Fenómenos de propagación en redes complejas' · Fenómenos de propagación en redes...

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Tesis Doctoral

Fenómenos de propagación en redesFenómenos de propagación en redescomplejascomplejas

Medus, Andrés Daniel

2015-06-12

Este documento forma parte de la colección de tesis doctorales y de maestría de la BibliotecaCentral Dr. Luis Federico Leloir, disponible en digital.bl.fcen.uba.ar. Su utilización debe seracompañada por la cita bibliográfica con reconocimiento de la fuente.

This document is part of the doctoral theses collection of the Central Library Dr. Luis FedericoLeloir, available in digital.bl.fcen.uba.ar. It should be used accompanied by the correspondingcitation acknowledging the source.

Cita tipo APA:

Medus, Andrés Daniel. (2015-06-12). Fenómenos de propagación en redes complejas. Facultadde Ciencias Exactas y Naturales. Universidad de Buenos Aires.

Cita tipo Chicago:

Medus, Andrés Daniel. "Fenómenos de propagación en redes complejas". Facultad de CienciasExactas y Naturales. Universidad de Buenos Aires. 2015-06-12.

❯❱ ❯

t ♥s ①ts ② trs

♣rt♠♥t♦ ís

♥ó♠♥♦s ♣r♦♣ó♥ ♥ rs ♦♠♣s

ss ♣rs♥t ♣r ♦♣tr tít♦

♦t♦r ❯♥rs ♥♦s rs ♥ ár ♥s íss

♣♦r

♥rés ♥ s

rt♦r ss r ♦ sr ♦rs♦

♦♥sr♦ st♦s r P♦ ♥ ♥♥♥

r r♦ ♣rt♠♥t♦ ís ② ❯

♥♦s rs

s♠♥♥ st tss st♠♦s ♦s ♠♣♦s ♥ó♠♥♦s ♣r♦♣ó♥ s♦r rss♦s ♠♦s ♥♦s ♦♥ s ♣ú ♥ ♣r♠r♦ ♦s ♥③♠♦s t♦ s♠♥ó♥ ♦♣♥ó♥ rs♣t♦ ♥ó♥ s♦r ♥r s♥ ♥tr ♣♦r r♣♦s ♠rs ♥ s♣írt ♠♦♦ ①r♦ ♣r s♠♥ó♥ tr ♥ ♥ s♦ ♦♥sr♠♦s ♦♥t ♥ó♥ ♦♠♦ ♥ s rtrísts q ♥♥ r♦ tr ♦s♥ts s♦s st♠♦s s ♦♥s♥s ♦♥r ♣qñ♦s r♣♦s♥t♥ó♥ ♥ s♠♥ó♥ ♦♣♥ó♥ ♦♥trr ♥ó♥ ②♦♥ ♦ ♥ ♦rtr ♥ ♥ ♥ ♥ ♣rtr ♥♦s ♥trs♠♦s ♣♦r s♦ ♥ r♣ ❱ír ♦♥tr sr♠♣ó♥ r♦ ② s♣♣rs ♦ r♥t rsr♠♥t♦ sr♠♣ó♥ ♥ ♣íss ♦♥ ♠♣s♣♦♥ ♥s ♦♠♦ ❯❯ ♥trr ② s ♥tr ♦tr♦s ♦str♠♦s q ♠♥s♠♦ s♠♥ó♥ tr qí ♥tr♦♦ ♣r♦♠ ♦r♠ó♥ ♦♠♥♦s ♦ strs ♥♦♥♦rs ss♣ts ♦♥trr sr♠♣ó♥ ♥r♠♥t♥♦ ♣r♦ r♦ts ú♥ ♥ q♦s s♦s ♦♥s ♠♣ ♦♥ ♠t 95% ♦rtr ♥ ♣r♦♣st ♣♦r r♥③ó♥ ♥

