coexist in the interval [Kc2 ; Kc

1 ].The maximal locking frequency M increases with K

during the first phase. In particular, for sufficiently largecoupling, M displays plateaus followed by jumps for largecoupling: this indicates that the oscillators frequencies i aregrouped in small clusters. Finally, for r 1 the frequency M

attains a maximal value. By reducing the coupling, followingnow protocol (II), M remains stuck to such a value for a largeK interval. Then M reveals a rapid decrease towards zero forsmall coupling K Kc

2 . In the next section, we will give aninterpretation of this behavior.

We will also perform a series of simulations with a differentprotocol (S), to test for the independence of the results reportedfor Kc

1 and Kc2 from the chosen initial conditions. In particular,

for a certain coupling K we consider an asynchronous initialcondition, and we perturb such a state by forcing all theoscillators with natural frequency |i | < ωS to be locked.Namely, we initially set their velocities and phases to zero, thenwe let the system evolve for a transient time TR followed by aperiod TW during which r and the other quantities of interestare measured. These simulations will be employed to identifythe interval of coupling parameters over which the coherentand incoherent solutions can be numerically observed.

In more detail, we fix the coupling K and we perform aseries of simulations for increasing ωS values, namely, fromωS = 0 to ωS = 3 in steps ωS = 0.05. For each simulationwe measure the order parameter r and whenever it is finitefor some ωS > 0, the corresponding coupling is associatedto a partially synchronized state; the smallest coupling forwhich this occurs is identified as Kc

2 (an example is reportedin Fig. 2). With the same procedure also Kc

1 can be measured,

2 3 4 5K

0

0.2

0.4

0.6

ωs2 3 4 5K

0

2000

4000

6000

8000

Ns

K2c

K1c

Incoherent State

Coherent State

Coherent State

Incoherent State

FIG. 2. (Color online) Minimal ωS giving rise to a state charac-terized by a finite level of synchronization (i.e., r > 0) as a functionof the coupling constant K . The inset reports the minimal numberNS of oscillators which should be initially locked in order to lead tothe emergence of a coherent state, as a function of K . The vertical(green) dot-dashed line refers to the estimated Kc

1 and the (blue)dashed line indicates the estimated Kc

2 . The data refer to simulationsperformed with protocol (S) for N = 16 000, m = 6, with TW = 2000and TR = 20 000.

in particular by estimating the minimal K for which theunperturbed asynchronous state (corresponding to ωS = 0)spontaneously evolves towards a partially synchronized state,as shown in Fig. 2. To give a statistically meaningful estimationof Kc

1 and Kc2 , we have averaged the results obtained for

various different initial conditions, ranging from 5 to 8, forall the considered system sizes and masses.

In principle, this approach cannot test rigorously for thestability of the coherent and incoherent states, since it dealswith a very specific perturbation of the initial state. However, aswe will show, the estimations of the critical couplings obtainedwith protocol (S) coincide with those given by protocols (I) and(II), thus indicating that the reported results are not criticallydependent on the chosen initial conditions.

III. MEAN-FIELD THEORY

In the fully coupled case Eq. (1) can be rewritten, byemploying the order parameter definition (2) as follows:

mθi + θi = i − Kr sin(θi − φ), (3)

which corresponds to a damped driven pendulum equation.This equation admits for sufficiently small forcing frequencyi two fixed points: a stable node and a saddle. At larger fre-

quencies i > P 4π

√Krm

a homoclinic bifurcation leadsto the emergence of a limit cycle from the saddle. The stablelimit cycle and the stable fixed point coexist until a saddlenode bifurcation, taking place at i = D = Kr , leads tothe disappearance of the fixed points, and for i > D onlythe oscillating solution is present. This scenario is correct forsufficiently large masses, and at small m one has a directtransition from a stable node to a periodic oscillating orbit ati = D = Kr [29].

Therefore for sufficiently large m there is a coexistenceregime where, depending on the initial conditions, the singleoscillator can rotate or stay quiet. How this single unit propertywill reflect in the self-consistent collective dynamics of thecoupled systems is the topic of this paper.

A. The theory of Tanaka, Lichtenberg, and Oishi

Tanaka, Lichtenberg, and Oishi (TLO) in their semi-nal papers [20,21] have examined the origin of the first-order hysteretic transition observed for Lorentzian and flat(bounded) frequency distributions g() by considering twodifferent initial states for the network : (I) the completelydesynchronized state (r = 0) and (II) the fully synchronizedone (r ≡ 1). Furthermore, in case I (II) they studied howthe level of synchronization, measured by r , varies due tothe increase (decrease) of the coupling K . In the first case theoscillators are all initially drifting with finite velocities 〈θi〉; byincreasing K the oscillators with smaller natural frequencies|i | < P begin to lock (〈θi〉 = 0), while the others continueto drift. This picture is confirmed by the data reported in Fig. 1,where the maximal value M of the frequencies of the lockedoscillators is well approximated by P . The process continuesuntil all the oscillators are finally locked leading to r = 1.

In case (II), TLO assumed that initially all the oscillatorswere already locked, with an associated order parameter r ≡ 1.

042905-3


60


Therefore, the oscillators can start to drift only when the stablefixed point solution will disappear, leaving the system onlywith the limit cycle solution. This happens, by decreasingK , whenever |i | D = Kr . This is numerically verified;indeed, as shown in Fig. 1, the maximal locked frequencyM remains constant until, by decreasing K , it encounters thecurve D , and then M follows this latter curve down to thedesynchronized state.

In both the examined cases there is a group of desyn-chronized oscillators and one of locked oscillators separatedby a frequency, P in the first case and D in the secondone. These groups contribute differently to the total level ofsynchronization of the system, namely,

r = rL + rD, (4)

where rL (rD) is the contribution of the locked (drifting)population.

For the locked population, one gets

rI,IIL = Kr

∫ θP,D

−θP,D

cos2 θg(Kr sin θ ) dθ, (5)

where θP = sin−1(P /Kr) and θD = sin−1(D/Kr) ≡ π/2.The drifting oscillators contribute to the total order param-

eter with a negative contribution; the self-consistent integraldefining rD has been estimated by TLO in a perturbativemanner by performing an expansion up to the fourth orderin 1/(mK) and 1/(m). Therefore the obtained expression iscorrect in the limit of sufficiently large masses, and it reads as

rI,IID −mKr

∫ ∞

P,D

1

(m)3g() d, (6)

where g() = g(−).By considering an initially desynchronized (fully synchro-

nized) system and by increasing (decreasing) K one can get atheoretical approximation for the level of synchronization inthe system by employing the mean-field expression (5), (6),

and (4) for case I (II). In this way, two curves are obtained in thephase plane (K,r), namely, rI (K) and rII (K). In the following,we will show that these are not the unique admissible solutionsin the mentioned plane, and these curves represent the lowerand upper bound for the possible states characterized by apartial level of synchronization.

Let us notice that the expression for rL and rD reportedin Eqs. (5) and (6) are the same for case (I) and (II), onlythe integration extrema have been changed. These are definedby the frequency which discriminates locked from driftingoscillators, which in case (I) is P and in case (II) D . Thevalue of these frequencies is a function of the order parameterr and of the coupling constant K , therefore by increasing(decreasing) K they change accordingly.

However, in principle one could fix the discriminating fre-quency to some arbitrary value 0 and solve self-consistentlyEqs. (4), (5), and (6) for different values of the coupling K .This amounts to solving the following equation:

∫ θ0

−θ0

cos2 θg(Kr0 sin θ ) dθ − m

∫ ∞

0

1

(m)3g() d = 1

K,

(7)with θ0 = sin−1(0/Kr0). Thus obtaining a solution r0 =r0(K,0), which exists provided that 0 D(K) = r0K .Therefore a portion of the (K,r) plane, delimited by thecurve rII (K), will be filled with the curves r0(K) obtainedfor different 0 values (as shown in Fig. 3 for fully coupledsystems and in Fig. 14 for diluted ones.). These solutionsrepresent clusters of NL oscillators for which the maximallocking frequency and NL do not vary upon changing thecoupling strength. These states will be the subject of numericalinvestigation of the next sections. In particular, we will showvia numerical simulations that for K > Kc

2 these states arenumerically observable within the portion of the phase spacedelimited by the two curves rI (K) and rII (K) (see Fig. 3 andFig. 14).

0 2 4 6 8 10K

0

0.2

0.4

0.6

0.8

1

r

1 2 3 4 5K

0

100

200

300

400

500

NL

(a)

KFG

0 2 4 6 8 10 12 14 16 18 20K

0

0.2

0.4

0.6

0.8

1

r

0 5 10 15 20

K0

100

200

300

400

500

NL

(b)

KFG

FIG. 3. (Color online) Average order parameter r vs the coupling constant K for m = 2 (a) and m = 6 (b). Mean-field estimates: the dashed(solid) red curves refer to rI = rI

L + rID (rII = rII

L + rIID ) as obtained by employing Eqs. (5) and (6) following protocol (I) (protocol (II)); the

(green) dot-dashed curves are the solutions r0(K,0) of Eq. (7) for different 0 values. The employed values from bottom to top are 0 = 1.21and 1.71 (a) and 0 = 0.79, 1.09, 1.31, and 1.79 (b). Numerical simulations: (blue) filled circles have been obtained by following protocol(I) and then (II) starting from K = 0 until KM = 10 (KM = 20) for mass m = 2 (m = 6) with steps K = 0.2 (K = 0.5); (orange) filledtriangles refer to simulations performed by starting from a final configuration obtained during protocol (I) and by decreasing the coupling fromsuch initial configurations. The insets display NL vs K for the numerical simulations reported in the main figures; the value of KG

F [Eq. (14)]is also reported in the two cases. The numerical data refer to N = 500, TR = 5000, and TW = 200.

042905-4


61


B. Linear stability limit for the incoherent solution

As a final aspect, we will report the results of a recenttheoretical mean-field approach based on the Kramers descrip-tion of the evolution of the single oscillator distributions forcoupled oscillators with inertia and noise [30,31]. In particular,the authors in Ref. [31] have derived an analytic expression forthe coupling KMF

1 , which delimits the range of linear stabilityfor the asynchronous state. In the limit of zero noise, KMF

1 canbe obtained by solving the following equation:

1

KMF1

= πg(0)

2− m

2

∫ ∞

−∞

g()d

1 + m22, (8)

where g() is an unimodal distribution with zero average andstandard deviation σ . In the limit m → 0 one recovers the valueof the critical coupling for the usual Kuramoto model [1],namely, KMF

1 (m ≡ 0) = 2/(πg(0)). For a Lorentzian distri-bution g() = σ/π (σ 2 + 2) an explicit expression for anyvalue of the mass can be obtained:

KMF1 = 2σ (1 + mσ ), (9)

which coincides with the one reported by Acebron et al. [30].For a Gaussian distribution it is not possible to find anexplicit expression for any m; however, one can derive thefirst corrective terms to the zero mass limit, namely,

KMF1 = 2σ

√2

π

1 +

√2

πmσ + 2

πm2σ 2

+√(

2

π

)3

− 2

πm3σ 3

+ O(m4σ 4). (10)

On the opposite limit one can analytically show that the criticalcoupling diverges as

KMF1 ∝ 2mσ 2 for mσ → ∞. (11)

It can be seen that this scaling is already valid for not toolarge masses; indeed, the analytic results obtained via Eq. (8)are very well approximated, in the range m ∈ [1:30], by thefollowing expression:

KMF1 2σ (0.64 + mσ ). (12)

This result, together with Eq. (9), indicates that for both theLorentzian and the Gaussian distribution the critical couplingdiverges, in the limit of vanishing mass, proportionally to σ ,and, for a finite inertia, linearly with the mass and the varianceσ 2 of the frequency distribution.

In the next section we will compare our numerical resultsfor various system sizes with the mean-field result (8).

C. Limit of complete synchronization

Complete synchronization can be achieved, in the idealcase of infinite oscillators with a distribution g() with infinitesupport, only in the limit of infinite coupling. However, in finitesystems an (almost) complete synchronization is attainablealready at finite coupling; to give an estimation of this effectivecoupling KFC one can proceed as follows. Let us estimate thepinning frequency P required to have a large percentage of

oscillators locked; this can be implicitely defined as, e.g.,∫ P

−P

g() d = 0.954, (13)

where by assuming r 1 one sets P 4π

√KFC

mand from

Eq. (13) one can derive the coupling KFC . For a Gaussiandistribution the integral reported in Eq. (13) amounts toconsidering two standard deviations, and therefore one gets

KGFC

(π

2

)2

mσ 2, (14)

while for a Lorentzian distribution g() = σπ

1σ 2+2 this corre-

sponds to

KLFC

(13.815π

4

)2

mσ 2. (15)

These results reveal that for increasing mass and variance σ 2

of the frequency distribution the system becomes harder andharder to fully synchronize and that to achieve the same levelof synchronization a much larger coupling is required for theLorentzian distribution (for the same m and σ ). Notice thatKFC and KMF

1 present the same scaling with m and σ 2 forsufficiently large inertia values.

IV. FULLY COUPLED SYSTEM

In this section we will compare the analytical results withfinite N simulations for the fully coupled system: a firstcomparison is reported in Fig. 3 for two different masses,namely, m = 2 and m = 6. We observe that the data obtainedby employing procedure (II) are quite well reproduced from themean-field approximation rII for both masses (solid red curvein Fig. 3). This is not the case for the theoretical estimationrI (dashed red curve), which for m = 2 is larger than thenumerical data up to a quite large coupling, namely, K 5;while for m = 6, a better agreement is observable at smallerK; however, now r reveals a stepwise structure for the datacorresponding to protocol (I). This stepwise structure at largemasses is due to the breakdown of the independence of thewhirling oscillators, namely, to the formation of locked clustersat nonzero velocities [20]. Therefore, oscillators join in smallgroups to the locked solution and not individually as happensfor smaller masses; this is clearly revealed by the behavior ofNL versus the coupling K as shown in the insets of Fig. 3(b).

A. Hysteretic behavior

As already mentioned, we would like to better investigatethe nature of the hysteresis observed by performing simu-lations accordingly following protocol (I) or protocol (II).In particular, we consider as an initial condition a partiallysynchronized state obtained during protocol (I) for a certaincoupling KS > K1, and then we perform a sequence ofconsecutive simulations by reducing the coupling at regularsteps K . Some examples of the obtained results are shownin Fig. 3, where we report r and NL measured duringsuch simulations as a function of the coupling (orange filledtriangles). From the simulations it is evident that the number

042905-5


62


0 2 4 6K

0

1

2

ΩMΩP

(I)

ΩD(II)

FIG. 4. (Color online) Maximal locking frequency M vs thecoupling constant K . The initial state is denoted by the filled circleat KI = 5. The solid (red) curve indicates the frequency

(I )P and the

dashed (green) curve the frequency (II )D . The numerical data refer to

N = 500, TR = 5000, TW = 200, m = 2, and K = 0.1–0.05.

of locked oscillators NL remains constant until the descendingcurve, obtained with protocol (II), is reached. On the otherhand r decreases slightly with K; this decrease can bewell approximated by the mean-field solutions of Eq. (7),

namely, r0(K,0) with 0 = P (Ks,rI (KS)) = 4

π

√KsrI

m; see

the green dot-dashed lines in Fig. 3 for m = 2 and m = 6.However, as soon as, by decreasing K , the frequency 0

becomes equal or smaller than D , the order parameter has arapid drop towards zero following the upper limit curve rII .

To better interpret these results, let us focus on a simplenumerical experiment. We consider a partially synchronizedstate obtained for KI = 5 with N = 500 oscillators, then wefirst decrease the coupling in steps K up to a couplingKF = 2, and then we increase again K to return to the initialvalue KI . During such cyclic simulation we measure M

for each examined state; the results are reported in Fig. 4.It is clear that initially M does not vary, and it remainsidentical to its initial value at KI = 5. Furthermore, alsothe number of locked oscillators NL remains constant. Themaximal locking frequency (as well as NL) starts to decreasewith K only after M has reached the curve

(II )D , then it

follows exactly this curve, corresponding to protocol (II), untilK = KF . At this point we increase again the coupling: themeasured M stays constant at the value

(II )D = 2 ∗ rII (KF ).

The frequency M starts to increase only after its encounterwith the curve

(I )P (K). In the final part of the simulation M

recovers its initial value by following this latter curve. Fromthese simulations it is clear that a synchronized cluster can bemodified by varying the coupling, only by following protocol(I) or protocol (II), otherwise the coupling seems not to haveany relevant effect on the cluster itself. In other words, allthe states (K,M ) contained between the curves

(II )D and

(I )P are reachable for the system dynamics; however, they are

quite peculiar.We have verified that the path connecting the initial state at

KI to the curve (II )D (K), as well as the one connecting KF to

the curve (I )P (K), is completely reversible. We can increase

(decrease) the coupling from KI (KF ) up to any intermediatecoupling value in steps of any size K and then decrease(increase) the coupling to return to KI (KF ) by performingthe same steps, and the system will pass exactly from the samestates, characterized for each examined K by the values of r

and M . Furthermore, as mentioned, there is no dependenceon the employed step K , apart the restriction that the reachedstates should be contained within the phase space portiondelimited by the two curves

(II )D and

(I )P . As soon as the

coupling variations would eventually lead the system outsidethis portion of the phase space, one should follow a hystereticloop to return to the initial state, similar to the one reported inFig. 4. Therefore, we can affirm that hysteretic loops of anysize are possible within this region of the phase space. Forwhat concerns the stability of these states, we can only affirmthat from a numerical point of view they appear to be stablewithin the considered integration times. However, a (linear)stability analysis of these solutions is required to confirm ournumerical observations.

B. Finite size effects

Let us now examine the influence of the system size onthe studied transitions; in particular we will estimate thetransition points Kc

1 (Kc2 ) by considering either a sequence

of simulations obtained accordingly to protocol (I) [protocol(II)] or asynchronous (synchronous) initial conditions and byaveraging over different realizations of the distributions of theforcing frequencies i.

The results for the protocol (I) [protocol (II)] simulationsare reported in Fig. 5 for sizes ranging from N = 500 up toN = 16 000. It is immediately evident that Kc

2 does not dependheavily on N , while the value of Kc

1 is strongly influenced bythe size of the system. Starting from the asynchronous state thesystem synchronizes at larger and larger coupling Kc

1 with anassociated jump in the order parameter which increases withN . Whenever the system starts to synchronize, then it followsreasonably well the mean-field TLO prediction, and this isparticularly true on the way back towards the asynchronousstate along the path associated to the protocol (II) procedure.However, TLO theory largely fails in giving an estimation ofKc

1 for large system sizes, as shown in Fig. 5.In the following, we will analyze if the reported finite-size

results, and in particular the values of the critical couplings Kc1

and Kc2 , depend on the initial conditions and on the simulation

protocols. For this analysis we focus on two masses, namely,m = 2 and m = 6, and we consider system sizes ranging fromN = 500 to N = 16 000. For each size and mass we evaluateKc

1 (Kc2 ) by following protocol (I) [protocol (II)], as already

shown in Fig. 5; furthermore now the critical couplings arealso estimated by considering random initial conditions andby applying the protocol (S).

The results are reported in Fig. 6 for four different valuesof the mass; it is clear, by looking at the data displayed inFigs. 6(c) and 6(d), that protocol (I) [protocol (II)] and protocol(S) give essentially the same critical couplings, suggesting thattheir values are not dependent on the chosen initial conditions.Furthermore, while Kc

2 reveals a weak dependence on N , Kc1

increases steadily with the system size. On the basis of ournumerical data, it seems that the growth slows at large N ,

042905-6


63


1 2 3 4 5 6 7 8K

0

0.2

0.4

0.6

0.8

1

r N=500N=1000N=2000N=4000N=8000N=16000

(a)

K1MF

0 2 4 6 8 10 12 14 16 18 20K

0

0.2

0.4

0.6

0.8

1

r N=500N=1000N=2000N=4000N=8000N=16000

(b)

K1MF

FIG. 5. (Color online) Average order parameter r vs the coupling constant K for various system sizes N : (a) m = 2 and (b) m = 6. The(red) solid and dashed curves are the theoretical estimates already reported in Fig. 3. The numerical data have been obtained by followingprotocol (I) and then protocol (II) from K = 0 up to KM = 10 (KM = 20) for mass m = 2 (m = 6) with K = 0.2 (K = 0.5). The verticaldashed (maroon) lines refer to KMF

1 reported in (8). Data have been obtained by averaging the order parameter over a time window TW = 200,after discarding a transient time TR 1000–80 000 depending on the system size; the larger TR have been employed for the larger N .

but we are unable to judge if Kc1 is already saturated to an

asymptotic value at the maximal reached system size, namely,N = 16 000.

To clarify this issue we compare our numerical results forKc

1 with the mean-field estimated KMF1 reported in Eq. (8).

The mean-field result is always larger than the finite-sizemeasurements; however, for small masses, namely, m = 0.8and m = 1.0, Kc

1 seems to approach this asymptotic valuealready for the considered number of oscillators, as shown inFigs. 6(a) and 6(b). Therefore, in these two cases we attemptto identify the scaling law ruling the approach of Kc

1 toits mean-field value for increasing system sizes. The resultsreported in Fig. 7 suggest the following power-law scaling:

[KMF

1 − Kc1 (N )

] ∝ N−γ (16)

with γ 0.22 for m = 1.0; an analogous scaling has beenfound for m = 0.8.

Let us now consider several different values of the massin the range 0.8 m 30; the data for the critical couplingsare reported in Fig. 8 for different system sizes ranging fromN = 1000 to N = 16 000. It is evident that Kc

1 grows withN for all masses, while Kc

2 varies in a more limited manner.In particular, the estimated Kc

2 shows an initial decrease withm followed by a constant plateau at larger masses [as shownin Fig. 8(b)]. A possible mean-field estimation for Kc

2 can begiven by the minimal value KII

m reached by the coupling alongthe TLO curve rII (K). This value is reported in Fig. 8(b)together with the finite-size data: at small masses KII

m givesa reasonable approximation of the numerical data, while atlarger masses it is always smaller than the finite size results,

0 4000 8000 12000 16000N1.5

2

2.5

3

3.5

Kc

0 4000 8000 12000 16000N1.5

2

2.5

3

3.5

Kc

(a)

(b)

0 4000 8000 12000 16000N1

2

3

4

5

Kc

0 4000 8000 12000 16000N1

2

3

4

Kc

(d)

(c)

FIG. 6. (Color online) The upper symbols refer to the critical couplings Kc1 (orange filled diamonds and red empty circles), while the lower

ones to Kc2 (cyan filled diamonds and blue empty circles) vs the system size N : (a) m = 0.8, (b) m = 1, (c) m = 2, and (d) m = 6. The filled

symbols refer to estimates performed with protocol (S), while empty symbols, in panels (c) and (d), have been obtained with protocol (I)[protocol (II)] for Kc

1 (Kc2 ). The dashed (black) lines in panels (a) and (b) are the mean-field values KMF

1 . This quantity is not reported inpanels (c) and (d) for clarity reasons, due to its large value, namely, KMF

1 = 5.31 for m = 2 and KMF1 = 13.27 for m = 6. For all panels the

data have been derived by averaging in time over a window TW = 2000 and over 8 (5) different initial conditions for the protocol (S) [protocol(I) and (II)]. For each simulation an initial a transient time TR 20 000 (TR 1000–80 000) has been discarded for protocol (S) [protocol (I)and (II)].