P♦r ♦tr ♣rt ♦♥sr♠♦s ♥ r♥t ♠♦♦ r s♦ ♦♥s ♦♣♥♦♥s ♣♥ ♣r♦♣rs t♥t♦ trés ♥s ♣rs♦♥s ♦♠♦ ♥♦♣rs♦♥s ♠♦♦ q st♦s út♠♦s ♥♦ ♣r♠t♥ ♦♥t♦ ♥r♠♥q sí s♠♥ó♥ ♦♣♥♦♥s s♠s♠♦ st♠♦s t♦ ♦♥srr rs ♦♥ s♠♥ó♥ ♦♣♥♦♥s s ♦♠♣ñ ♣♦r ♥ ♣r♦s♦ ♣tó♥ strtr trés r♦♥①♦♥♦ ♥s

s♥♦ ♥ó♠♥♦ ♥③♦ ♦rrs♣♦♥ ♣r♦♣ó♥ sr♠♣ó♥s♦r ♥ r s♦ Pr ♦ srr♦♠♦s ♥ ♠♦♦ ♥r♦r rs s♦s ♦♥ rtrísts ♦♠♣ts ♦♥ s r♣♦rts ♣r rs ♦t♥s trés ①♣r♠♥t♦s s♦s ♦ ♠♥t ♥áss rs ♦♥♥ ♦♠♦♦♦ ♦str♠♦s q rtrísts strtrs r t♥ ♣r♦ ♣rsst♥ sr♠♣ó♥ s♦r ♠s♠Prs rs s♦s s♠♥ó♥ ♦♣♥♦♥s ♥ó♥ ♠♦♦s s♦s ♥ ♥ts ♠♦♦s r♠♥t♦ rs ♦♠♣s ♣♠s

♣r♥ ♣♥♦♠♥ ♥ ♦♠♣① ♥t♦rs

strt♥ ts tss st② t♦ ①♠♣s ♦ s♣r♥ ♣♥♦♠♥ ♦♥ s♦ ♥t♦rs♦t rt t ♣ t ♥ t rst ♦♥ ♥②③ t t ♦ ts♣r ♦ ♦♣♥♦♥ rr♥ ♥t♦♥ ♦♥ ♠② r♦♣s ♦♠♣♦s♥ s♠♣s♦ ♥t♦r ❲ ♦♥sr t ♥t♦♥ ♦r ♦ ♠② r♦♣s s ♦♥ ♦t ♥♥ rtrsts ♦ tr tr rt ♥ t s♣rt ♦ ①r♦s♠♦ ♦r t st② ♦ tr ss♠♥t♦♥ ♥ ♠♥ s♦ts ❲ st② t♦♥sq♥s ♦ t t♦♥s ♦ s♠ ♥t♥t♦♥ r♦♣s ♥ t ss♠♥t♦♥♦ ♦♣♥♦♥s ♥st ♥t♦♥ ts ♦♥ ♥ ♦r ♥ ♣rtr r♥trst ♥ t s ♦ t ♥ ♥st ♠ss r ♥ ♠♠♣s t♦ t r♥t ♠ss rsr♥ ♥ ♦♥trs t s♣r t②♦ ♥s s s ❯ ♥♥ ♥ ❲s ♠♦♥ ♦trs ❲ s♦ tt t♠♥s♠ ♦ tr ss♠♥t♦♥ ♥tr♦ r ♣r♦♠♦ts t ♦r♠t♦♥ ♦♦♠♥s ♦r strs ♦ ♥♦♥♥t♦rs t♥ ss♣t t♦ ♠ss ♥rs♥ t♦trs ♣r♦t② ♥ ♥ t♦s ss r t ♦ ♦ 95% ♥♦r r♦♠♠♥ ② t ❲♦r t r♥③t♦♥ ❲ s

♦r♦r ♦♥sr r♥t ♦ ♦r s♦ ♥t♦r ♠♦ r ♦♣♥♦♥s♥ s♣r tr♦ ♦t ♣rs♦♥ ♥ ♥♦♥♣rs♦♥ ♥s s♦ tt t ttr ♦♦r t ss♠♥t♦♥ ♦ ♦♣♥♦♥s t ♦s ♥♦t ♦ t s♣r ♦ t ss ❲s♦ st② t t ♦ ♦♥sr♥ s♦ ♥t♦rs r t s♣r ♦ ♦♣♥♦♥ss ♦♠♣♥ ② strtr ♣tt♦♥ ♣r♦ss tr♦ t rr♥ ♦ ♥s