042905-7


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100 1000 10000 N

1

K1M

F -K1c

FIG. 7. (Color online) The differences KMF1 − Kc

1 (N ) (filled or-ange diamonds) are reported vs the system size N for m = 1. Thedashed (black) line is a power-law fit to the data: the differencevanishes, 5.27N−0.22 for m = 1 (b). The data for Kc

1 are the samereported in panel (b) in Fig. 6.

and it saturates to a constant value for m → ∞. These resultsindicate that finite size fluctuations destabilizes the coherentstate at larger coupling than expected from a mean-field theory.

On the other hand Kc1 appears to increase with m up to

some maximal value and then to decrease at large masses.However, this is clearly a finite size effect, since by increasingN the position of the maximum shifts to larger masses. Thefinite size curves Kc

1 = Kc1 (m,N ) are always smaller than the

mean-field result KMF1 [dashed orange line in Fig. 8(a)] for

all considered system sizes and masses. However, as shown inthe inset of Fig. 8(a), such curves collapse one over the otherif the variables are properly rescaled, suggesting the followingfunctional dependence:

ξ ≡ KMF1 − Kc

1 (m,N )

KMF1

= G

(m

Nγ

), (17)

where γ = 1/5. This result is consistent with the values ofthe scaling exponent γ found for fixed mass by fitting the

data with the expression reported in Eq. (16). However, weare unable to provide any argument to justify such scaling, andfurther analysis is required to interpret these results. A possiblestrategy could be to extend the approach reported in Ref. [32]for the finite-size analysis of the usual Kuramoto transition tothe Kuramoto model with inertia.

C. Drifting clusters

As already noticed in Ref. [20], for a sufficiently largevalue of the mass one observes that the partially synchronizedphase, obtained by following protocol (I), is characterizednot only by the presence of the cluster of locked oscillatorswith 〈θ〉 0, but also by the emergence of clusters composedby drifting oscillators with finite average velocities. This isparticularly clear in Fig. 9(a), where we report the data formass m = 6. By increasing the coupling K one observes forK > 3 the emergence of a cluster of whirling oscillators witha finite velocity |〈θ〉| 1.05; these oscillators have naturalfrequencies in the range |i | 0.15–0.25. The number ofoscillators in this secondary cluster NDC increases up toK 5, then it declines, and finally the cluster is absorbedin the main locked group for K 7. At the same timea second smaller cluster emerges characterized by a largeraverage velocity |〈θ〉| 1.6 (corresponding to larger |i | 0.27 − 0.34). This second cluster merges with the lockedoscillators for K 12.5, while a third one, composed ofoscillators with even larger frequencies |i | and characterizedby larger average phase velocity, arises. This process repeatsuntil the full synchronization of the system is achieved.

The effect of these extra clusters on the collective dynamicsis to induce oscillations in the temporal evolution of the orderparameter, as one can see from Fig. 9(b). In presence ofdrifting clusters characterized by the same average velocity(in absolute value), as for m = 6 and K = 5 in Fig. 9(b), r

exhibits almost regular oscillations, and the period of theseoscillations is related to the one associated to the oscillators inthe whirling cluster. This can be appreciated from Fig. 10(b),where we compare the evolution of the instantaneous velocity

0 10 20 30m

2

2.5

3

3.5

4

4.5

5

5.5

K1c

0 2 4m/N1/5

0.2

0.4

0.6

0.8

ξ

(a)

0 5 10 15 20 25 30m1.6

1.8

2

2.2

K2c

(b)

FIG. 8. (Color online) Critical couplings Kc1 (a) and Kc

2 (b) vs the mass m for different system sizes N , namely, N = 1000 (red diamond),2000 (blue circles), 4000 (green triangles), 8000 (magenta squares), and 16 000 (black asterisks). The dashed (orange) line in (a) is themean-field estimate KMF

1 , while the dashed (magenta) line in (b) is the value KIIm obtained by the TLO theory. The inset in panel (a) reports the

critical rescaled couplings ξ = (KMF1 − Kc

1 )/KMF1 as a function of m/N 1/5. The estimates have been obtained with protocol (S), by averaging

in time over a window TW = 2000–4000 and over eight different initial conditions. For each simulation an initial transient time TR 20 000has been discarded.

042905-8


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-0.4 -0.2 0 0.2 0.4 0.6Ωi

-4

-2

0

2

4

<dθ i/d

t>

K=15K=10K=5K=2.5

(a)

0 20 40 60 80time

0

0.2

0.4

0.6

0.8

1

r(t)

(b)

FIG. 9. (Color online) (a) Average phase velocity 〈θi〉 of the oscillators vs their natural frequencies i : (magenta) triangles refer to K = 2.5,(green) diamond to K = 5, (red) squares to K = 10, and (black) circles to K = 15. For each simulation an initial transient TR 5500 has beendiscarded and the estimates have been obtained with protocol (I), by averaging in time over a window TW = 5000. (b) Order parameter r(t)vs time for m = 6 and N = 500 and different coupling constants K: the (blue) solid curve corresponds to K = 1, the (magenta) dot-dashedline to K = 2.5, the (green) dashed line to K = 5, the (red) dashed line to K = 10, and the (black) solid line to K = 15. The data have beenobtained by employing protocol (I), and for each simulation an initial transient time TR 1500 has been discarded and data are averaged overa time TW = 5000.

θi for three oscillators and the time course of r(t). We considerone oscillator O1 in the locked cluster, and 2 oscillators O2

and O3 in the drifting cluster. We observe that these latteroscillators display essentially synchronized motions, whilethe phase velocity of O1 oscillates irregularly around zero.Furthermore, the almost periodic oscillations of the orderparameter r(t) are clearly driven by the periodic oscillationsof O2 and O3 [see Fig. 10(b)].

We have also verified that the amplitude of the oscillationsof r(t) (measured as the difference between the maximal rmax

and the minimal rmin value of the order parameter) and thenumber of oscillators in the drifting clusters NDC correlatesin an almost linear manner, as shown in Fig. 11(b). Thereforewe can conclude that the oscillations observable in the orderparameter are induced by the presence of large secondaryclusters characterized by finite whirling velocities. At smaller

masses (e.g., m = 2) oscillations in the order parameter arepresent, but they are much smaller and irregular (data notshown). These oscillations are probably due to finite sizeeffects, since in this case we do not observe any cluster ofdrifting oscillators in the whole range from an asynchronousto fully synchronized state.

The situation was quite different in the study reported inRef. [21], where the authors considered natural frequenciesi uniformly distributed over a finite interval and notGaussian distributed as in the present study. In that case, byconsidering an initially clusterized state, similar to what donefor protocol (S), r(t) revealed regular oscillations even formasses as small as m = 0.85. In agreement with our results, theamplitude of the oscillations measured in Ref. [21] decreasesby approaching the fully synchronized state (as shown inFig. 11). However, the authors in Ref. [21] did not relate the

-0.4 -0.2 0 0.2 0.4Ωi

-2

0

2

<dθ i/d

t>

O1

O2

O3

(a)

0 10 20 30 40time

0

0.5

1

1.5

2

r(t) O1

O3O2

(b)

FIG. 10. (Color online) (a) Average phase velocity 〈θi〉 of the oscillators vs the corresponding natural frequency i . The vertical dashedlines denote the three oscillators, O1, O2, and O3, whose dynamical evolution is shown in panel (b). (b) The black curve represents the orderparameter r vs time; the other curves refer to the time evolution of the phase velocities θ (t) of the three oscillators O1 (red dot-dashed curve),O2 (magenta solid line), and O3 (dashed blue curve). For each simulation an initial transient time TR 1000 has been discarded; the averagesreported in (a) have been obtained over a time window TW = 20 000. In both panels K = 5, m = 6, and N = 500.

042905-9


66


0 5 10 15 20K

0

0.2

0.4

0.6

0.8

1

r min

,r m

ax

(a)

0 5 10 15 20K

0

50

100

150

(b)

(rmax-rmin)*240

NDC

FIG. 11. (Color online) (a) Minima and maxima of the order parameter r as a function of the coupling constant K . The (blue) dashed linerefer to the theoretical estimate rI , as obtained by employing Eqs. (5) and (6). (b) The number of oscillators in the drifting clusters NDC (filledblack diamond) is reported vs the coupling K together with the amplitude of the oscillations of the order parameter rmax − rmin (empty redcircles) rescaled by a factor 240. For each simulation an initial transient time TR 1500 has been discarded. The estimates have been obtainedwith protocol (I), by averaging in time over a window TW = 5000, m = 6, N = 500.

observed oscillations in r(t) with the formation of driftingclusters.

As a final aspect, as one can appreciate from Fig. 5,for larger masses the discrepancies between the measuredr , obtained by employing protocol (I), and the theoreticalmean-field result rI increase. In order to better investigate theorigin of these discrepancies, we report in Fig. 11 the minimaland maximal value of r as a function of the coupling K andwe compare these values to the estimated mean-field valuerI . The comparison clearly reveals that rI is always containedbetween rmin and rmax, therefore the mean-field theory capturescorrectly the average increase of the order parameter, but it isunable to foresee the oscillations in r . A new version of thetheory developed by TLO in Ref. [20] is required in orderto include also the effect of clusters of whirling oscillators.A similar synchronization scenario, where oscillations in r(t)are induced by the coexistence of several drifting clusters, hasbeen recently reported for the Kuramoto model with degreeassortativity [33].

V. DILUTED NETWORKS

In this section we will analyze diluted networks obtainedby considering random realizations of the coupling matrix Ci,j

with the constraints that the matrix should remain symmetricand the in-degree should be constant and equal to Nc.2 Inparticular, we will examine if the introduction of the randomdilution in the network will alter the results obtained by themean-field theory and if the transition will remains hysteretic

2In particular, each row i of the coupling matrix Ci,j is generatedby choosing randomly a node m and by imposing Ci,m = Cm,i = 1;this procedure is repeated until Nc elements of the row are set equalto one. Obviously, before accepting a new link, one should verifythat in the considered row the number of links is smaller than Nc andthat this is true also for all the interested columns. Finally, we haveperformed an iterative procedure to ensure that all rows and columnscontain exactly Nc nonzero elements.

or not. For this analysis we limit ourselves to a single value ofthe mass, namely, m = 2.

Let us first consider how the dependence of the orderparameter r on the coupling constant K will be modified inthe diluted systems. In particular, we examine the outcomes ofsimulations performed with protocols (I) and (II) for a systemsize N = 2000 and different realizations of the diluted networkranging from the fully coupled case to Nc = 5. The results,reported in Fig. 12, reveal that as far N 125 (correspondingto the 94% of cut links) it is difficult to distinguish amongthe fully coupled situation and the diluted ones. The smallobserved discrepancies can be due to finite size fluctuations.For larger dilution, the curves obtained with protocol (II)reveal a more rapid decay at larger coupling. Therefore Kc

2increases by decreasing Nc and approaches Kc

1 as shown inFig. 12(b). The dilution has almost no effect on the curveobtained with protocol (I); in particular Kc

1 remains unchanged(apart fluctuations within the error bars) until the percentageof incoming links Nc/N reduces below 0.5%. For smallerconnectivities both Kc

1 and Kc2 shift to larger coupling and

they approach one another, indicating that the synchronizationtransition from hysteretic tends to become continuous. Indeed,this happens for N = 1000 and N = 500 [as shown in the insetof Fig. 12(b)]: for such system sizes we observe essentiallythe same scenario as for N = 2000, but already for in-degreesNc 5 the transition is no more hysteretic. This seems tosuggest that by increasing the system size the transition willstay hysteretic for vanishingly small percentages of connected(incoming) links. This is confirmed by the data shown inFig. 13, where we report the width of the hysteretic loop Wh,measured at a fixed value of the order parameter; namely,we considered r = 0.9. For increasing system sizes Wh,measured for the same fraction of connected links Nc/N ,increases, while the continuous transition, corresponding toWh ≡ 0, is eventually reached for smaller and smaller value ofNc/N . Unfortunately, due to the CPU costs, we are unable toinvestigate in details diluted systems larger than N = 2000.

Therefore, from this first analysis it emerges that the dilutedor fully coupled systems, whenever the coupling is properly

042905-10


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0 1 2 3 4 5 6K

0

0.2

0.4

0.6

0.8

1

r

Nc=NNc=5Nc=10Nc=15Nc=25Nc=50Nc=125Nc=250Nc=500Nc=1000

(a)

10 100 1000 10000Nc

1

2

3

4

5

10 100 N1

2

3

4

10 100 1000 N

2

4

6

8

10

(b)

K1c

K2c

N=2000 N=1000

N=500

FIG. 12. (Color online) (a) Average order parameter r vs the coupling constant K for diluted networks for various Nc: 5 (filled blackcircles), 10 (red squares), 15 (green diamond), 25 (blue up triangles), 50 (orange left triangles), 125 (turquoise down triangles), 250 (rightmagenta triangles), 500 (violet crosses), 1000 (empty maroon circles), 2000 (black asterisks). (b) Critical constants Kc

1 and Kc2 estimated for

various values of the in-degree. The numerical data refer to N = 2000; the upper inset refer to N = 1000, the lower one to N = 500. Forall simulations m = 2, TR = 10 000, and TW = 2000; each series of simulations have been obtained by following protocol (I) and then (II)starting from K = 0 until KM = 20 with steps K = 0.25. The reported data have been obtained by averaging over 10–20 different series ofsimulations, each corresponding to a different realization of the random network and of the distribution of the frequencies i. The error barsin panel (b) correspond to K/2.

rescaled with the in-degree, as in Eq. (1), display the samephase diagram in the (r ,K) plane even for very large dilution.In the following we will examine if the mean-field resultsobtained by following the TLO approach still apply to thediluted system. The comparison reported in Fig. 14 confirmsthe good agreement between the numerical results obtainedfor a quite diluted system (namely, with 70% of broken links)and the mean-field predictions (5) and (6). Furthermore, thedata reported in Fig. 14 show that also in the diluted caseall the states between the synchronization curves obtained

0.01 0.1 1Nc/N

0

0.5

1

1.5

2

2.5

3

Wh

N=500N=1000N=2000

0 3 6 9

K0

0.2

0.4

0.6

0.8

1

r

W

FIG. 13. (Color online) Width of the hysteretic loop Wh, mea-sured in correspondence of an order parameter value r = 0.9, as afunction of the percentage of connected links Nc/N . The (green)circles refer to N = 500, the (red) squares to N = 1000, and the(black) diamonds to N = 2000. The dashed lines refer to logarithmicfitting to the data in the range 0.01 < Nc/N 1. In the inset isgraphically explained how Wh has been estimated, starting fromone of the curves reported in Fig. 12(a). The data refer to the sameparameters and simulation protocols as in Fig. 12.

following protocol (I) and protocol (II) are reachable andnumerically stable, analogously to what shown in Sec. IV Afor the fully coupled system. These states, displayed as orangefilled triangles in Fig. 14, are characterized by a clustercomposed by a constant number NL of locked oscillators with

2 4 6 8K

0

0.2

0.4

0.6

0.8

1

r

0 2 4 6 8

K0

100

200

300

400

500

NL

FIG. 14. (Color online) Average order parameter r vs the cou-pling constant K for a diluted network with 70% of cut links.Mean-field estimates: the dashed (solid) red curves refer to rI =rIL + rI

D (rII = rIIL + rII

D ) as obtained by employing Eqs. (5) and (6)following protocol I [protocol (II)]; the (green) dot-dashed curvesare the solutions r0(K,0) of Eq. (7) for different 0 values. Theemployed values from bottom to top are 0 = 2.05, 1.69, and1.10. Numerical simulations: (blue) filled circles have been obtainedby following protocol (I) and then (II) starting from K = 0 untilKM = 20 with steps K = 0.5; (orange) filled triangles refer tosimulations performed by starting from a final configuration obtainedduring protocol (I) and by decreasing the coupling from suchinitial configurations. The insets display NL vs K for the numericalsimulations reported in the main figures. The numerical data refer tom = 2, N = 500, Nc = 150, TR = 5000, and TW = 200.

042905-11


68


frequencies smaller than a value M . The number of oscillatorsin the cluster NL remains constant by varying the couplingbetween the two synchronization curves (I) and (II). Finally,the generalized mean-field solution r0(K,0) [see Eq. (7)] isable, also in the diluted case, to well reproduce the numericallyobtained paths connecting the synchronization curves (I) and(II) (see Fig. 14 and the inset).

VI. A REALISTIC NETWORK: THE ITALIANHIGH-VOLTAGE POWER GRID

In this section, we examine if the previously reportedfeatures of the synchronization transition persist in a somehowmore realistic setup. As we mentioned in the introductiona highly simplified model for a power grid composed ofgenerators and consumers, resembling a Kuramoto model withinertia, can be obtained whenever the generator dynamics canbe expressed in terms of the so-called swing equation [23,24].The self-synchronization emerging in this model has beenrecently object of investigation for different network topolo-gies [25,34,35]. In this paper we will concentrate on theItalian high-voltage (380 kV) power grid (Sardinia excluded),which is composed of N = 127 nodes, divided in 34 sources(hydroelectric and thermal power plants) and 93 consumers,connected by 342 links [34]. This network is characterizedby a quite low average connectivity 〈Nc〉 = 2.865, due to thegeographical distributions of the nodes along Italy [36].

In this extremely simplified picture, each node can bedescribed by its phase φi(t) = ωACt + θi(t), where ωAC/2π =50 Hz or 60 Hz is the standard AC frequency and θi representsthe phase deviation of the node i from the uniform rotation atfrequency ωAC. Furthermore, the equation of motion for eachnode is assumed to be the same for consumers and generators;these are distinguished by the sign of a quantity Pi associatedto each node: a positive (negative) Pi corresponds to generated(consumed) power. By employing the conservation of energyand by assuming that the grid operates in proximity of theAC frequency (i.e., |θ | ωAC) and that the rate at which theenergy is stored (in the kinetic term) is much smaller thanthe rate at which is dissipated, the evolution equations for thephase deviations take the following expression [24]:

θi = α

⎡⎣−θi + Pi + K

∑j

Ci,j sin(θj − θi)

⎤⎦ . (18)

To maintain a parallel with the previously studied model (1),we have multiplied the left-hand side by a term α, whichin (18) represents the dissipations in the grid, while in (1) itcorresponds to the inverse of the mass. The parameter αK

now represents the maximal power which can be transmittedbetween two connected nodes. More details on the model arereported in Refs. [23,24,35]. It is important to stress that inorder to have a stable, fully locked state, as a possible solutionof (18), it is necessary that the sum of the generated powerequal the sum of the consumed power. Thus, by assuming thatall the generators are identical as well as all the consumers,the distribution of the Pi is made of two δ functions locatedat Pi = −C and Pi = +G. In our simulations we have setC = 1.0, G = 2.7353 and α = 1/6. This setup corresponds to

0 10 20 30K

0

0.2

0.4

0.6

0.8

1

r

FIG. 15. (Color online) Average order parameter r vs the pa-rameter K for the Italian high-voltage power grid network. The(blue) circles data have been obtained by following protocol (I) fromK = 1 up to KM = 40 with K = 1. The other symbols refer tosimulations performed following protocol (II) starting from differentinitial coupling KI down to K = 1, namely, (orange) trianglesKI = 10, (red) squares KI = 9, and (green) diamonds KI = 7.The dashed vertical (magenta) line indicates the value K = 9. Thereported data have been obtained by averaging the order parameterover a time window TW = 5000, after discarding an initial transienttime TR 60 000. The numerical data refer to α = 1/6, N = 127,〈Nc〉 = 2.865.

a Kuramoto model with inertia with a bimodal distribution ofthe frequencies.

As a first analysis we have performed simulations withprotocol (I) for the model (18) by varying the parameterK , and we have measured the corresponding average orderparameter r . As shown in Fig. 15 the behavior of r with K

is nonmonotonic. For small K the state is asynchronous withr 1/

√N , then r shows an abrupt jump for K 7 to a finite

value, and then it decreases reaching a minimum at K 9.For larger K the order parameter increases steadily with K

tending towards the fully synchronized regime.This behavior can be understood by examining the average

phase velocity of the oscillators 〈θi〉. As shown in Fig. 16,for coupling K < 7 the system is split in two clusters: onecomposed by the sources which oscillates with their properfrequency G and the other one containing the consumers,which rotates with average velocity −C. The oscillators in thetwo clusters rotate independently one from the other, thereforer 1/

√N . For K 7 the oscillators get entrained (as shown

in Fig. 16), and most of them are locked with almost zeroaverage velocity; however, a large part (50 out of 127) forma secondary cluster of whirling oscillators with a velocity〈θ〉 −0.127. This secondary cluster has a geographicalorigin, since it includes power stations and consumers locatedin the central part and south part of Italy, Sicily included. Thepresence of this whirling cluster induces large oscillations inthe order parameter [see Fig. 18(a)], reflecting almost regulartransitions from a desynchronized to a partially synchronizedstate. By increasing the coupling to K = 8 the two clustersmerge in a unique cluster with few scattered oscillators;however, the average velocity is small but not zero, namely,〈θ〉 −0.05 (as reported in Fig. 16). Therefore the average

042905-12


69


0 50 100-1

0

1

2

3

0 50 100-1

0

1

2

3

0 50 100-0.2

0

0.2

0 50 100

-0.05038

-0.05036

-0.05034

-0.05032

0 50 1000

1

2

3

0 50 100

-0.04

-0.02

0

0 50 1000123

0 50 1000123K=1 K=4 K=7

K=8 K=9 K=10

<dθ i/d

t>

Oscillator Index

FIG. 16. (Color online) Average phase velocity of each oscillator 〈θi〉 vs the oscillator index for different values of the coupling K . Theoscillators have been reordered so that the first 93 are consumers and the last 34 sources. The data have been obtained by employing protocol(I), starting from zero coupling K = 0 and with K = 1. For each simulation an initial transient time TR 5000 has been discarded and theaverage is taken over a window TW = 5000. The numerical data refer to the same parameters as in Fig. 15.