s♦♥ ♣♥♦♠♥♦♥ sss r ♦rrs♣♦♥s t♦ t s♣r ♦ ♠ss ♦♥ s♦ ♥t♦r ♦r ts ♠ ♦♣ ♠♦ ♦r s♦ ♥t♦rsr♦t t trs ♦♠♣t t t♦s r♣♦rt ♥ trtr ♦r ♥t♦rs♦t♥ tr♦ s♦ ①♣r♠♥ts ♦r r♦♠ ♦♥♥ s♦ ♥t♦rs s s ♦♦ ❲ s♦ tt t♦♣♦♦ rtrsts ♦ t ♥t♦r t ♠ss♣rsst♥ ♦♥ t t♦♣ ♦ t♠②♦rs s♦ ♥t♦rs s♣r ♦ ♦♣♥♦♥s ♥t♦♥ ♥ts ♠♦♥ ♦♠♣① ♥t♦rs r♦t ♠♦s ♣♠s

r♠♥t♦s

Psr♦♥ r♦s ñ♦s ♠♦ ♣r♥③ ♦ss q sr♦♥ ♥ ♦trs q♥♦ t♥t♦ ♣r♦ ♦ ♥ st ♠♥♦ P♦r ♦ ♦ rr rs♣rs♦♥s q r♦♥ st t♣ ♥ ①♣r♥ tr♥s♦r♠♦r

♥ ♣r♠r r r③♦ ♦ ♣♦r r♠ rt♦ s ♣rts sr♣♦ ♣♦r ♦r♥tr♠ ② ♣r♠tr♠ trr ♥ ♥ ♠♥t st♠♥t s♠♣r♣♥♥t ♦s ♣r♦②t♦s ♦♥ ♥tss♠♦ ② ♥rí ♥s ♦s♣♦ q r sts í♥s strá s♦r♣r♥♦ ② q ♥♥ ♥ t♦♦s st♦s ñ♦s trtr♦♦ ♠é ♣♦r s ♥♦♠r ♣ s♦ ♥ r♥ ①♣r♥ ♦♠♣rtr st♦sñ♦s ♥t♥s♦ tr♦ ♦♥ é rs ♣♦r t♦♦ ♦

r③♦ ♣rt♠♥t♦ ís ② t♦♦s ss ♠♠r♦s q tr♥♥♥s♠♥t ♣r q s ♦ss ♥♦♥♥ ás á ♥♦s sr♦s ② ♦ss ♥ s q t♦♦s ♠♦s ♦♥r ♦t♦ r♥r ♥ó♥ ♣ú rtt ② ①♥ s s♥ s ♥ s ♦ ①t♥s♦ st r♠♥t♦ ❯♥rs ♥♦s rs ♣♦r r♥r♠ ♣♦s str ♥ ♦r♠ rtt ② ♥♥r ♠ ♦t♦r♦

♥ st ♠♥♦ t srt r③r♠ ♦♥ ♥s ♣rs♦♥s q tr♦♥ ♠ ①st♥ é♠ ❯♥ s s r♦ tr♦ ♥ r♥ ♦♠♣ñr♦ ♦♥ ② ♠♦r ♠♦ t♥ r♥t ♦♠♦ ♠ ♠ás ①♥t ♣rs♦♥ ♠é♥ qr♦ rr ♥ r♠ ♣♦r ♦s rt♦s ♠♦♠♥t♦s s♦♥rs♦♥s t♥♦ós ② s ♥s♦♥s ①óts ♦♠♣rts ♠t ♦♦s♦r rt P rr♦ ♦tr ①♥t ♦♠♣ñr ♦♥ ② ♠ ♥és ♥ ♠ q s♠♣r st♦ ♥ ♦s ♠♦♠♥t♦s ís ♦s ♦♠♣ñr♦s ♦tr ♦♥ Prt♦ ♥ ② P♦ P♦ ♥③ ♣♦r ♣♦②♦ rá♥ PP

r③♦ ♠s ♣rs ② r♠♥ ♣♦r ♣♦②♦ ♠s tís ♠ r♥ ♠❱r♦ ♣♦r ♦♠♣ñr♠ ♦♥ s ♣rs♥ t♥t♦ r ♦♠♦ rt r♥t♦ ②á♥ ♣♦r str í ♥q ♥♦ ♥♦s ♠♦s ♠② s♦