0 50 100

-1

0

1

2

3

0 50 100

-1

0

1

2

3

0 50 100

-1

0

1

2

3

0 50 100

-1

0

1

2

3

0 50 100-0.2

-0.1

0

0.1

0 50 100-0.5

00.5

11.5

22.5

3

0 50 1000

1

2

3

K=2 K=3

K=4

K=6

K=1

K=7

Oscillator Index

<dθ i/d

t>

FIG. 17. (Color online) Average phase velocity of each oscillator 〈θi〉 vs the corresponding oscillator index, ordered following thegeographical distribution from northern Italy to Sicily. The panels refer to different couplings. The colored clusters indicate Italian regionswhich remain connected for all the considered simulations: red symbols refer to Piedmont and Liguria, green symbols to Veneto and FriuliVenetia Giulia, blue symbols to Campania and Apulia, and magenta symbols to Sicily. The data have been obtained by employing protocol(II) starting from KI = 12 with K = 1 down to K = 1. For each simulation an initial transient time TR 50 000 has been discarded and theaverages performed over a window TW = 5000. The numerical data refer to the same parameters as in Fig. 15.

042905-13


70


0 100 200 300 400 500time

0

0.1

0.2

0.3

0.4

0.5

r(t)

(a)

0 50 100 150 200time

0

0.1

0.2

0.3

0.4

0.5

r(t)

(b)

FIG. 18. (Color online) Order parameter r(t) vs time for the Italian high-voltage power grid network for different values of the parameterK . (a) The dotted (magenta) curve corresponds to K = 4, the dashed (black) line to K = 7, the solid (orange) line to K = 9, the dot-dashed(cyan) line to K = 10. The data have been obtained by employing protocol (I), and for each simulation an initial transient time TR 5000 hasbeen discarded. (b) The solid (magenta) curve corresponds to K = 1, the dashed (black) line to K = 4, the solid (orange) thick line to K = 6,the dot-dashed (cyan) line to K = 7. The data have been obtained by employing protocol (II), and for each simulation an initial transient timeTR 50 000 has been discarded. The simulations refer to the same parameters employed in Fig. 15.

value of the order parameter r decreases with respect to K = 7,where a large part of the oscillators was exactly locked. Up toK = 9, the really last node of the network, corresponding toone generator in Sicily connected with only one link to the restof the Italian grid, still continues to oscillate independentlyfrom the other nodes, as shown in Fig. 16. Above K = 9 allthe oscillators are finally locked in a unique cluster, and theincrease in the coupling is reflected in a monotonous increasein r , similar to the one observed in standard Kuramoto models(see Fig. 15).

By applying protocol (II) we do not observe any hystereticbehavior or multistability down to K = 9; instead for smallercoupling a quite intricate behavior is observable. As shown inFig. 17 starting from KI = 12 and decreasing the coupling insteps of amplitude K = 1, the system stays mainly in onesingle cluster up to K = 7, except the last node of the network,which already detached from the network at some larger K .

Indeed, at K = 7 the order parameter has a constant valuearound 0.2 and no oscillations. As shown in Fig. 18(b), bydecreasing the coupling to K = 6, wide oscillations emergein r(t) due to the fact that the locked cluster has split in twoclusters; the separation is similar to the one reported for K = 7in Fig. 16. By further lowering K , several small whirlingclusters appear and the behavior of r(t) becomes seeminglyirregular for 2 K 5 as reported in Fig. 18(b). An accurateanalysis of the dynamics in terms of the maximal Lyapunovexponent has revealed that the irregular oscillations in r(t)reflect quasiperiodic motions, since the measured maximalLyapunov is always zero for the whole range of the consideredcouplings. The presence of the inertial term, together with anarchitecture which favors a splitting based on the proximityof the oscillators, lead to the formation of several whirlingclusters characterized by different average phase velocities.The value of the order parameter arises as a combination of

0 100 200 300 400 500time

0

0.2

0.4

0.6

0.8

r(t)

0 30 60 90 120Oscillator Index

-0.4

-0.2

0

0.2

0.4

<dθ i/d

t>(a) (b)

FIG. 19. (Color online) Italian high-voltage power grid network with Gaussian distributed powers Pi with zero average and unitary standarddeviation. (a) Order parameter r(t) vs time for different values of the parameter K . The data have been obtained by employing protocol (I)starting from K = 0.5 to K = 20 with K = 0.5: the dot-dashed (red) line corresponds to K = 2, the dashed (green) line to K = 3, andthe solid (blue) curve to K = 4. (b) Average phase velocity for each oscillator vs the corresponding oscillator index, ordered following thegeographical distribution from northern Italy to Sicily for coupling K = 2. The clusters are colored as in Fig. 17. For each simulation an initialtransient of TR = 100 000 has been discarded, and the reported data have been mediated over a time window TW = 5000 and refer to α = 1/6.

042905-14


71


these different contributions, each corresponding to a differentoscillatory frequency. The splitting in different clusters isprobably also at the origin of the multistability observed forK < 7: depending on the past history the grid splits in clustersformed by different groups of oscillators, and this gives rise todifferent average values of the order parameter (see Fig. 15).

We have verified that the emergence of several whirlingclusters, with an associated quasiperiodic behavior of the orderparameter, is observable also by considering an unimodal(Gaussian) distribution of the Pi , as shown in Fig. 19. Thisconfirms that the main ingredients at the origin of thisphenomenon are the inertial term together with a short-rangeconnectivity. Thus the bimodal distribution, here employed,seems not to be crucial, and it can only lead to an enhancementof such an effect.

VII. CONCLUSIONS

We have studied the synchronization transition for a glob-ally coupled Kuramoto model with inertia for different systemsizes and inertia values. The transition from an incoherent tocoherent state is hysteretic for sufficiently large masses. Inparticular, the upper value of the coupling constant (Kc

1 ), forwhich an incoherent state is observable, increases with thesystem sizes for all the examined masses. The estimated finitesize value Kc

1 has a nonmonotonic dependence on the massm, exhibiting a maximum at some intermediate value of m.However, all the data obtained for different masses and sizescollapse onto an universal curve, whenever the normalizeddistance of Kc

1 with respect to its mean-field value [31] isreported as a function of the mass divided by N1/5. On theother hand, the coherent phase is attainable above a minimalcritical coupling (Kc

2 ), which exhibits a weak dependence onthe system size, and it saturates to a constant asymptotic valuefor sufficiently large inertia values.

Furthermore, we have shown that clusters of lockedoscillators of any size coexist within the hysteretic region.This region is delimited by two curves in the plane individ-uated by the coupling and the average value of the orderparameter. Each curve corresponds to the synchronization(desynchronization) profile obtained starting from the fullydesynchronized (synchronized) state. The original mean-field theory developed by Tanaka, Lichtenberg, and Oishiin 1997 [20,21] gives a reasonable estimate of both theselimiting curves, while a generalization of such a theory iscapable to reproduce all the possible synchronization anddesynchronization hysteretic loops. However, the TLO theorydoes not take into account the presence of clusters composedby drifting oscillators emerging for sufficiently large masses.

The coexistence of these clusters with the cluster of lockedoscillator induces oscillatory behavior in the order parameter.

The properties of the hysteretic transition have been exam-ined also for random diluted networks; the main properties ofthe transition are not affected by the dilution up to extremelyhigh values. The transition appears to become continuous onlywhen the number of links per node becomes of the order offew units. By increasing the system size the transition to thecontinuous case (if any) shifts to smaller and smaller values ofthe connectivity.

In this paper we focused on Gaussian distribution ofthe natural frequencies; however, we have obtained similarresults also for Lorentzian distributions. It would, however, beinteresting to examine how the transition modifies in presenceof nonunimodal distributions for the natural frequencies,like bimodal ones. Preliminary indications in this directioncan be obtained by the reported analysis of the self-synchronization process occurring in the Italian high-voltagepower grid, when the generators and consumers are mimickedin terms of a Kuramoto model with inertia [24]. In this casethe transition is largely nonhysteretic; probably this is due tothe low value of the average connectivity in such a network.Coexistence of different states made of whirling and lockedclusters, formed on a regional basis, is observable only forelectrical lines with a low value of the maximal transmissiblepower. These states are characterized by quasiperiodic oscil-lations in the order parameter due to the coexistence of severalclusters of drifting oscillators.

A natural continuation of the presented analysis wouldbe the study of the stability of the observed clusters oflocked and/or whirling oscillators in the presence of noise.In this respect, exact mean-field results have been reportedrecently for fully coupled phase rotors with inertia and additivenoise [31,37]. However, the emergence of clusters in suchsystems has been not yet addressed, neither on a theoreticalbasis nor via direct simulations.

ACKNOWLEDGMENTS

We acknowledge useful discussions with J. Almendral, M.Bar, I. Leyva, A. Pikovsky, J. Restrepo, S. Ruffo, and I Sendina-Nadal, and we thank M. Frasca for providing the connectivitymatrix relative to the Italian grid. Financial support has beenprovided by the Italian Ministry of University and Researchwithin the project CRISIS LAB PNR 2011-2013 and MINECO(Spain) within the project FIS2012-38949-C03-01. S.O. andA.T. thank the German Science Foundation DFG, within theframework of SFB 910 “Control of self-organizing nonlinearsystems,” for the kind hospitality offered at Physikalisch-Technische Bundesanstalt in Berlin.

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B 26 (2012).[35] M. Rohden, A. Sorge, D. Witthaut, and M. Timme, Chaos 24,

013123 (2014).[36] The map of the Italian high-voltage power grid can be seen

at the web site of the Global Energy Network Institute,http://www.geni.org, and the data here employed have beenextracted from the map delivered by the union for the coordina-tion of transport of electricity (UCTE), https://www.entsoe.eu/resources/grid-map/

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Explosive transitions to synchronizationin networks of phase oscillatorsI. Leyva1,2, A. Navas2, I. Sendina-Nadal1,2, J. A. Almendral2, J. M. Buldu1,2, M. Zanin2,3,4,D. Papo2 & S. Boccaletti2

1Complex Systems Group, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain, 2Center for Biomedical Technology,Universidad Politecnica de Madrid, 28223 Pozuelo de Alarcon, Madrid, Spain, 3Faculdade de Ciencias e Tecnologia,Departamento de Engenharia Electrotecnica, Universidade Nova de Lisboa, Portugal, 4Innaxis Foundation & Research Institute,Jose Ortega y Gasset 20, 28006, Madrid, Spain.

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject ofthe utmost interest. The occurrence of a first-order phase transition to synchronization of an ensemble ofnetworked phase oscillators was reported, so far, for very particular network architectures. Here, we showhow a sharp, discontinuous transition can occur, instead, as a generic feature of networks of phaseoscillators. Precisely, we set conditions for the transition from unsynchronized to synchronized states to befirst-order, and demonstrate how these conditions can be attained in a very wide spectrum of situations. Wethen show how the occurrence of such transitions is always accompanied by the spontaneous setting offrequency-degree correlation features. Third, we show that the conditions for abrupt transitions can be evensoftened in several cases. Finally, we discuss, as a possible application, the use of this phenomenon to expressmagnetic-like states of synchronization.

Many complex systems operate transitions between different regimes or phases under the action of a controlparameter. These transitions can be monitored using a global order parameter, a physical quantity (e.g.scalar, vector, …) accounting for the symmetry of the phases. Phase transitions can be of first or second

order according to whether the order parameter varies continuously or discontinuously at a critical value of thecontrol parameter. In complex networks theory1, phase transitions have been observed in the way the graphcollectively organizes its architecture (e.g. percolation2,3) and dynamical state (e.g. synchronization4–6).

Recently, Achlioptas et al.7 proposed a slight modification of the classical percolation models, leading to a first-order type transition, named ’’explosive percolation’’. In the model, the authors consider a preferential growthprocess for a network with fixed number of vertices. Edges are added one by one, by picking each time tworandomly selected possible edges, and deciding to keep the one having a lower value of the product of the size ofthe two components that edge is joining, this way delaying as much as possible the formation of a giant connectedcomponent. At the percolation threshold, a macroscopic part of the system (a giant cluster of size O(N)) becomesconnected. While the transition is found to be abrupt in Erdos-Renyi (ER) networks, heterogeneous structures asscale-free (SF) networks not always display explosive percolation, which crucially depends, instead, on the degreedistribution8,9. The continuous or discontinuous nature of explosive percolation is still a matter of debate10.

The existence of abrupt transitions has also been investigated for the synchronization of networked phaseoscillators5,6. In this context, a first-order transition has initially been described in the Kuramoto model11 with aparticular realization of a uniform frequency distribution (evenly spaced frequencies) and an all-to-all networktopology12. However, lately it has been shown that a positive correlation between the node degree and thecorresponding oscillator’s natural frequency may lead to a first-order transition also for SF networks13. Theselatter results have been further generalized to networked chaotic oscillators, and experimentally proved14.

In this report, we show that a sharp, discontinuous phase transition is not restricted to the above rather limitedand apparently opposite cases, but it constitutes, instead, a generic feature of the synchronization of networkedphase oscillators. Precisely, we initially give a condition for the transition from unsynchronized to synchronizedstates to be first-order, and demonstrate how such a condition is easy to attain in many circumstances, and for awide class of oscillators’ initial frequency distributions. We then show how such transitions are always accom-panied by the spontaneous emergence of frequency-degree correlation features, and discuss how the consideredcondition can even be softened in several cases. Finally, we illustrate, as a possible application, the option ofexpressing magnetic-like states of synchronization with the use of such transitions.

SUBJECT AREAS:DYNAMIC NETWORKS

EMERGENCE

STATISTICAL PHYSICS,THERMODYNAMICS ANDNONLINEAR DYNAMICS

OSCILLATORS

Received19 December 2012

Accepted21 January 2013

Published15 February 2013

Correspondence andrequests for materials

should be addressed toI.S.N. (irene.sendina@

urjc.es)

SCIENTIFIC REPORTS | 3 : 1281 | DOI: 10.1038/srep01281 1


74

ResultsOur results are obtained with a network of N Kuramoto11 oscillators,which is described by:

dwi

dt~vizd

XN

i~1

aij sin wjwi

, ð1Þ

where wi is the phase of the ith oscillator (i 5 1, …, N), vi is itsassociated natural frequency [drawn from a generic frequency dis-tribution p(v)], d is the coupling strength, and aij are the elementsof the adjacency matrix that uniquely defines the nodes’ interactions.

The classical order parameter for the system given by equation (1)

is r tð Þ~ 1N

XNl~1eiwl tð Þ

, and the level of phase synchronization can

be monitored by looking at the value of S 5 Ær(t)æT, where Æ…æT

denotes a time average with T ? 1. Furthermore, for each oscillatori, we denote by N ið Þ the set of oscillators linked to it.

As the coupling strength d increases, system (1) undergoes a phasetransition from the unsynchronized S*1

ffiffiffiffiNp

to a synchronous(S , 1) state, where all oscillators ultimately acquire the same fre-quency. For this phase transition to display a first-order feature, wehave to avoid that any oscillator behave as the core of a clusteringprocess, where its neighbors begin to aggregate to the synchronousstate smoothly and progressively, as in the classical routes describedin Ref.15.

Along this paper, we realize such a condition by explicitly impos-ing certain constraints in the frequency differences between eachnode i and the whole set N ið Þ of oscillators belonging to its neigh-borhood. A first general constraint can be set as follows:

(C1) for each oscillator i, all nodes j belonging to N ið Þ satisfyj vi – vj j. cc.

Notice that condition (C1) is tantamount to imposing a minimalvalue for the frequency difference between linked nodes of thenetwork.

We now fix N 5 500, and illustrate the synchronization route forseveral frequency distributions p(v), when condition (C1) is met in afrequency gap-conditioned (FGC) random network (see Methodssection for the details on the network construction). In particular,we use the adjacency matrix obtained after the FGC network is con-structed to simulate system (1), and monitor the state of the networkby gradually increasing the value of d in steps dd 5 10–4, from d 5 0.At each step, a long transient is discarded before the data are acquiredfor further processing. Moreover, insofar as we are looking for a first-order phase transition (and thus for an expected associated hyster-esis), simulations are also performed in the reverse way, i.e. startingfrom a given dmax (where the network is phase synchronized), andgradually decreasing the coupling by dd at each step. In what follows,the two sets of numerical trials are termed forward and backward,respectively.

Figure 1 reports the results obtained by setting p(v) as a uniformfrequency distribution in the interval [0, 1]. Panels a and b of Fig. 1show S as a function of d. In particular, panel a (b) illustrates the caseof a fixed mean degree Ækæ 5 40 (of a fixed frequency gap c 5 0.4), andreports the results of the forward and backward simulations at dif-ferent values of c (Ækæ). A first important result is the evident first-order character acquired, in all cases, by the transition for sufficientlyhigh values of c.

A second relevant result is the spontaneous emergence of degree-frequency correlation features associated to the passage from a sec-ond- to a first-order phase transition. While such a correlation wasimposed ad hoc in Refs.13,14, here condition (C1) creates for eachoscillator i a frequency barrier around vi, where links are forbidden.

Figure 1 | Explosive transition to synchronization. Phase synchronization level S (see text for definition) vs. the coupling strength d, for different

values of the gap c at Ækæ 5 40 (panel a), and for different values of the average degree Ækæ at c 5 0.4 (panel b). In both panels, the continuous (dashed)

lines refer to the forward (backward) simulations. See the Methods section for the construction procedure of the networks. In panels c and d, the degree ki

that each node achieves after the network construction is completed (upper plots) and the average of the natural frequencies Ævjæ for j[N ið Þ(bottom plots) are reported vs. the node’s natural frequency vi, for Ækæ 5 100 and frequency gaps c 5 0.0 (panel c), and c 5 0.4 (panel d). The red solid line

in panel d is a sketch of the theoretical prediction f(v) (see text). Panels e and f show S (color coded according to the color bar) in the parameter space

(d, c) for (e) Ækæ 5 20 and (f) Ækæ 5 60. The horizontal dashed lines mark the separation between the region of the parameter space where a second-order

transition occurs (below the line) and that in which the transition is instead of the first order type (above the line). The yellow striped area delimits the

hysteresis region.

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75

The final degree ki is proportional to the total probability for thatoscillator to receive connections from other oscillators in the net-work, and therefore to 1

Ðvizcvic p v0ð Þdv0. This is shown in panels c

and d of Fig. 1, where the degree ki that each node achieves aftercompleting the FGC network construction is reported as a functionof its natural frequency vi, for Ækæ 5 100. Precisely, the upper plot ofpanel c refers to the case c 5 0 in which no degree-frequency cor-relation is present. In the upper plot of panel d, instead, we report thecase c 5 0.4 (a value for which a first-order phase transitionoccurs) and the (conveniently normalized) function f vð Þ~1Ð

vzcvc p v0ð Þdv0, with p(v) 5 1 for v g [0, 1], and p(v) 5 0 else-

where, which gives evidence of the emergence of a very pronouncedV-shape relationship between the frequency and the degree of thenetwork’s nodes. By further inspecting the average frequency of eachoscillator’s neighborhood, comparison between the lower plots ofpanels c and d manifests that condition (C1) leads to the emergenceof a bipartite-like network where low frequency oscillators are mainlycoupled to high frequency oscillators.

Finally, panels e and f of Fig. 1 report S in the d – c space, for Ækæ 5

20 and Ækæ 5 60, and show that the rise of a first order phase trans-ition is, indeed, a generic feature in the parameter space. The hori-zontal dashed lines in panels e and f mark the values of cc, separatingthe two regions where a second-order transition (below the line) anda first-order transition (above the line) occurs. Remarkably, whileincreasing Ækæ of the network facilitates the occurrence of the explos-ive transition, as both the values of c and d for which the first-orderphase transition takes place decrease, it also shrinks the width of thehysteresis (the yellow striped area in panels e and f). This is consistentwith the results of Ref.12 for an all-to-all connected case, where in thelimit N R ‘ a first-order phase transition has been predicted in theabsence of hysteresis. It is worth noticing that similar scenarios ofsynchronization were observed in the past for chains of periodic16

and chaotic phase coherent17 oscillators. In this latter cases, syn-chronization was shown to appear or vanish in two ways: a softtransition (without cluster formation) for chains with very smallfrequency mismatches between the elements, and a hard transitionfor rather long chains with relatively large frequency mismatches.

We verified that fulfillment of condition (C1) leads to a first-ordertransition for a very wide class of distributions of the oscillators’natural frequencies. The panels a and b of Fig. 2 show, for instance,

the case of the asymmetrical Rayleigh distribution, conveniently re-

scaled to the interval [0, 1], given by p vð Þ~ v

s2ev2

2s2 (see the inset in

panel a of Fig. 2), with s 5 0.25. The results highlight the presence ofan abrupt transition (Fig. 2, panel a), together with the emergence ofa clear frequency-degree correlation, well described, again, by thefunction f(v) (Fig. 2, panel b).

We now move to discussing several ways for even softening con-dition (C1), thus making it even more generally applicable, while stillkeeping the abrupt character of the transition. For the first extension,we consider again the case of a uniform frequency distribution in theinterval [0, 1]. In this case, condition (C1) can be relaxed as follows:

(C2) for each oscillator i, all nodes j belonging to N ið Þ satisfyjvi – Ævjæj . cc, where Æ…æ indicates the average value over theensemble N ið Þ.

The new condition is tantamount to softening condition (C1) tothe local mean field of frequency differences in the neighborhood ofeach network node. The panels c and d of Fig. 2 report the results ofnetworks obtained with a modified construction procedure, in whichpairs of randomly selected nodes are now linked if the value of jvi –Ævjæj (averaged over the set of nodes j already linked to node i, and theone candidate to be further linked), exceeds a gap c. Again, an explos-ive transition occurs (Fig. 2, panel c) in correspondence with theemergence of frequency-degree correlations (Fig. 2, panel d).

Furthermore, it is worth noticing that a strict application of con-dition (C1) for non uniform frequency distributions implies thatoscillators at different frequencies would in general have a differentnumber of available neighbors in the network. For this reason, anatural extension of condition (C1) is to consider a frequency-dependent gap c(v) defined by

Ðvzcvcp v0ð Þdv0~Z. Panels e and f

of Fig. 2 report the case of a Gaussian distribution limited to thefrequency range [0, 1], centered at v 5 0.5, and given by

p vð Þ~ 1

sffiffiffiffiffiffi2Np e

v0:5ð Þ22s2 , with s 5 0.13. The gap condition for the

construction of the network is now to fix the value of Z, and accept

the pairing of nodes when vivj

w 12

c við Þzc vj

. The generic

case is again that an explosive transition is obtained, with pro-

Figure 2 | Extension to different frequency distributions and network construction rules. (Top row) S vs. d resulting from the forward (continuous lines)

and backward (dashed lines) simulations of system (1) for different frequency distributions or network construction rules, and (bottom row) the

corresponding distribution of the final node degree ki vs. the corresponding oscillator’s natural frequency. (a)-(b) Rayleigh distribution for c 5 0.3. In panel

b, the red solid line depicts the theoretical prediction f(v); (c)-(d) uniform frequency distribution, but network constructed accordingly to the local mean

field condition (see text) for c 5 0, and c 5 0.4. In panel d c 5 0.4; (e)-(f) Gaussian distribution with Z 5 0.7. The insets in panels a and e report the

corresponding distribution p(v). See the text and the Methods section for the details on the specific construction procedure used in each case. In all cases, Ækæ5 60.




76

nounced frequency-degree correlation features, as long as p(v) issymmetrical.

An alternative way to explore the frontier between first- and sec-ond-order transitions is to start from the case of Ref.12 (an all to allnetwork configuration with evenly spaced frequencies of the oscilla-tors), and to study the robustness of explosiveness under removal oflinks by means of two pruning processes: a random pruning, wherelinks are randomly chosen and removed, and a preferential pruningthat removes links with a probability proportional to the inverse ofthe frequency difference of the connected nodes, so that close fre-quency oscillators are more probable to be disconnected (see theMethods section for details).

The results are shown in Fig. 3. For the random pruning process(panel a of Fig. 3), one can easily see that, as the connectivity of thenetwork becomes low enough (i.e. as more and more links are prunedin a way that Ækæ progressively decreases) the transition graduallypasses from first- to second-order. On the contrary, for the preferen-tial pruning process (panel b of Fig. 3), the system maintains theexplosive behavior due to the higher probability to uphold connec-tions associated to large frequency mismatches, although the size ofthe discontinuity decreases. Once again, inspection of the nodes’degree and of the mean frequencies of the nodes’ neighborhoods(panel c of Fig. 3) reveals the spontaneous emergence of correlationbetween topology and dynamics for the preferential pruning process.

DiscussionIn conclusion, we demonstrated the validity and generality of severalconditions for the occurrence of abrupt phase transitions in networksof phase oscillators. Our study generalizes all previous results, andextends the possibility of encountering first-order phase transitionsto a large variety of network topologies, as well as to a large varietyof frequency distributions of the oscillators. This suggests prac-tical methods for engineering networks able to display criticalphenomena, and the emergence of dynamical abrupt transitions intheir macroscopic states. Furthermore, the evidence for the emer-gence of frequency-degree correlations in connection with theseabrupt transition, may shed light on the mechanisms underlyingthe relationship between topology and dynamics in many real-worldsystems.

One interesting application is expressing magnetic-like states ofsynchronization in such an ensemble of networked oscillators, pro-vided that the coupling strength is set inside the hysteresis region ofthe first-order phase transition. For this purpose, one can againconsider the case of an initial uniform distribution of the oscilla-

tors’ frequencies, and modify system (1) as follows:dwi

dt~viz

Dp sin wpwi

zd

PNi~1 aij sin wjwi

, where Dp is the strength

of a unidirectional connection to an external pacemaker (equal for alloscillators in the network), and wp is the phase of the pacemaker

Figure 3 | Explosive synchronization in networks with evenly spaced natural frequencies. S vs. d resulting from the forward (continuous lines) and

backward (dashed lines) simulations of system (1), for different values of the average degree Ækæ (see legend in panel a) after performing a random

pruning (a) and a preferential pruning (b) of links in an all-to-all connected network (see the text and the Methods section for the networks

construction). Natural frequencies are evenly spaced in the interval [0, 1]. In panel c, the degree ki that each node achieves after the network pruning is

completed (upper plots) and the average of the local natural frequency Ævjæ for j[N ið Þ (bottom plots) are reported vs. the node’s natural frequency vi for

the preferential pruning process with Ækæ 5 10.

Figure 4 | Magnetic-like states of synchronization. (a) Time evolution of the parameter r(t) under a pacemaker forcing for a value d 5 0.004 outside the

hysteresis region (upper plot), and a value d 5 0.009 within the hysteresis region (bottom plot). vp 5 1.0 and Dp 5 0.0005 (bottom red line), Dp 5 0.005

(middle blue line), and Dp 5 0.02 (top black line). The pacemaker is active from t 5 50 to t 5 350, as marked by the vertical dashed lines. (b) Colormap of S 5

Ær(t)æt.350 (coded as indicated in the color bar), showing the region of the parameter space Dp-vp where the magnetic-like state of synchronization is

maintained after removal of the pacemaker. The initial frequencies of the oscillators are taken from a uniform distribution in the interval [0, 1]. c 5 0.49 and

Ækæ 5 40.




77

obeyingdwp

dt~vp. Initially, the system is let to freely evolve into the

unsynchronized regime, with Dp 5 0. The pacemaker is thenswitched on, and Dp is selected so that all oscillators are entrainedto the pacemaker phase. At a subsequent time, the pacemaker isswitched off again, and the state of the system is monitored. Theresults are illustrated in Fig. 4. Precisely, panel a shows that settingd outside (inside) the hysteresis region produces a final state thatrelaxes to the original unsynchronized behavior (that stays perma-nently in a synchronized configuration for sufficiently large Dp

values). Panel b depicts the regions of the parameter space Dp – vp

for which these magnetic-like states can ultimately be produced.

MethodsAlgorithms for the networks’ construction. Frequency Gap-conditioned (FGC)random networks. The considered networks result from the following procedure:i) we assign natural frequencies vi, drawn from a distribution p(v), to the Noscillators; ii) we randomly pick a pair (i, j) of oscillators, and form a link betweenthem only if the value of jvi – vjj exceeds a given gap c; iii) we repeat point ii) until thedesired number of links L in the graph is formed. After a final check on theconnectedness of the resulting network, the procedure yields Erdos-Renyi-like

topologies with an average degree kh i: 2LN

.

Random and preferential pruning. The considered networks result from thefollowing procedure: i) we start from an all-to-all network configuration, and assignto the N oscillators natural frequencies vi evenly spaced spanning the interval [0, 1],

i.e. vi~1

N1i1ð Þ; ii) from the total number of links

N2

N1ð Þ, N2

N1 kh ið Þlinks are pruned to produce a network with desired mean degree Ækæ. The linksremoval can either be performed randomly, or selectively by assigning to each link a

pruning probability pij~1

vivj

.

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3. Li, G., Braunstein, L. A., Buldyrev, S. V., Havlin, S. & Stanley, H. E. Transport andpercolation theory in weighted networks. Phys. Rev. E 75, 045103 (2007).

4. Boccaletti, S., Kurths, J., Osipov, G., Valladares, D. L. & Zhou, C. S. Thesynchronization of chaotic systems. Phys. Rep. 366, 1 (2002).

5. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D. U. Complexnetworks: Structure and dynamics. Phys. Rep. 424, 175 (2006).

6. Arenas, A., Dıaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. S. Synchronizationin complex networks. Phys. Rep. 469, 93 (2008).

7. Achlioptas, D., D’Souza, R. M. & Spencer, J. Explosive percolation in randomnetworks. Science 323, 1453 (2009).

8. Cho, Y. S., Kim, J. S., Park, J., Kahng, B. & Kim, D. Percolation transitions inscale-free networks under the Achlioptas process. Phys. Rev. Lett. 103, 135702(2009).

9. Radicchi, F. & Fortunato, S. Explosive percolation in scale-free networks. Phys.Rev. Lett. 103, 168701 (2009).

10. Grassberger, P., Christensen, C., Bizhani, G., Son, S.-W. & Paczuski, M. Explosivepercolation is continuous, but with unusual finite size behavior. Phys. Rev. Lett.106, 225701 (2011).

11. Kuramoto, Y. Chemical oscillations, waves and turbulence (Springer, 1984).12. Pazo, D. Thermodynamic limit of the first-order phase transition in the Kuramoto

model. Phys. Rev. E 72, 046211 (2005).13. Gomez-Gardenes, J., Gomez, S., Arenas, A. & Moreno, Y. Explosive

synchronization transitions in scale-free networks. Phys. Rev. Lett. 106, 128701(2011).

14. Leyva, I., Sevilla-Escoboza, R., Buldu, J. M., Sendina-Nadal, I.,Gomez-Gardenes, J., Arenas, A. et al. Explosive first-order transition tosynchrony in networked chaotic oscillators. Phys. Rev. Lett. 108, 168702 (2012).

15. Gomez-Gardenes, J., Moreno, Y. & Arenas, A. Paths to synchronization oncomplex networks. Phys. Rev. Lett. 98, 034101 (2007).

16. Osipov, G. V. & Sushchik, M. M. Synchronized clusters and multistability inarrays of oscillators with different natural frequencies. Phys. Rev. E 58, 7198(1998).

17. Osipov, G., Pikovsky, A., Rosenblum, M. & Kurths, J. Phase synchronizationeffects in a lattice of nonidentical Rssler oscillators. Phys. Rev. E 55, 2353 (1997).

AcknowledgmentsWork supported by Ministerio de Educacion y Ciencia, Spain, through grantsFIS2009-07072 and from the BBVA-Foundation within the Isaac-Peral program of Chairs.Authors acknowledge also the R&D Program MODELICO-CM [S2009ESP-1691], and theusage of the resources, technical expertise and assistance provided by supercomputingfacility CRESCO of ENEA in Portici (Italy).

Author ContributionsIL, AN, ISN and SB devised the model and designed the study. IL and ISN carried out thenumerical simulations. IL, ISN, JAA, JMB, DP, and MZ analyzed the data and prepared thefigures. AN, MZ, DP, and SB wrote the main text of the manuscript.

Additional informationCompeting financial interests: The authors declare no competing financial interests.

License: This work is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivs 3.0 Unported License. To view a copy of thislicense, visit http://creativecommons.org/licenses/by-nc-nd/3.0/

How to cite this article: Leyva, I. et al. Explosive transitions to synchronization in networksof phase oscillators. Sci. Rep. 3, 1281; DOI:10.1038/srep01281 (2013).




78


Explosive synchronization in weighted complex networks

I. Leyva,1,2 I. Sendina-Nadal,1,2 J. A. Almendral,1,2 A. Navas,2 S. Olmi,3,4 and S. Boccaletti31Complex Systems Group, Univ. Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

2Center for Biomedical Technology, Univ. Politecnica de Madrid, 28223 Pozuelo de Alarcon, Madrid, Spain3CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy

4INFN - Sezione di Firenze and CSDC, via Sansone 1, 50019 Sesto Fiorentino, Italy(Received 25 July 2013; published 14 October 2013)

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subjectof the utmost interest. Given a set of phase oscillators networking with a generic wiring of connections anddisplaying a generic frequency distribution, we show how combining dynamical local information on frequencymismatches and global information on the graph topology suggests a judicious and yet practical weightingprocedure which is able to induce and enhance explosive, irreversible, transitions to synchronization. We reportextensive numerical and analytical evidence of the validity and scalability of such a procedure for differentinitial frequency distributions, for both homogeneous and heterogeneous networks, as well as for both linear andnonlinear weighting functions. We furthermore report on the possibility of parametrically controlling the widthand extent of the hysteretic region of coexistence of the unsynchronized and synchronized states.

DOI: 10.1103/PhysRevE.88.042808 PACS number(s): 89.75.Hc, 89.75.Kd, 89.75.Da, 64.60.an

I. INTRODUCTION

One of the most significant challenges of present-dayresearch is bringing to light the processes underlying the spon-taneous organization of networked dynamical units. When anetwork passes from one to another collective phase under theaction of a control parameter, the nature of the associated phasetransition is disclosed by the behavior of the order parameterat criticality. In complex networks’ theory [1,2] such phasetransitions have been observed in the way a graph collectivelyorganizes its architecture through percolation [3–5], and itsdynamical state through synchronization, both for continuous[6,7] and discrete [8] dynamical systems.

Abrupt transitions to synchronized states of networkedphase oscillators were initially reported in a Kuramotomodel [9] for a particular realization of a uniform frequencydistribution (evenly spaced frequencies) and an all-to-allnetwork topology [10]. Later on, the same finding was alsodescribed for both periodic [11] and chaotic [12] phaseoscillators in the yet particular condition of a heterogeneousdegree distribution with positive correlations between the nodedegree and the corresponding oscillator’s natural frequency.Recently, Ref. [13] introduced a more general frameworkwhere explosive synchronization (ES) is obtained in weightednetworks, where weights are selected to be proportional to theabsolute value of the frequency of the oscillators in a way thatproduces positive correlations between the node input and thefrequency of the oscillator.

The weighting procedure proposed in Ref. [13] inherentlyasymmetrizes each link of the network, favoring the interactiondirections from lower to higher frequencies. In this work, wepropose an alternative general framework for ES in complexnetworks, based on a weighting procedure which instead keepsthe symmetric nature of the links. The method is inspired byour recent study of Ref. [14], where it is shown that ES canbe obtained for any given frequency distribution, providedthe connection network is constructed following a rule offrequency disassortativity; that is, that the synchronizationclustering formation is prevented avoiding close frequencies

to couple, in a network generation scheme ruled by dynamicalproperties, as the Achlioptas rule [4] works for the structuralcase in explosive percolation.

We here deal with the more general case of a networkwith given frequency distribution and architecture, and weshow that a weighting procedure on the existing links, whichcombines information on the frequency mismatch of the twoend oscillators of a link with that of the link betweenness,has the effect of inducing or enhancing ES phenomena forboth homogeneous and heterogeneous graph topologies, aswell as any symmetric or asymmetric frequency distribution.In addition, we show the general scaling properties of theobtained transition and provide analytical arguments in supportof our claims.

II. MODEL AND NUMERICAL RESULTS

Without lack of generality, our reference is a network G ofN Kuramoto [9] phase oscillators, described by

dθi

dt= ωi + σ

〈k〉N∑

i=1

αij sin(θj − θi), (1)

where θi is the phase of the ith oscillator (i = 1, . . . ,N), ωi

is its associated natural frequency drawn from a frequencydistribution g(ω), σ is the coupling strength, 〈k〉 is the graphaverage connectivity (〈k〉 ≡ 2L

N, with L being the total number

of links), and

αij = aij |ωi − ωj |α (2)

is the weighted link for nodes i,j , with aij being the elementsof the adjacency matrix that uniquely defines G and α beinga constant parameter which eventually modulates the weight.The strength of the ith node (the sum of all its links weights)is then si = ∑

j αij . The classical order parameter for system

(1) is r(t) = 1N

| ∑Nj=1 eiθj (t)|, and the level of synchronization

can be monitored by looking at the value of R = 〈r(t)〉T , with〈· · ·〉T denoting a time average over a conveniently large timespan T .

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I. LEYVA et al. PHYSICAL REVIEW E 88, 042808 (2013)

As the coupling strength σ increases, system (1) undergoesa phase transition at a critical value σc from the unsynchronized(R ∼ 1/

√N ) to the synchronous (R = 1) state, where all

oscillators ultimately acquire the same frequency. In thefollowing, we will describe the nature of such a transitionas a function of the rescaled order parameter σ/〈k〉. As forthe stipulations followed in our simulations, the state of thenetwork is monitored by gradually increasing σ in stepsδσ = 0.0005, starting at σ = 0. Whenever a step δσ is made,a long transient (200 time units) is discarded before the dataare recorded and processed. Moreover, as we are focusingon abrupt, irreversible transitions (and thus on expectedassociated hysteretic phenomena), we perform the simulationsalso in the reverse way, i.e., starting from a given value σmax

(where R = 1), and gradually decreasing the coupling by δσ

at each step. In what follows, the two sets of numerical trialsare termed forward and backward, respectively.

A. Homogeneous networks

We first report our results for the case of homogeneousgraph topologies. For this purpose, we consider Erdos-Renyi(ER) random networks [15] of size N , and we describe howan explosive transition is induced for sufficiently large valuesof 〈k〉 and irrespectively on the specific frequency distributiong(ω). Figure 1(a) reports the results for N = 500 and severalfrequency distributions g(ω) within the range [0,1]. For thesimplest case of uniform frequency distribution g(ω) = 1,

0 0.5 1 1.5 2 2.5 30

0.5

1

σ/ k

R

(a)α = 0 – Uniform

α = 1 – Uniform

α = 1 – Gauss

α = 1 – Bimodal

α = 1 – Rayleigh

α = 1 – Semi-Gauss

0 0.2 0.4 0.6 0.8 10

20

40

ωi

si

(b)

FIG. 1. (Color online) (a) Synchronization transitions for N =500 ER networks, 〈k〉 = 30, for unweighted case (α = 0) (bluesquares), and linearly weighted cases (α = 1) with several frequencydistributions within the range [0,1]: uniform, Gaussian, Gaussian-derived, Rayleigh, and semi-Gaussian. Solid and dashed lines referto the forward and backward simulations, respectively. (b) Nodestrengths si (see text for definition) vs natural frequencies ωi , forthe unweighted (dark blue squares) and weighted (light blue circles)networks reported in panel (a). Solid line is proportional to theanalytical prediction (ω − a

2 )2 + 14a

in the thermodynamical limit ofour model, with a = 1 the width of the uniform frequency distribution(see text for more details).

while the unweighted network [α = 0 in Eq. (2)] displays asmooth, second-order-like transition to synchronization [darkblue curve in Fig. 1(a)], the effect of a linear weighting (α = 1)is that of inducing a sharp transition in the system, with anassociated hysteresis in the forward (solid line) and backward(dashed line) simulations. This drastic change in the natureof the transition is independent of the frequency distributiong(ω), as long as they are defined in the same frequency range[0,1] as shown in Fig. 1(a). The results are identical forsymmetric distributions (homogeneous, Gaussian, a bimodaldistribution derived from a Gaussian) and for asymmetricfrequency distributions (Rayleigh, a Gaussian centered at 0 butjust using the positive half). See details of the used frequencydistributions in Ref. [16].

Figure 1(b) accounts for the existence of a parabolicrelationship between the strengths and the natural frequenciesof the oscillators associated with the passage from a smoothto an explosive phase transition. This relationship has beenobtained analytically [see Eq.(7)] in the thermodynamical limitof the Kuramoto model and perfectly fits the numerical resultsshown as a solid line in Fig. 1(b). It has to be remarkedthat, while in Ref. [11] degree-frequency correlation featureswere imposed to determine explosiveness in the transition tosynchronization, here the effect of the weighting is to letthese correlation features between topology and dynamicsspontaneously emerge, with the result of shaping a bipartite-like network where low- and high-frequency oscillators are theones with maximal overall strength.

Further information about the nature and scaling propertiesof the transition induced by the linear weighting procedureis gained from Fig. 2, where it is shown the dependence ofthe scaled critical coupling σc/〈k〉 on the average connectivity〈k〉 and on the network size N . Precisely, Fig. 2(a) showsthat, independently of N , a dynamical bifurcation exists at〈k〉 ∼ 17, corresponding to the passage from a second- toa first-order-like phase transition. For the latter regime, thetwo branches expanding from 〈k〉 17 are associated withthe hysteresis in the forward and backward simulations. Therelative independence on N can be explained consideringthat an important condition for ES to occur is that eachnode neighborhood must represent a statistically significantsample of the network frequencies up to give a close enoughapproximation to the global mean frequency, and therefore thesynchronization frequency. To reach this target, the requiredsampling size n for a given population size N is usuallycalculated with the following formula [17]:

n = N

1 + C2(N − 1),

where C := 2e/zα/2, e is the error allowed, 1 − α is theconfidence level, and zα/2 is the upper α/2 percentagepoint of the standard normal distribution. Aside from thetechnical details, the important feature in the expression isthat the sampling size converges to a finite value, even foran infinite population. This is exactly what Fig. 2(a) shows.Once the mean degree is large enough, each node has aneighborhood assuring that its neighbor frequency averageis statistically accurate. Precisely, Fig. 2(a) suggests thatC ≈ 0.24, indicating that, for mean degrees greater than ∼17,each node has a sufficiently large neighborhood independently

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EXPLOSIVE SYNCHRONIZATION IN WEIGHTED COMPLEX . . . PHYSICAL REVIEW E 88, 042808 (2013)

10 20 30 40 501.2

1.3

1.4

k

c/k

(a) N=200

N=500N=1000

0 500 1000 1500 2000

1.2

1.4

1.6

N

c/k

(b)

ForwardBackward

σσ

FIG. 2. (Color online) Critical scaled couplings σc/〈k〉 at theonset of synchronization (forward simulation) and desynchronization(backward simulation) using a linear weighting procedure ij (α =1) as a function of (a) 〈k〉 for several ER network sizes N , andof (b) N in all-to-all coupled networks. In panel (a) the verticaldashed line marks the passage from a smooth to an explosive phasetransition. In both panels (a) and (b) the upper and lower branchescorrespond to forward and backward simulations, respectively. Eachdot accounts for an average of at least 20 independent runs of uniformfrequency distributions. Horizontal dashed lines in panel (b) are closeto the analytical values defining the range of the hysteresis in thethermodynamical limit for the Kuramoto model (see explanation intext). Frequencies are uniformly distributed in the range [0,1].

of the population size N . Figure 2(b) shows how the scaledcritical couplings defining the hysteresis of the ES transitionconverge to constant values for the Kuramoto model (all-to-allcoupling) when N increases which are quite close to thoseobtained in the thermodynamical limit of the Kuramoto modeldiscussed in the analytical section.

Furthermore, the weighting procedure inducing ES is quitegeneral, as a large family of detuning dependent functions canbe used. As an example, Fig. 3 describes the case of nonlinearweighting procedures; that is, α = 1 in Eq. (2). There, we setagain N = 500 and 〈k〉 = 30 and consider both ER graphs[Fig. 3(a)] and a regular random network [Fig. 3(b)], i.e.,a network where each node has exactly the same numberof connections (ki = 〈k〉 = 30) with the rest of the graph.This latter case has been obtained by a simple configurationmodel [18], imposing a δ-Dirac degree distribution. The resultsin Fig. 3 show that the generic nonlinear function of thefrequency mismatch given by Eq. (2) is able to induce ESin both topologies, and that the effect of a superlinear (α > 1)weighting [sublinear (α < 1) weighting] is that of enhancing(reducing) the width of the hysteretic region.

B. Heterogeneous networks

So far, we have considered only homogeneous degreedistributions. In order to properly describe the passage from ahomogeneous to a heterogeneous degree distribution, we rely

0 1 2 3 4 50

0.5

1(a)

σ/ k

R

0 1 2 3 4 5 6 70

0.5

1

σ/ k

R

(b)

α=0.1α=0.5α=1α=2α=3

FIG. 3. (Color online) Synchronization transitions for ER net-works, N = 500, uniformly distributed frequencies in the [0,1] range,and nonlinear weighting functions α

ij . Both plots consider severalα values, from sublinear to superlinear weighting [see legend inpanel (b)]. (a) ER networks, 〈k〉 = 30, (b) regular random networks,k = 30. In all cases, forward and backward simulations correspondrespectively to solid and dashed lines.

on the procedure introduced in Ref. [19]. Such a technique,indeed, allows constructing graphs with the same averageconnectivity 〈k〉 and grants one the option of continuouslyinterpolating from ER to scale-free (SF) networks [20], bytuning a single parameter 0 p 1. With this method,networks are grown from an initial small clique by sequentiallyadding nodes up to the desired graph size. Each newly addednode has a probability p of forming random connectionswith already existing vertices, and a probability 1 − p offollowing a preferential attachment rule [20] for the selectionof its connections. As a result, the limit p = 1 induces an ERconfiguration, whereas the limit p = 0 corresponds to a SFnetwork with degree distribution P (k) ∼ k−3.

Let us set N = 1000 and 〈k〉 = 30 and, after the networkconstruction, let us randomly distribute the oscillators’ fre-quencies in the interval [0,1] and use again a linear weightingfunction ij = aij |ωi − ωj |. The comparative results arereported in Fig. 4 for different networks ranging from a com-pletely regular random network to networks with increasingdegree of heterogeneity (check the degree distributions in theinset of Fig. 4) and it is easy to see that heterogeneity inthe degree distribution actually opposes the onset of explo-sive synchronization. Actually, the latter is only suppressedwhen the second moment of the degree distribution diverges(SF). A similar qualitative scenario (not shown) is obtainedalso for different frequency distributions, network sizes, and(superlinear or sublinear) weighting functions, allowing oneto conclude that heterogeneous degree distributions requirea different weighting approach, where the information onfrequency mismatch has to be properly combined with localor global information on the network topology.

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0 1 2 30

0.2

0.4

0.6

0.8

1

σ/ k

R

RRp=1. (ER)p=0.2p=0. (SF)

30 100100

101

102

k

P(k)

FIG. 4. (Color online) Explosive synchronization vs degree ofheterogeneity. Synchronization transitions as a function of thecoupling strength for linearly weighted networks (α = 1) with thesame average connectivity 〈k〉 = 30 and different networks: a regularrandom (RR) network with homogeneous degree (blue circles),and networks constructed using the model referenced in the text,p = 1 (ER, red squares), p = 0.2 (green triangles), and p = 0 (SF,black diamonds). In all cases, forward and backward simulationscorrespond respectively to solid and dashed lines. Inset shows log-logplot of the four corresponding degree distributions.

The problem closely resembles what was called, in pastyears, the paradox of heterogeneity [21] where increasingthe heterogeneity in the connectivity distribution of an un-weighted network led to an overall deterioration of synchrony,despite the associated reduction of the network’s shortestpath. That paradox was lately solved by proving optimalsynchronization conditions when proper weighting proceduresare implemented on the graph’s links accounting for eitherlocal [22] or global [23] information on the specific networktopology. Therefore, in analogy with what was reported inRef. [23], we consider a new weighting function

ij = aij |ωi − ωj |

β

ij∑j∈Ni

β

ij

, (3)

with β being a parameter and ij being the edge betweennessassociated with the link aij [24], defined as the number ofshortest paths between pairs of nodes in the network that runthrough that edge.

The results are reported in Fig. 5(a). While the caseβ = 0 (black triangles, already shown in Fig. 4) correspondsto a smooth transition, the effect for β = 0 in Eq. (3) ishighly nontrivial. Precisely, moderate (positive or negative)values of β establish in system (1) an abrupt transition tosynchronization. However, increasing β beyond a critical valueleads system (1) to display again a smooth and reversiblecharacter of the transition.

In its turn, Fig. 5(b) reports the hysteresis’ area [the area ofthe plane (R,σ/〈k〉) covered by the hysteretic region] as a func-tion of β, obtained by an ensemble average over 10 differentforward and backward simulations of system (1) together withthe weighting function (3), each one starting from a differentrealization of the uniform frequency distribution. The plotreveals the existence of an optimal condition around β = 0.5where the width of the hysteresis is maximized. Therefore, β

can be seen as an operational parameter through which one cancontrol and regulate the width and extent in σ of the hysteresis

1 1.2 1.4 1.6 1.80

0.5

1

σ

R

(a)

β =-2.0β =0.5β =2.0β =3.0α = 1

−2 −1 0 1 2 3 40

0.05

0.1

0.15

0.2

β

hyst

are

a

(b)

FIG. 5. (Color online) (a) Synchronization transitions for SFnetworks using different schemes of coupling weighting. In blacktriangles, the link between nodes i and j is weighted using α

ij

with α = 1 (as in Fig. 4), while the rest of cases refer to theweighting function ij of Eq. (3), with the values of the β parametergiven in the legend. In all cases, forward and backward simulationscorrespond, respectively, to continuous and dashed lines. (b) Area ofthe hysteretic region vs β. Each point is an average of 10 differentsimulations, each one starting from a different realization of thefrequency distributions. In all cases, 〈k〉 = 30, N = 1000, and naturalfrequencies are uniformly distributed in the interval [0,1].

associated with the irreversible nature of ES. The latter canbe of interest for controlling the range of coupling strengthfor which system (1) can be used to originate magnetic-likestates of synchronization, i.e., situations in which an originallyunsynchronized configuration, once entrained to a given phaseby an external pacemaker acting for a limited time lapse, isable to permanently stay in a synchronized configuration [14].

III. ANALYTICAL RESULTS

In order to study the onset and nature of the explosivetransition, we must analytically examine the behavior ofthe system in the thermodynamic limit. Let us considerthe paradigmatic case in which N oscillators form a fullyconnected graph, as the original Kuramoto model, but withweights ij = |ωi − ωj |. Then, the dynamical equations are

θi = ωi + σ

N

N∑j=1

ij sin(θj − θi),

for i = 1, . . . ,N .By considering the following definitions:

1

N

N∑j=1

ij sin θj := Ai sin φi,

1

N

N∑j=1

ij cos θj := Ai cos φi,

042808-4


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the dynamical equations are usually expressed [25] in termsof trigonometric functions as

θi = ωi + σAi sin(φi − θi).

While these transformations are the same as those usedin the original Kuramoto model, now there is an explicitdependence on i in the quantities Ai and φi . In order to continueour analysis, we will then assume some mild approximations.

In the corotating frame, the phases must verify ωi =σAi sin(θi − φi) to have a static solution (i.e., θi = 0), whichin the thermodynamic limit reads

ω = σAω sin(θω − φω). (4)

The definition of Aω and φω implies that

F (ω) := Aω sin φω =∫

g(x)|w − x| sin θ (x)dx,

whose second derivative verifies

F ′′(ω) =∫

g(x)2δ(w − x) sin θ (x)dx = 2g(ω) sin θ(ω),

using the distributional derivative of the signum function.Likewise, if we consider

G(ω) := Aω cos φω =∫

g(x)|w − x| cos θ (x)dx,

its second derivative verifies

G′′(ω) = 2g(ω) cos θ (ω).

Then, Eq. (4) takes the form

2

σg(ω)ω = F ′′(ω)G(ω) − F (ω)G′′(ω). (5)

Let us work out F (ω) and G(ω). When all oscillators areclose to synchronization, we can assume that cos θ (x) ≈ R,thus

G(ω) ≈ R

∫g(x)|w − x|dx = Rs(ω),

where s(ω) is just the strength of a node with intrinsicfrequency ω. Therefore, Eq. (5) can be approximated by

2

Rσg(ω)ω = F ′′(ω)s(ω) − F (ω)s ′′(ω), (6)

which is a second-order ordinary differential equation (ODE)whose integration yields F (ω). Notice that when s(ω) is arather involved function, Eq. (6) is already an approximation,and we can just consider a polynomial expansion in ω to obtainan analytical expression of F (ω).

For instance, given a uniform distribution g(ω) in theinterval [−a/2, + a/2], the resulting strength is a second-order polynomial,

s(ω) = a

[(ω

a

)2

+ 1

4

], (7)

which perfectly fits our numerical simulations [see Fig. 1(b)],even though it has been deduced for a complete graph. Then,

the integration of Eq. (6) results in

F (ω) = a

[1 + 4

(ωa

)2]arctan

(2wa

) − (2 + π )wa

(4 + π )σR,

using the initial condition F (0) = 0, since g(ω) is a symmetricfunction [thus F (ω) is an odd function], and the consistencyequation

F (ω) =∫

g(x)|ω − x| sin θ (x)dx =∫ |ω − x|

2F ′′(x)dx.

Therefore, since F ′′(ω) = 2g(ω) sin θ(ω), we find that

sin θ (ω) = 1

σRH

(2ω

a

),

where

H (z) := 4

4 + π

[z

1 + z2+ arctan(z)

].

To determine how the order parameter R depends on thecoupling constant σ , we use

R =∫

g(x) cos θ (x)dx =∫

g(x)√

1 − sin2 θ (x)dx, (8)

which is an implicit equation in R. When σR 2+π4+π

≈ 0.72,sin θ (x) 1 for all x, which means that all oscillators arefrequency locked, so

R =∫ a

2

− a2

g(x)

√1 −

[1

σRH

(2x

a

)]2

dx.

When σR 2+π4+π

, only those oscillators with frequency in theinterval [−ω∗,ω∗] are locked, where

ω∗ := a

2H−1(σR),

thus

R =∫ a

2 H−1(σR)

− a2 H−1(σR)

g(x)

√1 −

[1

σRH

(2x

a

)]2

dx.

Hence, if we define

(μ) :=

1 if μ 2+π4+π

H−1(μ) if 0 μ < 2+π4+π

and

I (μ) :=∫ (μ)

0

√1 −

[1

μH (z)

]2

dz,

Eq. (8) takes the formμ

σ= I (μ), (9)

where μ = σR. Therefore, given a coupling constant σ , thevalue of R is computed by solving this implicit equation in μ.Notice that, geometrically, the solutions are the points wherethe straight line passing through the origin with slope 1/σ

intersects I (μ).The main feature characterizing I (μ) is its inflection point

at 2+π4+π

, at which the curve changes from being concave upto concave down [see Fig. (6)]. This implies that, depending

042808-5


83


0 0.5 1 1.50

0.5

1

μ

I(μ

)

(a)

I(μ)

μ/σc1

μ/σc2

0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1(b)

R

σ

FIG. 6. (Color online) (a) I as a function of μ = σR (solid curve)as given by Eq. (9). The dashed lines are those straight lines whoseintersection with I marks the backward (σ c

1 ) and forward (σ c2 ) critical

points of ES for an all-to-all connected network and for a uniformfrequency distribution. (b) The corresponding synchronization orderparameter R as a function of the coupling strength. Solid (dashed)curves correspond to the stable (unstable) solution. Dotted verticallines mark the region of hysteresis defined by σ c

1 and σ c2 in panel (a).

on σ , there are three qualitatively different type of solutions.When σ is small, we have the trivial solution R = 0 since thestraight line and I (μ) only intersect at μ = 0. This situationchanges when σ is such that the slope of the straight line istangent to I (μ) [i.e., when σ = 1.03, corresponding to thered dashed line in Fig. 6(a)]. When σ is greater than thisvalue, we enter into the region where the hysteresis takes placesince, now, there are three values of R, two of them are stablesolutions (R = 0 and R ≈ 1) and the third one is a unstablesolution [see Fig. 6(b)]. The solution R ≈ 1 appears thereforeabruptly, due to the existence of the inflection point. Thisbehavior changes when the slope of the straight line is tangentto I (0) [i.e., when σ = 1.43, corresponding to the blue dashedline in Fig. 6(a)], which is the point where the stable solution

R = 0 collapses with the unstable one, becoming unstable [seeFig. 6(b)]. Notice that the numerical values obtained for theKuramoto model for large N in Fig. 2(b) are quite close tothose predicted by the theory.

IV. CONCLUSIONS

In conclusion, we have introduced a weighting procedurebased on the link frequency mismatch and on the link be-tweenness to induce an explosive transition to synchronizationin a generic complex network of phase oscillators and for ageneric distribution of the frequencies. As a consequence ofthis procedure, topological and dynamical correlation featuresspontaneously emerge, with the result of shaping a bipartite-like network where frequency disassortativity prevails.

In this scenario, the passage from a smooth to an abrupttransition has found to be fully rescalable, and criticallydepends only on the average connectivity and not on thenetwork size.

In addition, we analytically proved that our weightingprocedure yields a first-order-like transition whose hysteresisextent is calculated. Moreover, the theoretical framework al-lows for a geometrical interpretation of the explosive transitionin which the weighting imposes a multivalued Kuramoto phaseorder parameter, in contrast with the classical model.

The present results could provide significant insights intothe study of real complex networks such as power gridswhich can be modeled as networks of phase oscillators whosecoupling may depend on the dynamics of the nodes [26].

ACKNOWLEDGMENTS

The authors acknowledge Alessandro Torcini for manyfruitful discussion on the subject and the computationalresources and assistance provided by CRESCO, the cen-ter of ENEA in Portici, Italy. Financial support from theSpanish Ministerio de Ciencia e Innovacion (Spain) underprojects FIS2011-25167, FIS2009-07072, and of Comunidadde Madrid (Spain) under project MODELICO-CM S2009ESP-1691, are also acknowledged. S. O. acknowledges the MIURproject CRISISLAB PNR 2011-2013.

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[15] P. Erdos and A. Renyi, Publ. Math. Debrecen 6, 290 (1959).[16] The details of the used distributions are: (i) Gaussian g(ω) =

(1/a√

2π )e− (ω−0.5)2

2a2 with a = 0.23, (ii) a bimodal distribution de-

rived from a Gaussian g(ω) = (1/a√

2π )e− ω2

2a2 if ω < 0.5, and

g(ω) = (1/a√

2π)e− (ω−1.0)2

2a2 , (iii) Rayleigh distribution g(ω) =(ω/b2)e− ω2

2b2 with b = 10 and normalized to the [0, 1] range, (iv)distribution derived from a Gaussian centered at 0 but just using

the positive half g(ω) = (1/a√

2π )e− (ω−0.5)2

2a2 with ω > 0, namedas semi-Gaussian in Fig. 1(a).

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Effects of degree correlations on the explosive synchronization of scale-free networks

I. Sendina-Nadal,1,2,* I. Leyva,1,2 A. Navas,2 J. A. Villacorta-Atienza,2 J. A. Almendral,1,2 Z. Wang,3,4 and S. Boccaletti5,6

1Complex Systems Group, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain2Center for Biomedical Technology, Universidad Politecnica de Madrid, 28223 Pozuelo de Alarcon, Madrid, Spain

3Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China4Center for Nonlinear Studies, Beijing–Hong Kong–Singapore Joint Center for Nonlinear and Complex Systems (Hong Kong)and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

5CNR–Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy6Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv, Israel

(Received 10 August 2014; revised manuscript received 27 January 2015; published 26 March 2015)

We study the organization of finite-size, large ensembles of phase oscillators networking via scale-freetopologies in the presence of a positive correlation between the oscillators’ natural frequencies and thenetwork’s degrees. Under those circumstances, abrupt transitions to synchronization are known to occurin growing scale-free networks, while the transition has a completely different nature for static randomconfigurations preserving the same structure-dynamics correlation. We show that the further presence ofdegree-degree correlations in the network structure has important consequences on the nature of the phasetransition characterizing the passage from the phase-incoherent to the phase-coherent network state. While highlevels of positive and negative mixing consistently induce a second-order phase transition, moderate values ofassortative mixing, such as those ubiquitously characterizing social networks in the real world, greatly enhancethe irreversible nature of explosive synchronization in scale-free networks. The latter effect corresponds to amaximization of the area and of the width of the hysteretic loop that differentiates the forward and backwardtransitions to synchronization.

DOI: 10.1103/PhysRevE.91.032811 PACS number(s): 89.75.Hc, 89.75.Kd, 89.75.Da, 64.60.an

I. INTRODUCTION

During the last 15 years, network theory has successfullyportrayed the interaction among the constituents of a varietyof natural and man-made systems [1,2]. It was shown thatthe complexity of most real-world networks (RWNs) canbe reproduced, in fact, by a growth process that eventuallyshapes a highly heterogeneous [scale-free (SF)] topology in thegraph’s connectivity pattern [3]. Furthermore, such a SF degreedistribution affects, in its turn, in a non-negligible way almostall the dynamical processes taking place over RWNs [1].

Actually, and distinct from the degree distribution, manyother important properties account for the fine details of thestructure of any RWN, mostly due to particular forms ofcorrelation (or mixing) among the network vertices [4]. Thesimplest case is the degree correlation [5], in which the networkconstituents tend to interact according to their respective de-grees. Remarkably, nontrivial forms of degree correlation havebeen (experimentally and ubiquitously) detected in RWNs,with social networks displaying typically an assortative mixing(i.e., a situation in which each network’s unit is more likely toconnect to other nodes with approximately the same degree),while technological and biological networks exhibiting adisassortative mixing (which takes place when connectionsare more frequent between vertices of fairly different degrees).Both the assortative and disassortative mixing properties areknown to considerably affect the organization of the networkinto collective dynamics, such as synchronization [6–8],epidemic spreading [9], and network controllability [10].

*Corresponding author: [email protected]

Possibly the most studied emerging collective dynamicsin SF networks is synchronization [1,11], as such a stateplays a crucial role in many relevant phenomena like, forinstance, the emergence of coherent global behaviors in bothnormal and abnormal brain functions [12], the food webdynamics in ecological systems [13], or the stable operation ofelectric power grids [14–16]. In particular, it has been recentlyshown that the transition to the graph’s synchronous evolutionmay have either a reversible or an irreversible discontinuousnature. The former case is what is traditionally investigatedin coupled oscillators, where a second-order phase transitioncharacterizes the continuous passage from the incoherent tothe coherent state of the network [17,18]. The latter, instead,corresponds to a discontinuous transition, called explosive syn-chronization (ES) [19,20]. ES based on Kuramoto oscillatorshas rapidly become a subject of enormous interest [19–28].While originally it was suggested that ES was due to a positivecorrelation between the natural frequencies of oscillators andthe degrees of nodes [20], more recent studies have proposedthe unifying framework of a mean field, where the effectivecouplings are conveniently weighted [22,27,29]. Yet onlypreliminary studies exist on the effect of degree mixing onES [30–33], and their evidence is still not conclusive andsometimes also conflicting.

In this paper, we focus on ES of coupled phase oscillatorsin two different models of SF networks, in the presenceof a positive correlation between the node’s degree and itsassociated oscillator’s natural frequency, and show that thedegree mixing has important effects on the nature of thephase transition characterizing the passage from the phase-incoherent to the phase-coherent network state. In particular,we will first show that growing and static SF networks havingthe same degree distribution display in fact different explosive

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I. SENDINA-NADAL et al. PHYSICAL REVIEW E 91, 032811 (2015)

transitions and, second, we hypothesize that this dissimilarityis due to the presence of some sort of degree mixing. Actually,our evidence is that there is an optimal level of assortativity forwhich the width and area of the hysteretic region associatedwith ES is maximal, thus magnifying the irreversible nature ofthat transition.

II. THE MODEL

To this purpose, let us start by considering a network ofN coupled phase oscillators whose phases θi (i = 1, . . . ,N)evolve according to the Kuramoto model [17]:

dθi

dt= ωi + σ

N∑i=1

aij sin(θj − θi), (1)

where ωi is the natural frequency of the ith oscillator. Oscilla-tors interact through the sine of their phase difference, and arecoupled according to the elements of the network’s adjacencymatrix aij , with aij = 1 if oscillators i and j are coupled, andaij = 0 otherwise. The strength of the coupling is controlledby the parameter σ , by increasing which one eventually (i.e.,above a critical value of the coupling) promotes the transitionto the coherent state, where all phases evolve in a synchronousway [17,34].

Following the changes in the level of synchronizationamong oscillators as the coupling strength increases is tan-tamount to monitoring the classical order parameter s(t) =1N

| ∑Nj=1 eiθj (t)| [17]. Indeed, the time average of s(t), S =

〈s(t)〉T , over a large time span T assumes values ranging fromS ∼ 0 (when all phases evolve independently) to S ∼ 1 (whenoscillators are phase synchronized).

Typically, Eq. (1) give rise to a second-order phasetransition from S 0 to S 1 for a unimodal and evenfrequency distribution g(ω), with a critical coupling at σc = 2/

[πg(ω = 0)] for the case of all-to-all connected oscillators [35],and σ ′

c = σc〈k〉〈k2〉 for the case of a complex network with first

and second moments of the degree distribution 〈k〉 and 〈k2〉,respectively [11]. However, in the last few years it was pointedout that a different scenario (ES) can arise, featuring an abrupt,first-order-like transition to synchronization, and associatedwith the presence of a hysteretic loop [19–23]. In this lattercase, the forward (from S 0 to S 1) and backward (fromS 1 to S 0) transitions occur in a discontinuous wayand for different values of the coupling strength, in this waymarking an irreversible character of the phase transition, whichis of particular interest at the moment of engineering (orcontrolling) magneticlike states of synchronization [23].

III. EXPLOSIVE SYNCHRONIZATION DEPENDENCEON THE SF MODEL

We concentrate on ES in growing and static SF networks,when a microscopic relationship between the structure and thedynamical properties of the system is imposed. In particular,and following the approach of Ref. [20], we will choose adirect proportionality between the frequency and the degreedistribution [g(ω) = P (k)], implying that each network’soscillator is assigned a natural frequency equal to its degree,

ωi = ki , where ki = ∑j aij is the number of neighbors of the

oscillator i in the network.As for the stipulations followed in our simulations, SF

growing networks are constructed following the procedureintroduced in Ref. [36]. Such a technique, indeed, allowsconstruction of graphs with the same average connectivity〈k〉, and grants one the option of continuously interpolatingfrom Erdos-Renyi (ER) [37] to Barabasi-Albert (BA) [3] SFtopologies, by tuning a single parameter 0 α 1. With thismethod, networks are grown from an initial small clique ofsize N0 > m, by sequentially adding nodes, up to the desiredgraph size N . Each newly added node then establishes m newlinks, having a probability α of forming them randomly withalready existing vertices, and a probability 1 − α of followinga preferential attachment (PA) rule for the selection of itsconnection. When the latter happens, a generalization of theoriginal PA rule [3] is used that includes an initial and constantattractiveness A for each of the network’s sites, so that theattractiveness of node i (the probability that such a node hasto receive a connection) is Ai = A + ki [38]. The result of theabove procedure is that the limit α = 1 induces an ER config-uration, whereas the limit α = 0 corresponds to a SF networkwith degree distribution P (k) ∼ k−γ , with γ = 2 + A/m

(when A = m, γ = 3, and the BA model is recovered).With the aim of inspecting whether ES depends on

the chosen SF network model, we further comparativelyconsider ensembles of networks displaying the very sameSF distributions as those obtained with the PA algorithmdescribed above (α = 0) but this time we construct the SFtopology by means of the so called configuration model(CM) [39,40], a randomized realization of a given networkwhere the node degree distribution remains intact. In bothcases, we set N = 103, 〈k〉 = 6, and distribute the oscillators’frequencies so as to determine a direct correlation with thenode degree (ωi = ki) and, therefore, both network modelsdisplay identical frequency distributions.

The results are shown in Fig. 1, and reveal a dramaticdependence of the ES behavior on the underlying SF networkmodel used, despite both having exactly the same P (k). Inparticular, the top row of Fig. 1 reports the order parameter S

when σ is gradually increased in steps of δσ (forward tuning,solid line), and also in the reverse way, i.e., departing froma network state where S = 1 and gradually decreasing thecoupling by δσ at each step (backward tuning, dashed line).The different areas of hysteresis displayed by the PA (leftpanel) and CM (right panel) networks seem to indicate thata crucial condition to obtain a strong irreversibility in ESis having an underlying growth process through which theSF topology is shaped. In order to statistically characterizesuch critical behavior, the bottom panels of Fig. 1 accountfor the probability density functions of the area of hysteresis(left panel) and the largest difference in S(σ ) (right panel) forboth SF models. Averages are obtained from the simulationof 400 PA networks (and their corresponding CM networkrealizations). The values of the hysteretic areas are much largerfor PA than for CM networks up to the point that the ES for thelatter is almost absent (the most likely values for the hysteresisarea and Smax are very small).

As a first conclusion, we can affirm that, despite having thesame P (k) and therefore the same g(ω), the two classes of

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1 1.2 1.4 1.60

0.5

1

σ

S area ofhysteresis sync.

jump

PA

1 1.2 1.4 1.60

0.5

1

σ

S sync.jump

CM

0 0.1 0.20

0.1

0.2

area of hysteresis

PD

F

PACM

0 0.5 10

0.2

0.4

sync. jump

PD

F

PACM

FIG. 1. (Color online) Comparative results on ES between SFnetworks belonging to two different ensembles: preferential attach-ment (PA) and configuration model (CM). Top row: Forward (solidlines) and backward (dashed lines) synchronization curves for PA(left panel) and CM (right panel) SF networks with exactly the samedegree distribution. The area of the hysteresis is depicted as a blueshaded area in the PA-SF network while the synchronization jumpsare marked in both cases. Bottom row: Probability density functionsof the area of hysteresis (left panel) and of the synchronization jumps(right panel) for 400 PA (CM) network realizations. In all cases,N = 103, 〈k〉 = 6, γ = 2.4, and natural frequencies ωi = ki . TheCM network realizations are constructed using the degree sequencesof the PA networks.

SF networks exhibit a different explosive behavior. Supportedby the evidence provided in Fig. 1, we hypothesize thatthis difference is related to the presence of two-point degreecorrelations P (k,k′). One customary way to quantify theamount of degree correlation with a single parameter is byusing the Pearson correlation coefficient r between the degreesof all nodes at either ends of a link, which can be calculatedas in Ref. [4]:

r = L−1 ∑i jiki − [

L−1 ∑i

12 (ji + ki)

]2

L−1∑

i12

(j 2i + k2

i

) − [L−1

∑i

12 (ji + ki)

]2 ,

where ji and ki are the degrees of the nodes at the ends of theith link, with i = 1, . . . ,L. Actually, one has that −1 r 1,with positive (negative) values of r quantifying the level ofassortative (disassortative) mixing of the network. We recallhere that the BA model does not exhibit any form of mixing inthe thermodynamic limit [r → 0 as (log2 N )/N for N → ∞[4,41] and that a random CM produces networks that are highlydisassortative.

IV. THE EFFECT OF THE DEGREE MIXING

In the following, we study the impact of increasing ordecreasing the assortativity mixing on a network with a givendegree sequence ki from a power-law distribution k−γ .In order to generate SF networks with given and tunablelevels of degree mixing, we follow an adjusted version of

the degree-preserving [42] rewiring algorithm of Xulvi-Brunetand Sokolov [43], but similar and fully consistent results areobtained using other procedures to impose degree mixing, suchas simulated annealing [44] or with prescribed correlations [4].For each one of the growing and static SF networks, we choosea pair of links at random and monitor the degrees of the fournodes at the ends of such links. The links are then rewired insuch a way that the two largest- and the two smallest-degreenodes become connected provided that none of those linksalready exist in the network (in which case the rewiring stepis aborted and a new pair of links is selected). Repeating sucha procedure iteratively results in progressively increasing theassortativity of the network, in that more and more connectednodes of the network will display a similar degree. Conversely,if the rewiring is operated in a way to determine that thelargest- (second-largest-) and the smallest- (second-smallest-)degree nodes are connected, the resulting network becomesprogressively dissasortative.

In this way, we first generate a PA network of sizeN = 5×103 with a given mean degree 〈k〉 = 2m and slopeγ as previously described, and produce the correspondingrandom CM network realization. Then we check whether thosenetworks are uncorrelated, that is, whether r = 0. If not (whichis always the case due to finite-size effects), we perform the linkrewiring procedure until the networks are neutral (i.e., with nodegree correlations). Finally, we take these resulting configu-rations as our PA and CM reference networks, and further per-form on them the link rewiring procedure in order to produce anensemble of networks, all of them having the same degree dis-tribution, but different values of the assortativity coefficient r .

Figure 2 illustrates the effect of the imposed degreemixing on ES. Extensive numerical simulations of Eq. (1)were performed at various values of r , and for SF networkswith different slopes γ ranging from 2.4 to 3.0, and thesame mean degree 〈k〉 = 6. The most relevant result is that,regardless of the specific SF network model, the hysteresisof the phase transition is highly enhanced (weakened) forpositive (negative) values of the assortative mixing parameter,and that there is an optimal positive value of r where theirreversibility of the phase transition is maximum. Notice thatthe enhancement is far more pronounced in PA (left panel)than in CM (right panel) netoworks. Moreover, as the slopeof the power law of the degree distribution becomes steeper(large values of γ ), the enhancement produced by a positivedegree mixing gradually vanishes and the optimum pointslightly shifts to higher values of r . Notice, finally, that nullvalues of the hysteretic area indicate that the transition haslost its irreversible character, so that degree mixing can turnan explosive irreversible phase transition into a second-order,reversible one.

The nonmonotonic behavior of the area of hysteresis isfurther exemplified by looking at the synchronization curvesS shown in the bottom panels of Fig. 2. The left (right) panelshows the forward and backward synchronization transitionsfor three values of the assortative (disassortative) mixing.From the results shown in the right panel it is evident that anincreasing level of disassortative mixing reduces the thresholdof the forward transition (which is consistent with the gene-ral claims of Refs. [6–8] that disassortativity favors thenetwork’s synchronizability), but it progressively reduces

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−0.2 0 0.20

0.2

0.4

0.6

0.8

r

area

of h

yste

resi

s

PAγ=2.4γ=2.6γ=2.8γ=3.0

−0.2 0 0.20

0.2

0.4

0.6

0.8

r

area

of h

yste

resi

s

CMγ=2.4γ=3.0

1 2 30

0.5

1

σ

S

r=0.012r=0.096r=0.150

1 1.5 20

0.5

1

σ

S

r=−0.012r=−0.096r=−0.150

FIG. 2. (Color online) ES as a function of the degree mixing.Top row: Area of the hysteretic region delimited by the forwardand backward synchronization curves vs the Pearson correlationcoefficient r for PA (left panel) and CM (right panel) networkswith different values (reported in the legend) of the exponent γ ofthe degree distribution P (k) ∼ k−γ . Each point is an average overten different simulations. Bottom row: Forward (solid lines) andbackward (dashed lines) synchronization curves for PA networks(γ = 2.4) displaying different levels of assortative (left panel) anddisassortative (right panel) mixing. Curves are coded accordingly tothe specific value of the parameter r (reported in the legend of eachpanel). In all cases, networks are SF with N = 5 × 103, 〈k〉 = 6, andωi = ki .

the hysteretic area associated with ES, until eventually asecond-order reversible transition is recovered. In contrast, theeffects of assortativity (left panel) are seemingly nontrivial: thethreshold for the forward synchronization has an increasingtrend with r > 0, but the area of hysteresis appears to widenfor intermediate values of r .

Further information can be gathered from inspection ofFig. 3, where we illustrate the effect of varying the mean

0 0.1 0.20

0.2

0.4

r

area

of h

yste

resi

s <k>=6<k>=10<k>=20<k>=40

0 0.1 0.20

0.2

0.4

r

area

of h

yste

resi

s α =0.0α =0.1α =0.2α =0.3

FIG. 3. (Color online) Area of hysteresis as a function of r forPA networks with different amounts of heterogeneity. In the leftpanel, the different curves correspond to different values of the meandegree 〈k〉 (reported in the legend), while in the right panel 〈k〉 = 6and the network heterogeneity is varied by means of increasing theparameter α (see the legend for the color and symbol code of thedifferent reported curves), from pure SF (α = 0) to α = 0.3 (α = 1corresponds to a pure ER network). In all cases, N = 103, γ = 2.4,and ωi = ki , and each point is an average of ten simulations.

degree 〈k〉 (left panel), and the level of heterogeneity α (rightpanel) in PA networks of smaller size (N = 103). In theleft panel it is observed that already at r = 0 (uncorrelatednetworks), increasing the mean degree results in narrowingthe area of hysteresis, with the consequence that the phasetransition becomes smoother and smoother, until eventuallyES is lost. For generic values of r , as 〈k〉 increases, the curvesof the area of hysteresis are attenuated and shifted to highervalues of r . Regarding the effect of the heterogeneity in thenetwork’s connectivity (right panel), moving from pure PAnetworks (α = 0) to slightly larger values of α causes rapiddeterioration of the enhancement of hysteresis. However, apositive degree mixing can still turn a second-order phasetransition (for r = 0) into an abrupt and irreversible one at avalue of r ∼ 0.1 when α = 0.2.

V. DISCUSSION

From the results reported in the top row of Fig. 2 one clearlysees how growing PA networks present a larger hysteresisarea than static CM networks for any value of r , althoughboth classes of network models display an enhancement ofirreversibility in connection with an increase in the degree-degree correlation. Figure 4 provides further evidence of howES is achieved for a particular value of γ in terms of themaximum gap in the order parameter Smax (left panel) andthe critical coupling strengths σ

f wc and σbk

c (right panel) mark-ing, respectively, the forward (solid symbols) and backward(hollow symbols) transition points. As a function of the degreemixing, the curves obtained for growing PA (red circles) andstatic CM (blue triangles) SF networks have slightly differenttrends. This figure makes it more apparent that the criticalcoupling for the forward transition decreases almost linearlyas the mixing increases in the region r < 0 [6–8], while forr > 0, the dependence of σ

f wc on r is clearly nonlinear. This

suggests that specific pronounced topological mesoscales ariseat those values of r which influence the forward transition,having the effect of obstructing the otherwise increasing trend

−0.2 0 0.21

2

3

r

σ c

σ CM

σ CM

σ PA

σ PA

−0.2 0 0.20

1

r

Δσc

CM

PA

−0.2 0 0.20

0.2

0.4

0.6

0.8

r

sync

. jum

p

σ CM

σ CM

σ PA

σ PA

FIG. 4. (Color online) Behavior of the critical parameters char-acterizing the synchronization transition. Critical coupling strengths(left panel) and synchronization jumps Smax of the order parameter(right panel) at the synchronization transitions during the forward(solid symbols) and backward (hollow symbols) continuations forgrowing PA () and static CM () SF networks as a function of thedegree mixing r . In all cases, N = 5 × 103, 〈k〉 = 6, and γ = 2.4.The vertical dashed line marks r = 0. The inset of the left panelreports the corresponding width of the hysteresis curves, calculatedas σc = |σf w

c − σ bkc | for PA () and CM () SF networks, with

“fw” and “bk” indicating “forward” and “backward,” respectively.

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EFFECTS OF DEGREE CORRELATIONS ON THE . . . PHYSICAL REVIEW E 91, 032811 (2015)

of σf wc . In contrast, in the backward continuations (hollow

symbols) the relationship between σc and r exhibits a changeof slope only at r = 0. The inset in the left panel of Fig. 4reports the corresponding widths of the hysteresis curves,calculated as σc = |σf w

c − σbkc |, and underlines once again

the existence of a maximum for σc(r), in correspondencewith the maximum in the area of the hysteresis reported inFig. 2. Regarding the behavior of the maximum gap of theorder parameter at the forward transition (right panel of Fig. 4),it displays a plateau at large values within the interval of r

where the ES still holds, and the abruptness of the transitiondeteriorates for large values of |r|.

Figure 4 then clarifies that the enhancement of the hysteresisis associated with a moderate increase in the degree-degreecorrelation, recovering a second-order transition for largevalues of positive and negative r . This nontrivial effect canbe understood by examining the inner mechanism of thefrequency-degree correlation. Explosive transitions result froma frustration in the path to synchronization [45]. In the case ofER networks, where the path to synchronization starts frommultiple seeds homogeneously distributed in the network,this frustration can be induced by imposing a gap in thefrequency differences of each pair of nodes. The larger isthe gap frequency, the higher is the frustration (explosivity)of the system, which shows a positive correlation betweenthe explosive character of the system and the width of thehysteresis [23]. In the case of general SF networks, thispath starts from the hubs, leading to the synchronization ofthe system by progressively recruiting nodes [46]. However,under positive frequency-degree correlations this frustrationis induced by an emergent frequency gap existing betweenhubs and their neighbors. Therefore, frustrating the path tosynchronization in SF networks is tantamount to isolatingthe influence of the hubs in the system. In this way,the more connected the network is through the hubs, themore explosive the transition becomes once the hubs areisolated.

The above arguments can be quantified by evaluating thenode betweenness centrality, which computes the fraction ofall shortest paths passing through each node of the network.Figure 5 shows the mean betweenness (left panel) for thecore made of the set of the first three higher-degree nodes(for the remaining nodes the betweenness does not changesignificantly) and the mean distance from the hub to the rest ofthe network (right panel), for both PA (red and solid symbols)and CM (blue and hollow symbols) networks. PA networksare comparatively more connected through the hubs than CMstatic networks within the region of degree-degree correlationwhere explosive behavior is observed. Therefore, in the case ofthe CM there are more paths connecting the network that do notnecessarily pass through the hubs, allowing progressive localsynchronization and thus reducing the explosive character ofthe transition and the associated hysteresis width. This is ofcourse due to the specific characteristics of the hubs in eachnetwork model. While a growing PA network starts from an all-to-all connected seed, for the static CM network the hubs arerandomly distributed in the network, as their natural degree-degree correlations reveal: r 0 for growing PA networks andr = −0.19 for the CM networks.

−0.2 0 0.21

1.5

2x 106

r

betw

eene

ss c

entr

ality

PA CMγ=2.4γ=2.6

γ=2.4γ=3.0

−0.2 0 0.2

2.8

3

3.2

r

hub

mea

n di

stan

ce PA CMγ=2.4γ=3.0

γ=2.4γ=3.0

FIG. 5. (Color online) Dependence of the network structuralproperties on the degree mixing. Betweenness centrality of the firstthree higher-degree hubs (left panel) and mean network distance ofthe hub of the network (right panel) as a function of the Pearsoncorrelation coefficient r for PA (red and solid symbols) and CM(blue and hollow symbols) SF networks with different values of theexponent γ of the degree distribution P (k) ∼ k−γ as indicated in thelegend. Each point is the average of ten network realizations withN = 5 × 103 and 〈k〉 = 6.

According to Ref. [25], σc increases with the degree of themain hub for uncorrelated SF networks in the limit of smallmean degree networks, where the role of the hubs is certainlydominant. Therefore, we suggest that a small increase in thedegree-degree correlation promotes the connectivity of thehubs, leading to the emergence of a core with a larger effectivedegree, which increases the hysteresis width accordingly.However, further increase of the assortativity or dissassorta-tivity enhances the modularity of the network, thus breakingthe dominant role of hubs by over- or underconnecting them.This is reflected by the decrease of the core’s betweenness forlarge positive and negative values of r for both growing andstatic networks (see again Fig. 5).

VI. SUMMARY

In summary, we have reported simulations of the dynamicsof networked ensembles of phase oscillators whose interac-tions are mediated by a scale-free topology of connections,and for which a positive correlation exists between eachoscillator’s natural frequency and the corresponding nodedegree. Our results allow us to conclude that the furtherpresence of degree-degree mixing in the network structurehas crucial consequences for the nature of the phase transitionaccompanying the emergence of the phase-coherent state of thenetwork. In particular, we have shown that high levels of bothpositive and negative mixings consistently produce a second-order phase transition, whereas moderate values of assortativemixing magnify the irreversible nature of ES in both static andgrowing SF networks. When related to the fact that nontrivialforms of degree correlation indeed ubiquitously characterizethe structure of real-world SF networks, our results may beof relevance for understanding why real-world biological andtechnological networks organize themselves on topologicalstructures that tend to avoid explosive synchronization phe-nomena (which are there usually associated with pathologicalstates of the networks), whereas social network topologiesactually favor the explosive and irreversible emergence ofsynchronous states.

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ACKNOWLEDGMENTS

The authors acknowledge financial support from the Min-isterio de Economıa y Competitividad (Spain) under ProjectsNo. FIS2012-38949-C03-01 and No. FIS2013-41057-P, andfrom INCE Foundation (Project No. 2014-011). Z.W. ac-

knowledges the National Natural Science Foundation of China(Grant No. 11005047). The authors also acknowledge the com-putational resources, facilities, and assistance provided by the“Centro computazionale di RicErca sui Sistemi COmplessi”(CRESCO) of the Italian National Agency ENEA.

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Effective centrality and explosive synchronization in complex networks

A. Navas,1 J. A. Villacorta-Atienza,2 I. Leyva,1, 3 J. A. Almendral,1, 3 I. Sendina-Nadal,1, 3 and S. Boccaletti4, 5

1Center for Biomedical Technology, Univ. Politecnica de Madrid, 28223 Pozuelo de Alarcon, Madrid, Spain2Department of Applied Mathematics, Facultad de Ciencias Matematicas, Universidad Complutense, Madrid, Spain

3Complex Systems Group & GISC, Univ. Rey Juan Carlos, 28933 Mostoles, Madrid, Spain4CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy

5Embassy of Italy in Israel, Trade Tower, 25 Hamered St., 68125 Tel Aviv, Israel

Synchronization of networked oscillators is known to depend fundamentally on the interplay be-tween the dynamics of the graph’s units and the microscopic arrangement of the network’s structure.For non identical elements, the lack of quantitative tools has hampered so far a systematic studyof the mechanisms behind such a collective behavior. We here propose an effective network whosetopological properties reflect the interplay between the topology and dynamics of the original net-work. On that basis, we are able to introduce the “effective centrality”, a measure which quantifiesthe role and importance of each network’s node in the synchronization process. In particular, weuse such a measure to assess the propensity of a graph to synchronize explosively, thus indicatinga unified framework for most of the different models proposed so far for such an irreversible transi-tion. Taking advantage of the predicting power of this measure, we furthermore discuss a strategyto induce the explosive behavior in a generic network, by acting only upon a small fraction of itsnodes.

PACS: 89.75.Hc, 05.45.Xt, 87.18.Sn, 89.75.-k

One of the most intriguing processes in complex net-works’ dynamics is synchronization: the spontaneous or-ganization of the network’s units into a collective dy-namics. This phenomenon is known to be related to adelicate interplay between the topological attributes ofthe network and the main features of the dynamics ofeach graph’s unit [1–3]. The conditions for synchroniza-tion in complex networks have been addressed by meansof different approaches. For identical units, one of themost successful tools is, for instance, the Master Stabil-ity Function [4], which rigorously shows how the spectralproperties of the graph influence the stability of synchro-nization [1]. However, the general case of non-identicalunits is far more complicated, and often needs a numer-ical approach, where the topology-dynamics relationshipcan only be investigated within specific scenarios [5–9].

Such a connection between structure and dynamics ofa network is of particular importance in the case of therecently reported explosive synchronization (ES), an irre-versible and discontinuous transition to the graph’s syn-chronous state. Originally, ES was described in all-to-all coupled ensembles of Kuramoto oscillators [10] for aspecific distribution of natural frequencies [11]. Lateron, various kinds of degree-frequency correlations werefound to be able to induce ES in networks of periodicand chaotic oscillators [12–15], or neural networks [16]. Inaddition, other microscopic mechanisms were proposed,based on diverse coupling strategies [17–19], or by intro-ducing adaptive dynamics in a fraction of the network’sunits [20].

In this Letter, we propose the use of an effective topo-logical network whose structure explicitly reflects the in-terplay between the topology and dynamics of the orig-inal system. On that basis, we introduce the effectivecentrality as a measure to quantify the role of each node

in the synchronization process. This measure allows usto provide a general explanation of the mechanisms un-derlying ES and revisit the main scenarios where such abehavior was previously reported. Finally, we formulatea criterion to induce explosive transitions by acting onlyon a small fraction of the network’s nodes.

We start by considering a network of N phase oscilla-tors, whose instantaneous phases evolve in time accordingto the Kuramoto model [10]:

θi = ωi +σ

N

N∑

j=1

Aij sin(θj − θi) i = 1, ..., N, (1)

where θi is the phase of the i-th oscillator, ωi its nat-ural frequency (chosen from a generic, known, distribu-tion g(ω)) and σ the coupling strength. The topologyof the network is encoded in the adjacency matrix A(Aij = 1 if node i is linked to node j, and Aij = 0 oth-erwise). The degree of a node is ki =

∑j Aij . The level

of synchronization is measured by the order parameterr = 1

N 〈|∑Ni=1 e

θi |〉T , where |.| and 〈.〉T denote moduleand time average, respectively. Along this Letter, thenetwork size is fixed to N = 1, 000 and the natural fre-quencies ωi are randomly drawn from a uniform distribu-tion in the interval [−0.5, 0.5], unless otherwise specified.

As the coupling strength σ gradually increases, system(1) experiences a transition from an incoherent (r ' 0)to a synchronized state (r ' 1), a process often referredto as path to synchrony (PTS) [21]. In heterogeneousnetworks, the PTS is mainly dominated by the most con-nected nodes (or hubs) which actually act as synchroniza-tion seeds, and progressively recruit the other network’snodes. At variance, in homogeneous networks, the PTS ischaracterized by the emergence of coherent clusters grow-


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ing around multiple synchronization seeds. Along thePTS, the first nodes that locally synchronize generallycorrespond to pairs of connected oscillators whose natu-ral frequencies are closer, whereas the globally synchro-nized state emerges around those with natural frequen-cies close to the synchronization frequency Ωs. Whiletraditionally attention has been focused on the naturaldynamics of each node, recent works [18, 19] have shownthe importance of frequency detuning in the process ofsynchronization, motivating a different approach, wherethe links prevail over the nodes themselves. Hence, wepropose the frequency detuning ∆ωij ≡ |ωi−ωj | betweeneach pair of nodes as the key dynamical feature for thedetermination of the PTS. To formalize our idea, let usintroduce a change of variables rie

iΨi = 1N

∑j∈Γi

eiθj ,where ri(t) is a local order parameter and Γi is the setof neighbors of the node i. Substituting into Eq. (1) weobtain

θi = ωi + σri sin (Ψi − θi) . (2)

It naturally follows that the velocity difference is Φij =

θi−θj = ωi−ωj+σ [ri sin(Ψi − θi)− rj sin(Ψj − θj)]. As

synchronization implies Φij = 0, the set of links throughwhich synchronization may take place must fulfill

∆ωij ≤ σ (ri + rj) , (3)

which in fact relates local synchronization to the fre-quency detuning associated to the links, being thosepairs of nodes with large detuning harder to synchro-nize. To further show the role of frequency detun-ing, we investigate how a modification of the adjacencymatrix by a certain function of the detuning affects thePTS, i. e. A → Af (∆ω). Considering a first orderapproximation, it results in Aij → Aij (1 + δ∆ωij) forf(0) = 1 and f ′(0) = δ. Figure 1 reports the synchro-nization transition curves for (a) Erdos-Reyni (ER) witha mean degree 〈k〉 = 30 [26] and (b) scale-free (SF) with〈k〉 = 12 networks generated by the Barabasi-Albert al-gorithm [27]. It can be seen an enhancement (frustra-tion) of the synchronization transition as δ is increased(decreased). Hence, positive (negative) values of δ po-tentiate (weaken) the strength of the couplings accordingwith their ∆ωij . Notice that such a local perturbationof the adjacency matrix is more effective in promotingor delaying the PTS than a global perturbation of thesame mean equally acting on all links as shown by thecorresponding dashed lines in Fig. 1.

Since we are concerned about extracting the dynam-ical backbone of the network composed by the seeds ofsynchronization and according to the above results, weintroduce the following effective adjacency matrix in or-der to amplify their role:

Cij ≡ Aij(

1− ∆ωij∆ωmax

), (4)

0 0.5 10

0.5

1

σ

r

(a)

δ = −0 . 5

δ = 0

δ = 0 . 5

δ 〈Δωi j 〉 = −0 . 5

δ 〈Δωi j 〉 = 0 . 5

0 0.5 1σ

(b)

FIG. 1. (Color online). Synchronization transition curves(black circles) compared with the modified adjacency matrixAij (1 + δ∆ωij) in (a) ER, 〈k〉 = 30 and (b) SF 〈k〉 = 12networks for positive and negative δ values. Red triangles(δ > 0) and blue squares (δ < 0) show how a local perturba-tion enhances/frustrates the synchronization more efficientlythan a global perturbation over all links (dashed lines, whereδ∆ωij → δ〈∆ω〉, being 〈∆ω〉 the average over nonzero valuesof the detuning matrix).

where we have chosen f ′(0) = −1/∆ωmax, being ∆ωmaxthe maximum possible detuning present in the system inorder to guarantee Cij ≥ 0. Within this specific choiceof f(∆ω), Cij results in an effective topological networkthat exhibits the structure of the original one but en-hancing those more synchronizable pairs of nodes (i.e.those with small detuning) according to Eq. (3). Weremark that, although we are here referring to the Ku-ramoto model of Eq. (1), Eq. (4) can be applied muchbroadly to any kind of oscillator’s ensemble which canbe associated to a set of well defined natural frequenciesωi. Indeed, numerical results (not reported here) indi-cate that the main conclusions we will draw for Eq. (1)are in fact valid also for networks of chaotic oscillators,provided that the power spectrum of each unit is pro-nouncedly peaked around a unique frequency, i.e. theunits are strongly phase coherent [22].

Now, in order to quantify the role of each node in thesynchronization process, we extract the most importantnodes in the effective network defined by C, i.e. we calcu-late the standard eigenvector centrality measure [1, 23–25] of C, obtaining the effective centrality vector ΛC.The i-th component ΛCi ≥ 0 provides a measure of theimportance of the node i in the effective network andquantifies its potential to behave as a seed of synchro-nization.

In order to show that the effective network really allowsto predict the importance of each node in the PTS, weproceed with confronting our approach with a dynamicalexploration of the system for the case of ER networks.Precisely, we calculate a local synchronization matrixS = Sij = Aij |〈ei∆θij 〉t| [21] where ∆θij = θi − θjand σ ∈ [0, 0.7]. The eigenvector centrality of S, denotedby ΛS, provides the actual synchronization centrality ofeach node in the synchronization process.

In Fig. 2 we sort the nodes according to the increas-ing value of the corresponding centrality and we report


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3

0 0.1 0.2 0.3 0.4 0.5 0.6

40

60

80

100

σ

% P

redic

tion

FIG. 2. (Color online). Average percentage of coincidencebetween the third of the nodes with the highest (red trian-gles) and lowest (blue circles) dynamical ΛS

i and effective ΛCi

centralities (solid symbols) and the corresponding percentagebetween dynamical ΛS

i and topological ΛAi centralities (empty

symbols). Calculations are performed on ER networks with〈k〉 = 50, and refer to 10 realizations of the network’s topologyand frequency distribution (see text for details).

the percentage of coincidence between the third of thenodes with the highest (lowest) synchronization ΛSi andeffective ΛCi centralities (solid symbols), and the corre-sponding percentage of coincidence between the third ofthe nodes with the highest (lowest) synchronization ΛSiand topological ΛAi centralities (empty symbols). It canbe seen that the ranking based on ΛCi is able to predict upto 80% of the nodes with the highest (lowest) dynamicalcentrality, while the topological centrality only detectsat most the 50%. According to ΛCi the maximum of pre-dictability is reached around the synchronization thresh-old and decreases rapidly due to the homogenization ofthe synchronization matrix S for overcritical couplings,while the predictability according to ΛAi is approximatelyconstant until it increases to 100% when Sij = 1 (S = A),that is, in the trivial completely synchronous state.

As effective centrality reveals itself as a suitable mea-sure to characterize the PTS, we move on elucidatinghow this quantity helps us also to understand the micro-scopical mechanisms underlying explosive synchroniza-tion (ES). As it has been remarked in the introduction,ES can be induced when topology and dynamics are re-lated in several specific ways [11–15, 17–20, 28]. Almostall these methods are based on a manipulation of theadjacency matrix and/or the links weights, such that inEq. (1) Aij is replaced by a certain matrix Ωij whichusually correlates the structural and dynamical featuresof the network.

To show how the different procedures impact the ef-fective centrality vector, we compare the ΛC associ-ated with the non explosive case and the correspond-

ing ΛC when the explosive method is applied, that is,Cij = Ωij(1 − ∆ωij

∆ωmax). Results for some of the different

methods are condensed in Fig. 3 for ER (left panels) and

SF (right panels) networks. In each panel, ΛCi (red dots)are plotted together with ΛCi (cyan dots) to show howthe explosive method actually modifies the effective cen-

FIG. 3. (Color online). Effective centrality of explosive net-

works (ΛCi , red clouds) versus that of non-explosive networks

(ΛCi , cyan clouds) for ER (left panels) and SF (right pan-

els) topologies (see text for definitions). (a) and (b) accountfor the weighted method of Ref. [18] with a uniform naturalfrequency distribution in [−0.5, 0.5]. (c) and (d) correspondto the weighted method of Ref. [28] with the same uniformfrequency distribution (see text for the methods’ description).Panel (e) is the same as in (d) but natural frequencies are herechosen from the positive values of a Gaussian distribution, ina way that ES is no longer induced. Finally, (f) corresponds toa SF network where the degree-frequency correlation methodfrom Ref. [12] is applied and the same uniform frequencydistribution as for panels (a-d) is used. The inset in (f) is azoom centered on the highest degree nodes. All data refer toaverages over 100 realizations.

trality vector and, therefore, the dynamical role of eachnode. Nodes are sorted in ascending order of ΛCi . Inall the cases where the structural and dynamical corre-lations introduced through Ωij successfully lead to ES

(Figs. 3(a-d,f)), there is an increase (decrease) of ΛCi ofthose nodes whose ΛCi was low (high), that is, the weight-

ing method produces a flattening of ΛCi . In this way, thepotential ability of the nodes to behave as seeds of lo-cal synchronization is frustrated until a certain couplingstrength is reached. Only once the coupling strength is


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4

large enough, the rest of the network fulfills the condition(3), and therefore a sudden transition to synchronizationtakes place.

Figure. 3(a-b) correspond to the method described inRef. [18], where ES is achieved choosing Ωij = Aij |ωi −ωj | for ER networks (a) and Ωij = Aij |ωi−ωj |lij/

∑j lij

for SF networks (b), being lij the edge betweenness [1].Figures 3(c-d) show, instead, the case Ωij = Aij |ωi|/kiproposed in Ref. [28] for uniform frequency distributionscentred in zero. It is easy to see that the above increase-decrease compensation is fulfilled for both ER (c) and SF(d) networks. As expected, in SF networks the modifi-cation affects mainly the hubs, thwarting their dynami-cal influence as seeds, and frustrating the PTS. Finally,Fig. 3(f) reports the case of ES induced in SF networksby imposing a frequency-degree correlation ωi = ki [12].Here the effect is focused on the hubs (see the inset),whose effective centrality is now strongly decreased, whilethe imposed correlation does not produce a substantial

difference between ΛC and ΛC for the rest of the nodes.There are however cases when, even if the structure anddynamics are correlated, ES does not occur. For instance,Fig. 3(e) reports the same case presented in Fig. 3(c) butfor positive definite frequencies. And indeed, for this fre-quency distribution, it is seen that the weighting methodfails to flatten sufficiently ΛC (the red horseshoe cloud),with the consequence that ES fails to emerge as well.

An important application of our effective centrality isthe possibility of engineering an efficient strategy to pro-duce ES in a generic network by only acting upon a smallfraction of its nodes, according to a given ranking defin-ing their role as synchronization seeds. We here test fourpossible rankings: i) the effective ranking based on ΛCi ,ii) the distance ranking, that sorts the nodes according to

the distance ∆ΛCi = |ΛCi −ΛCi |, iii) the topology ranking,based on ΛAi and, finally, iv) a random ranking, which isused for comparison. These specific rankings are actuallysuggested by the characteristic PTS occurring in both ERand SF networks, where the increase-decrease conditionand the dominant role of hubs constitutes, respectively,the essential feature (see Fig. 3).

Figure 4 reports the jump in the discontinuous syn-chronization transition curve for all the rankings as afunction of the % of perturbed nodes affected by theweighting methods of Refs. [12, 18]. The discontinuityjump is calculated as max[r(σ + δσ) − r(σ)], the maxi-mal difference found in the order parameter between twoconsecutive σ values. For homogeneous ER networks(Fig. 4(a)), we choose the first %p of the nodes sortedby the corresponding ranking and weight their links asΩij = Aij |ωi − ωj |/Ωij , where Ωij is the mean of thenon-zero elements of Ω [31]. The rest of the nodes re-main with the original adjacency. The effective ranking(black squares) indicates that a significant explosive ef-fect is obtained in the network already for just 6% of

0 10 20 30 400

0.3

0.6(a)

% perturbed nodes

Max

. Jum

p

0 10 20 30 400

0.4

0.8

% perturbed nodes

(b)

FIG. 4. (Color online). Maximum jump size in the synchro-nization curve σ(r) vs. fraction of perturbed nodes chosenalong several rankings: effective (back squares), distance (reddots), topology (magenta diamonds) and random (blue as-terisks). (a) ER networks 〈k〉 = 30, weighting method fromRef. [18]. (b) SF networks 〈k〉 = 6 imposing ωi = ki forthe selected nodes as in Ref. [12]. In all cases, data refer toaverages over 50 realizations.

the nodes but decreases afterwards, whereas the distanceranking (red circles) requires up to 15% to get an equiv-alent jump, inducing a complete explosive transition forpercentages above 30%. In comparison, using a randomranking it is necessary to manipulate at least the 40% ofthe nodes, while the topological ranking is not able toinduce ES in this interval.

For SF networks (Fig. 4(b)), we use instead the degree-frequency correlation as is [12], setting ωi = ki for thecorresponding first %p of nodes along the ranking. Inthis case, both the effective (black squares) and topologi-cal (magenta diamonds) rankings clearly outperforms thedistance ranking (red circles) by only affecting the 10%of the nodes. In comparison, the random ranking is notable to induce ES even above the 40%.

The differences between ER and SF cases are due to thedifferent ways the seeds spread in the network. In the ERcase, the effective ranking optimally performs for smallpercentages since it focuses on the seeds of synchroniza-tion. As soon as this percentage increases, the nodes withthe lowest ΛC are not longer captured, and the increase-decrease condition is not fulfilled. As there are multiplerandomly distributed seeds, the distance ranking is onlyslightly better than the random one, as both satisfy theincrease-decrease condition once the percentage is largeenough. In the SF case the topology is determinant asthe seeds are just a few hubs, allowing to induce ES act-ing upon a very small fraction of the nodes of the net-work, whereas the random targeting is definitely not thesuitable choice.

In conclusion, we have introduced an effective net-work whose topological properties quantitatively charac-terize the PTS of networked oscillators, allowing a deeperknowledge of the synchronization process. In particular,this approach allows to reveal the inner mechanisms be-neath ES, which is shown to be rooted in a frustrationof the PTS. Finally, it also allows to control such be-havior locally, since we have the means to identify andisolate those seeds involved in the emergence of synchro-


95

5

nization.

Authors acknowledge the computational resourcesand assistance provided by CRESCO, the supercom-putacional center of ENEA in Portici, Italy. Workpartly supported by the Spanish Ministerio de Cienciae Innovacion under projects FIS2012-38949-C03-01 andFIS2013-41057-P, and by the INCE Foundation underthe project INCE2014-011.

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96

CAPITULO 4

CONCLUSIONES

A lo largo de esta tesis hemos estudiado el fenomeno de la sincroniza-cion explosiva desde dos puntos de vista: a traves de un enfoque descriptivo,mediante el estudio de diferentes metodos capaces de inducir dicho com-portamiento, y a traves de un enfoque conceptual, proponiendo una seriede herramientas que permiten explicar y comprender los mecanismos res-ponsables de la sincronizacion explosiva. Se puede, pues, afirmar que los 3objetivos propuestos en la introduccion han sido adecuadamente tratados enlos artıculos del Capıtulo 3.

O.1 Estudiar diversos mecanismos que inducen sincronizacion ex-plosiva.

Se han propuesto dos metodos diferentes para inducir explosividad en re-des complejas. En el primer caso (Leyva et al., 2012), y como complementodel comportamiento explosivo en redes de tipo SF, se propone un metodopara generar redes aleatorias explosivas a traves de la construccion de to-pologıas con una condicion particular: que los enlaces entre los nodos i y jsolo pueden existir si las frecuencias naturales de ambos nodos cumplen lacondicion |ωi − ωj| ≥ γ, siendo γ un parametro del modelo. En el segundometodo (Leyva et al., 2013b), se generaliza este mecanismo para adaptarloa cualquier tipo de red con tal de introducir una particular matriz de pesossegun la red sea homogenea (ER) o heterogenea (SF). De esta forma, hemospropuesto un metodo general para inducir explosividad en cualquier red, sinrestricciones sobre sus propiedades topologicas particulares. Gracias a ello

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CAPITULO 4. CONCLUSIONES

se concluye que un elemento fundamental para la sincronizacion de redesde osciladores es el detuning entre las frecuencias naturales de cada par denodos, i. e. ∆ωij = |ωi − ωj|. En particular, encontramos que al imponeruna disasortatividad en las frecuencias naturales, i. e. frecuencias altas venpreferentemente frecuencias bajas, aparece un fenomeno de explosividad, sibien en el caso de redes SF es necesario introducir un factor extra asociadoa la conectividad de los nodos.

O.2 Entender y unificar los fundamentos de las transiciones explo-sivas.

A la hora de comprender en mayor profundidad los mecanismos inherentesa la sincronizacion explosiva hemos llevado a cabo dos enfoques complemen-tarios:

En primer lugar, (Navas et al., 2015) propone una red efectiva que repre-senta la interaccion combinada de la conectividad de la red y de la afinidaddinamica de sus elementos a traves de la diferencia en frecuencias naturales(O.1). De esta forma, es posible determinar que nodos son mas relevantes enel proceso de sincronizacion, siendo en general aquellos que estan bien conec-tados con su entorno y que a su vez tienen vecinos con frecuencias naturalesproximas. Una vez caracterizada la importancia de cada nodo en la ruta a lasincronizacion, exploramos como esta cambia al aplicar los diversos metodosque inducen una transicion explosiva. De este modo, se concluye que en todoslos casos la sincronizacion explosiva es el resultado de una frustracion de laruta a la sincronizacion, llevada a cabo aislando dinamicamente los nodosmas importantes en el proceso de sincronizacion a la par que se refuerzan losmenos centrales.

En segundo lugar, (Sendina-Nadal et al., 2015) explora numericamente ladependencia de la transicion explosiva en redes heterogeneas con las correla-ciones de grado. Lo primero que se muestra es como la eficiencia del metodode correlacion grado-frecuencia en redes SF depende fuertemente del modoen que se construye la red, siendo mucho menos efectivo si esta se construyesegun el modelo de configuracion (que genera redes disasortativas) que si sehace siguiendo el modelo de Barabasi-Albert (redes neutras). A continuacion,se vincula esta propiedad al hecho de que, en el primer caso, los hubs de lared estan tıpicamente desconectados entre sı, mientras que en el modelo deBarabasi-Albert la red se construye a partir de una componente o nucleo ini-cial totalmente conexa. El analisis de la betweenness de los nodos revela como

98


los hubs constituyen los centros de paso mas importantes de la interaccionentre pares de nodos de la red, siendo mayor en el modelo de Barabasi-Albertdebido a la presencia de dicho nucleo. En conclusion, y de acuerdo con losprincipios inherentes a la sincronizacion explosiva anteriormente expuestos,el grado de conectividad de los hubs (las semillas de sincronizacion en redesSF) determina la efectividad del aislamiento dinamico de dichas semillas alimponer una correlacion grado-frecuencia.

O.3 Describir los principios subyacentes a la histeresis que tıpica-mente exhiben las transiciones irreversibles.

Los resultados expuestos en (Olmi et al., 2014) permiten arrojar luz sobrelos fenomenos de irreversibilidad asociados a la sincronizacion explosiva. Enel modelo de Kuramoto de segundo orden, la histeresis aparece al introducirexplıcitamente en la dinamica un termino de inercia que rompe la simetrıa delproceso de sincronizacion respecto al incremento/decremento del parametrode acoplamiento σ. Este termino se puede interpretar como una perturbacionde las frecuencias naturales, siendo la distribucion de frecuencias naturalesperturbada distinta en el regimen asıncrono que en el regimen sıncrono, dondeel termino de inercia se vuelve despreciable. En conclusion, la ruptura de lasimetrıa de la distribucion de frecuencias naturales entre las fases sıncrona yasıncrona parece ser la responsable de la aparicion de la irreversibilidad ensistemas de osciladores acoplados.

Trabajo futuro

Si bien los principios fundamentales relativos a la explosividad parecenrazonablemente comprendidos, aun queda por estudiar el alcance real de es-te fenomeno, especialmente en areas de investigacion donde una transicionabrupta puede resultar crıtica, como en ingenierıa o en neurociencia. En par-ticular, los episodios de epilepsia se manifiestan mediante una sincronizacionsubita y espontanea de todas las areas del cerebro, y los tratamientos actua-les consisten en hacer pequenas descargas sobre ciertas zonas del cerebro conla intencion de interrumpir o evitar el proceso de sincronizacion global. Noobstante, los criterios sobre que zonas afectar son difıciles de determinar, yen muchos casos, son determinantes para la vida del paciente. Por tanto, enel trabajo con datos reales de pacientes epilepticos serıa interesante aplicarlos conocimientos sobre deteccion de semillas de sincronizacion y transiciones

99


explosivas. En particular, se podrıa elaborar un metodo mas preciso a la horade seleccionar que zonas del cerebro deben ser tratadas a fin de interrumpirlos brotes epilepticos.

Por otro lado, en las transiciones explosivas la histeresis aparece con fre-cuencia. Por tanto, podemos preguntarnos si existe algun tipo de mecanismoanalogo al de ruptura de la simetrıa de la distribucion de frecuencias entrelos estados sıncrono y asıncrono (O.3), sabiendo que a diferencia del casoinercial, la distribucion de frecuencias naturales en el modelo de Kuramotoexplosivo es igual en ambos estados, y por tanto dicha ruptura debe tenerun caracter mas sutil.

A continuacion desarrollamos un tratamiento original del problema, quesi bien no ofrece resultados finales, establece al menos un punto de partida in-tuitivo sobre los fenomenos que inducen la irreversibilidad en las transicionesexplosivas:

De acuerdo con la seccion (2.3.1), el acoplamiento σ necesario para sin-cronizar un par de osciladores es proporcional a ∆ω. Ası mismo, en (Navaset al., 2015) se generaliza este resultado al caso de redes complejas, ponien-do de relieve una vez mas la importancia de ∆ω en la caracterizacion dela ruta a la sincronizacion. Esta fuerte influencia de las frecuencias natura-les de los vecinos sobre la dinamica de un nodo sugiere explorar si existealguna dependencia con las frecuencias naturales de los vecinos de cada os-cilador en el modelo con y sin transicion explosiva. Para ello, estudiaremosla distribucion fN del promedio de las frecuencias naturales de los vecinosde cada nodo, i. e. ωi = (

∑j Aijωj)/ki. En el caso de una matriz de adya-

cencia pesada Wij, el promedio sera ponderado de acuerdo a la expresionωi = (

∑jWijωj)/(

∑jWij). De fN podemos deducir como sera la distribu-

cion del promedio de frecuencias instantaneas fI por continuidad: en el estadototalmente incoherente (σ = 0) ambas distribuciones coinciden, mientras queen el caso totalmente sıncrono (σ σc), fI es una delta de Dirac centradaen la frecuencia de sincronizacion Ωs. Por tanto, en los casos no explosivos,la transicion entre ambos estados implica que fI se transforma suavementede fN a δ(ω − ΩS), mientras que en el caso explosivo, fI ' fN para σ < σc yfI = δ(ω − ΩS) para σ > σc.

En los paneles superiores de la Fig. 4.1 se observa como, para el modelopesado propuesto por (Leyva et al., 2013a), el histograma de fN es bimodaltanto en redes ER como SF, y por tanto el histograma de fI es a su vezbimodal en el estado asıncrono y aproximadamente una delta de Dirac en el

100


−0.5 0 0.5 −0.5 0 0.5

ER SF

-0.5 -0.50.5 0.50 0

ω ω

ω

ω

f ( )

f ( )

Figura 4.1: Histograma de fN en transiciones explosivas inducidas con elmetodo del pesado (paneles superiores) y no explosivas (paneles inferiores)para topologıas ER y SF. La lınea vertical roja marca el valor de la frecuenciade sincronizacion Ωs.

0 20 40 60 80 −0.5 0 0.5

ωf ( )

ω ω

Figura 4.2: Histograma de fN en transiciones explosivas inducidas con elmetodo de correlacion grado-frecuencia (panel izquierdo) y no explosivas(panel derecho) para redes SF. La lınea vertical roja marca el valor de lafrecuencia de sincronizacion Ωs.

101


estado sıncrono, estando ambos centrados en la frecuencia promedio Ωs. Encomparacion con el caso no explosivo (Fig. 4.1, paneles inferiores), el histo-grama de fN es unimodal, advirtiendose una clara ruptura de simetrıa de ladistribucion entre ambos estados. Mientras en el caso continuo el maximode la distribucion fI evoluciona a una delta de Dirac estrechandose de formaprogresiva a partir de fN , en el modelo pesado los dos maximos de la distri-bucion convergen a un maximo intermedio de forma abrupta, y por tanto lainfluencia dinamica sobre los nodos del sistema no es la misma en un sentidoque en el otro. Finalmente, para el metodo de correlacion grado-frecuencia(Gomez-Gardenes et al., 2011), el histograma de fN (Fig. 4.2, panel izquier-do) es unimodal al igual que en el caso no explosivo, donde se ha usado unadistribucion de frecuencias naturales aleatoria (Fig. 4.2, panel derecho). Noobstante, los maximos de ambas distribuciones no estan centrados en la mis-ma frecuencia, siendo tan solo el caso no explosivo aquel cuyo maximo secentra en la frecuencia de sincronizacion Ωs.

De todo esto se concluye que la histeresis esta asociada a una ruptura desimetrıa en la distribucion fI (concretamente en la localizacion de sus maxi-mos) entre el estado asıncrono y el estado coherente. La particular forma dela distribucion de fI es debida a fN , que actua a modo de potencial efectivodel estado asıncrono, donde los maximos de la distribucion fN son analogos alos estados de equilibrio. Ası mismo, el potencial efectivo del estado sıncronoes el correspondiente a la delta de Dirac, y por tanto la solucion de equilibrioes la frecuencia de sincronizacion Ωs. De esta forma, al incrementar el acopla-miento del sistema, el metodo explosivo frustra la sincronizacion a traves delpotencial efectivo, retrasandola, mientras que al disminuir el acoplamientodesde el estado sıncrono, el potencial efectivo tiene la simetrıa propia de lastransiciones continuas, induciendo ası la aparicion de histeresis.

Por ultimo, es significativo que este fenomeno ya aparezca cuando la dis-tribucion de frecuencias naturales es bimodal (Martens et al., 2009). Ası mis-mo, cuando la distribucion de frecuencias naturales es uniforme (Pazo, 2005)no hay ruptura de simetrıa, lo cual concuerda con los resultados numericosdel problema, donde la discontinuidad aparece en ausencia de histeresis. Esteresultado apoya la hipotesis de que la explosividad y la histeresis son procesosindependientes, si bien en el caso explosivo la condicion suficiente dada por elmetodo pesado (Leyva et al., 2013b) implica a su vez la condicion necesariapara la aparicion de la histeresis.

102

APENDICE A

VARIEDADESINVARIANTES

Para un tratamiento riguroso de la teorıa de bifurcaciones es necesariollevar a cabo un estudio del comportamiento asintotico de las soluciones delsistema. El objetivo es desarrollar una serie de herramientas matematicasque nos permitan generalizar los conceptos de punto fijo o ciclo lımite ensistemas n-dimensionales, preparando ası el terreno formal de la teorıa debifurcaciones y desarrollando las herramientas necesarias para el estudio dela estabilidad de soluciones no hiperbolicas. Para ello, introduciremos unaserie de definiciones que nos permitan concretar el concepto de atractor engeneral y facilitar el estudio de su estabilidad en particular.

Definicion (Conjunto invariante.) Se dice que un conjunto S ⊂ Rn esinvariante bajo el campo vectorial definido por x = f(x) si para cualquierx0 ∈ S tenemos x(t, 0, x0) ∈ S para todo t ∈ R, siendo x(t, t0, x0) la trayec-toria que pasa por x0 en t = t0 y x(0, 0, x0) = x0.

Es decir, las trayectorias que empiezan en un conjunto invariante permanecenen dicho conjunto para todo su futuro (conjunto positivamente invariante)y para todo su pasado (conjunto negativamente invariante). El ejemplo mastrivial de conjunto invariante es una curva integral, pues cualquier condicioninicial sobre dicha curva es un punto de la unica solucion que la define. Noobstante, nos interesan especialmente aquellos conjuntos invariantes que con-tienen puntos fijos.

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APENDICE A. VARIEDADES INVARIANTES

Definicion (Conjunto Atractor.) Se dice que un conjunto invariante ce-rrado A ⊂ Rn es un conjunto atractor si existe un cierto entorno U de A talque ∀t ≥ 0, φ(t, U) ⊂ U y ∩t>0φ(t, U) = A, donde φ es el flujo asociado alcampo vectorial x = f(x).

La primera condicion implica que existe un cierto entorno U tal que el futuro(t ≥ 0) de las soluciones inicialmente contenidas en el tambien esta contenidoen U . De esta forma, podemos definir el atractor como el espacio de fasescomun a dichas soluciones. Esta definicion en terminos de ∩t>0φ(t, U) esta es-pecialmente disenada para adaptarse al caso de orbitas cerradas, donde, adiferencia de los puntos fijos (en los cuales U y A coinciden), el interior deU no pertenece al atractor, siendo la unica region comun la propia orbita enla frontera. Por tanto, U se conoce como la region de atraccion, definiendola cuenca de atraccion como ∪t≤0φ(t, U), es decir, el pasado del conjunto desoluciones que convergen sobre A.

A partir del concepto de atractor, podemos generalizar las ideas sobreestabilidad de puntos fijos siguiendo la misma filosofıa:

Definicion (Variedad invariante.) Se dice que un conjunto invarianteS ⊂ Rn es una variedad invariante Cr (r ≥ 1) si S tiene la estructura deuna variedad diferenciable Cr.

En este punto, conviene explicar brevemente que entendemos por una va-riedad diferenciable. Ası pues, las variedades son la generalizacion de lassuperficies m-dimensionales embebidas en Rn, de forma que localmente man-tengan una estructura euclidiana. Esta condicion garantiza que, al menos enuna region de la variedad, siempre podemos establecer una carta o sistemade coordenadas cartesiano. Atendiendo a la necesidad de generalizar la geo-metrıa analıtica sin depender del sistema de coordenadas, las variedades sedefinen en base a espacios topologicos, que al estar formados por una co-leccion de entornos abiertos, no necesitan distancias (ni por tanto coordena-das) para establecer relaciones de vecindad. Adicionalmente, una estructuradiferenciable garantiza la existencia de una familia de cartas que recubrencompletamente la variedad y tal que la aplicacion de cambio de cartas es Cr

para garantizar que el cambio de coordenadas esta bien definido (cabe recor-dar que el Jacobiano de la transformacion depende de derivadas parciales deprimer orden). Por tanto, siempre podemos aplicar las reglas del calculo mul-tivariable (definido en espacios euclideanos) sobre una variedad diferencial,dado que siempre podemos definir un conjunto de cartas compatibles paracubrirla entera, garantizando que las funciones definidas sobre la variedad

104


diferenciable seran a su vez funciones diferenciables.

Este contexto permite generalizar y simplificar el estudio de la estabilidadde un punto fijo x ∈ Rn del sistema no lineal x = f(x). Para ello, considera-mos una vez mas el sistema lineal asociado y = Ay, donde A ≡ Df(x). Ental caso, podemos descomponer Rn en la suma directa de tres subespaciosvectoriales lineales Es, Eu y Ec definidos en sus correspondientes bases:

Es = e1, ..., es,Eu = es+1, ..., es+u, (A.1)

Ec = es+u+1, ..., es+u+c,

donde e1, ..., es son los autovectores de A correspondientes a los autovalorescon parte real negativa, es+1, ..., es+u son los autovectores de A correspon-dientes a los autovalores con parte real positiva y es+u+1, ..., es+u+c son losautovectores de A correspondientes a los autovalores con parte real cero. Portanto, Es, Eu y Ec son los subespacios estable, inestable y central respec-tivamente, tal que Es ⊕ Eu ⊕ Ec = Rn con s + u + c = n. De hecho, lostres son variedades diferenciales invariantes del campo vectorial y = Ay ypor tanto toda solucion cuyas condiciones iniciales esten contenidas en Es,Eu o Ec permanecera en ella para todo t. De esta forma, el estudio de laestabilidad del punto fijo x (correspondiente a y = 0) del sistema lineal sereduce al estudio de dichas variedades diferenciales invariantes. ¿Que sucedecuando consideramos el sistema no lineal de partida? En este caso, x = f(x)se puede expresar como

y = Ay +R(y), y ∈ Rn, (A.2)

donde R(y) = O(|y|2) contiene todos los terminos no lineales. Por simplici-dad, adoptamos el sistema de referencia que diagonaliza la parte lineal de(A.2) en bloques tales que

α

βγ

=

As 0 00 Au 00 0 Ac

αβγ

, (A.3)

donde As es una matriz s × s cuyos autovalores tienen parte real negati-va, Au es una matriz u × u cuyos autovalores tienen parte real positiva yAc es una matriz c × c cuyos autovalores tienen parte real cero. Ası mis-mo, α = (α1, ..., αs)

T , β = (βs+1, ..., βs+u)T y γ = (γs+u+1, ..., γs+u+c)

T . Deesta forma, los s primeros autovectores de la matriz por bloques forman la

105


base del subespacio vectorial lineal estable asociado al sistema de coorde-nadas α, los u siguientes autovectores forman la base del subespacio vec-torial lineal inestable asociado al sistema de coordenadas β y los ultimosc autovectores forman la base del subespacio vectorial lineal central aso-ciado al sistema de coordenadas γ. El cambio de base viene dado por lamatriz T = (e1, ..., es, es+1, ..., es+u, es+u+1, ..., es+u+c) de autovectores apro-piadamente ordenados, tal que y = Ty′, donde y′ = (α, β, γ)T . Por tanto,

y′ = T−1y = T−1ATy′ + T−1R(Ty′), (A.4)

es decir,

T−1AT =

As 0 00 Au 00 0 Ac

,

T−1R(Ty′) =

Rs

Ru

Rc

.

(A.5)

Sustituyendo (A.5) en (A.4) obtenemos finalmente

α = Asα +Rs(α, β, γ),

β = Auβ +Ru(α, β, γ),

γ = Acγ +Rc(α, β, γ).

(A.6)

En base a esto, podemos definir el analogo a las variedades estable, ines-table y central en el caso no lineal. Para ello, supongamos que (A.6) esCr, con r ≥ 2. Entonces, el punto fijo (α, β, γ) = 0 posee localmente unavariedad diferenciable Cr s-dimensional estable W s

loc(0), una variedad dife-renciable Cr u-dimensional inestable W u

loc(0) y una variedad diferenciableCr c-dimensional central W c

loc(0), tal que todas coinciden y son tangentes alos respectivos subespacios invariantes del campo vectorial linealizado en elorigen. Gracias a su estructura diferenciable, podemos definir una serie defunciones Cr que nos permiten determinar localmente la forma de W s

loc(0),W uloc(0) y W c

loc(0). Ası pues, para |α|, |β| y |γ| suficientemente pequenos, secumple respectivamente que

W sloc(0) = (α, β, γ) ∈ Rs × Ru × Rc|β = hsβ(α), γ = hsγ(α);

Dhsβ(0) = 0, Dhsγ(0) = 0W uloc(0) = (α, β, γ) ∈ Rs × Ru × Rc|α = huα(β), γ = huγ(β);

Dhuα(0) = 0, Dhuγ(0) = 0W cloc(0) = (α, β, γ) ∈ Rs × Ru × Rc|α = hcα(γ), β = hcβ(γ);

Dhcα(0) = 0, Dhcβ(0) = 0

106


Como se puede ver, cada variedad diferenciable esta expresada localmenteen sus variables propias. De esta forma, W s

loc(0) depende unicamente de lasvariables α a traves de las funciones hsβ y hsγ, proyectando el flujo vectorialno lineal sobre el subespacio vectorial no lineal asociado a los autovalores conparte real negativa. Ası mismo, la condicion Dhsβ(0) = 0, Dhsγ(0) = 0 implicaque el plano tangente de W s

loc(0) en el origen no tiene componentes β ni γ, ypor tanto coincide con el subespacio vectorial lineal Es(α). El mismo analisisse extrapola a W u

loc(0) y W cloc(0).

Queda ahora mostrar la utilidad formal y practica de semejante apara-to matematico. En un entorno del punto fijo, las trayectorias contenidas enW sloc(0) y W u

loc(0) tienen el mismo comportamiento asintotico que aquellastrayectorias contenidas en Es y Eu respectivamente. Por tanto, el estudiode la estabilidad de puntos fijos hiperbolicos en sistemas n-dimensionales sereduce a estudiar las variedades invariantes del sistema linealizado. No obs-tante, es en el estudio de la estabilidad de puntos fijos no hiperbolicos dondeeste formalismo nos proporciona una nueva forma de enfrentar el problemade la estabilidad. Ası pues, cuando eliminar los terminos no lineales se vuelveineficaz, la existencia de una variedad central permite reducir la dimensiona-lidad del sistema proyectando el campo vectorial no lineal sobre W c

loc(0). Estatecnica es especialmente util cuando u = 0, es decir, Rn = W s

loc(0)⊕W cloc(0),

es decir, donde coexiste un subespacio de soluciones asintoticamente establescon la variedad central, pues la dinamica del sistema converge rapidamen-te a la variedad central debido al decaimiento exponencial de las solucionescontenidas en W s

loc(0). En base a esto, desarrollaremos a continuacion unmetodo que permite estudiar la estabilidad de puntos fijos no hiperbolicoscuya importancia se volvera manifiesta en la siguiente seccion, dedicada a lateorıa de bifurcaciones.

Para ello, consideramos x = f(x) en un sistema de coordenadas tal que

x = Ax+ f(x, y),

y = By + g(x, y), (x, y) ∈ Rc × Rs,(A.7)

donde

f(0, 0) = 0, Df(0, 0) = 0,

g(0, 0) = 0, Dg(0, 0) = 0.(A.8)

Este sistema de referencia permite separar la parte lineal de la no lineal,siendo A una matriz c × c cuyos autovalores tienen parte real cero y B unamatriz s× s cuyos autovalores tienen parte real negativa, de forma que Ec⊕

107


Es = Rn. En estas condiciones, la variedad central se puede expresar en unentorno suficientemente pequeno de x como

W cloc(0) = (x, y) ∈ Rc × Rs|y = h(x), h(0) = 0, Dh(0) = 0.

Por tanto, la dinamica del sistema no lineal restringida a la variedad diferen-ciable central en un entorno del punto fijo resulta ser

γ = Aγ + f(γ, h(γ)), γ ∈ Rc, (A.9)

donde la variable γ se utiliza para recalcar que hemos proyectado el cambiovectorial sobre la variedad central, siendo esta, en general, una hipersuper-ficie no lineal. Por tanto, encontrar una expresion analıtica de la variedadcentral se reduce a hallar la funcion y = h(x). A su vez, conocer la estabili-dad del sistema reducido en un entorno de γ = 0 es suficiente para conocerla estabilidad del sistema general en un entorno del punto fijo (x, y) = (0, 0).Este resultado es particularmente valido para puntos fijos y orbitas cerradas(i. e. orbitas periodicas, homoclınicas o heteroclınicas) siempre y cuando seencuentren suficientemente cerca del origen o sean orbitas suficientementepequenas respectivamente, garantizando ası que estos conjuntos invariantesse encuentran sobre la variedad diferenciable central. El metodo para obtenerh(x) se ilustra en el siguiente ejemplo:

Sea el campo vectorial bidimensional

x = x2y − x5,

y = −y + x2.

El origen es un punto punto fijo y el Jacobiano asociado es

J(0, 0) =

(0 00 −1

),

donde J ya esta diagonalizado y por tanto λ1,2 = 0,−1. Segun el analisis deestabilidad lineal, el origen es un punto fijo estable, si bien no asintoticamenteestable. Veamos como el calculo de la variedad central permite descubrir queesto no es cierto. Dado que hay un autovalor nulo, la variedad central esunidimensional, y por tanto vendra dada localmente por la funcion y = h(x).Vamos a establecer ahora la condicion necesaria para que h(x) determinela variedad central. En un entorno suficientemente pequeno del origen, lavariedad central ha de ser tangente a Ec. Por tanto, y dado que el autovalornulo esta asociado a la componente x, el espacio tangente asociado a la

108


variedad central, y = Dh(x)x ha de ser el mismo que el espacio tangenteasociado a Ec, y = −y + x2 = −h(x) + x2. Igualando ambas ecuaciones,

Dh(x)x = Dh(x)(x2h(x)− x5) = −h(x) + x2,

o lo que es lo mismo,

x2Dh(x)h(x)− x5Dh(x) + h(x)− x2 = 0.

Para resolver esta ecuacion diferencial, usaremos el metodo de soluciones porseries de potencias. Por tanto, desarrollando en serie de Taylor la variedadcentral y sabiendo que h(x) debe cumplir h(0) = Dh(0) = 0, se obtieneh(x) = ax2 + bx3 +O(x4). Sustituyendo,

x2(2ax+ 3b2 +O(x3))(ax2 + bx3 +O(x4))−−x5(2ax+ 3b2 +O(x4)) + ax2 + bx3 − x2 +O(x4) =

= ax2 + bx3 − x2 +O(x4) =

= (a− 1)x2 + bx3 +O(x4) = 0.

Por tanto, a = 1 y b = 0, y la expresion de la variedad central resultah(x) = x2+O(x4). Conocida la variedad central, podemos proyectar el campovectorial sobre ella, quedando reducido a

x = x4 +O(x4),

y = O(x4).

La componente y desaparece, como cabıa esperar al ser ortogonal Ec, mien-tras que el origen resulta ser inestable (o medio estable segun algunas termi-nologıas), pues una pequena perturbacion en x esta asociada a una velocidadsiempre positiva y a un flujo en la direccion positiva del eje x.

Por ultimo, formalicemos este resultado para el caso mas general de cam-pos vectoriales con parametros. Si recuperamos la descomposicion inicial delproblema, la presencia de un parametro µ ∈ Rp se puede introducir comouna variable extra en el sistema:

x = Ax+ f(x, y, µ),

µ = 0,

y = By + g(x, y, µ), (x, µ, y) ∈ Rc × Rp × Rs.

(A.10)

Por tanto, el punto fijo sera ahora (x, y, µ) = (0, 0, 0), el Jacobiano corres-pondiente a (A.10) tendra c+p autovalores con parte real cero y la reduccion

109


sobre la variedad central y = h(x, µ) central tomara la forma

u = Au+ f(u, h(u, µ), µ),

µ = 0, (u, µ) ∈ Rc × Rp.(A.11)

La condicion de tangencialidad se expresa como

y = Dxh(x, µ)x+Dµh(x, µ)µ = Bh(x, µ) + g(x, h(x, µ), µ). (A.12)

Sustituyendo una vez mas (A.10) en (A.12), se llega finalmente a la ecuacionque que debe satisfacer h(x, µ) para ser la variedad central:

Dxh(x, µ) [Ax+ f(x, h(x, µ), µ)]−Bh(x, µ) + g(x, h(x, µ), µ) = 0. (A.13)

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Documents

TRANSICIONES IRREVERSIBLES EN REDES COMPLEJAS · en redes complejas (Pikovsky et al., 2002; Osipov et al., 2007). Este proce-so, inherente a sistemas con un gran numero de elementos