Pr ♥ ♦ ♠ás ♠♣♦rt♥t st tss stá r ♠ ♦♠♣ñr rs ♠♦r ♣♦r str s♠♣r ♣r ♣♦②r♠ ② s♦st♥r♠ ♥♦♥♦♥♠♥t ♣♦r r♠ ♣③ ② ú♥ ♠ás ♠♣♦rt♥t ♣♦r ♥sñr♠ í í í rt r

❮♥ ♥r

♥tr♦ó♥ s♠♥ó♥ ♥ó♥ ♦♥tr

❱♥ó♥ ② rs s♦s ♦♥tt♦s ♣rs♦♥s ② ♥♦♣rs♦♥s

♦♥t ♥ó♥ ♦♠♦ ♥ rs♦ tr ♦♦ ①r♦

Pr♦♣s státs ② ♥á♠s s rs s♦s Prsst♥ sr♠♣ó♥

♥s ♣rsst♥ ♦♥ stró♥ t♠♣♦s t♦ strtr r s♦ ♥ ♣rsst♥

strtr tss

r ♥tr♦ó♥ s rs ♦♠♣s s státs ♣r♦♣s t♦♣♦ós

stró♥ r♦ st♥ ♠ ♥tr ♥♦♦s ♦♥t str③ó♥ r♦ ♠♦ ♣r♠r♦s ♥♦s strtr ♦♠♥s

só♥ t♦♣♦ó rs ♦♠♣s s t♦rs s ♥♦ Pqñ♦ s rs s

s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r stát ♦♦ r s♦ ♦♦ s♠♥ó♥ tr ② ♥ó♥

s♠♥ó♥ ♦♣♥ó♥ rs♣t♦ ♥ó♥ ♦♠♥t♦s ♥t♥ó♥

sr♣ó♥ ♦rít♠ ♠♦♦ r♥só♥ ♥♦rs ♥♦♥♦rs ♥áss r♦ts sr♠♣ó♥ ♥áss strs

♥♥ s♦r str ♠á①♠♦

❮♥ ♥r

strs ♠á①♠♦s Pr♦♣s t♦♣♦ós

t ♦rtr ♥

♦♥s♦♥s ♣r♠♥rs

s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ ♥r③♦

♦♥tt♦s ♣rs♦♥s ② ♦♣♥ó♥

♥s ♣tt♦s ♠♥s♠♦ r♦♥①♦♥♦

sr♣ó♥ ♦rít♠ ♠♦♦ ♥r③♦

stát t♦s tr♦♥ ♥s

♦r♠ó♥ strs

strs ♠á①♠♦s

ó♥ s ♠♣ñs ♥ó♥

♦♦ ♥r③♦ ♦♥ ♠♣♦ ①tr♥♦

rr♦ ♥ q ② φ

ó♥ φ s♦r ♦s ♠♦♦s ♦♥ r stát ② ♣tt

♥♥ φ ♥ t♦♣♦♦í ♦s strs ♠á①♠♦s

♦♦ ♥r③♦ ♦♥ t ♦rtr

♦♥s♦♥s ♣r♠♥rs

♥ró♥ rs s♦s ♦♥ s ♠♣ír

t s♦ ♦♠♦ ♥r♦r tr♦♥s strtrs

♦♦ st♦ást♦ ♥r③♦

t♦s ♠♠♦r ♥♦♠r♦♥♦s rr trá♦

♦r♠ó♥ ♥ít ♠♦♦

P♦ó♥ ♦♥st♥t γ = 0

P♦ó♥ r♥t

♦rr♦♥s r♦ t♦ ♦r♥

r♦ ♠♦ ♣r♠r♦s ♥♦s

♦♥t str③ó♥

s♦s st♦

Pr♦♣s t♦♣♦ós

♥s♠♦ ♥ rs rs

Prsst♥ sr♠♣ó♥ s♦r rs s♦s

♠♦♦

r s♦

ss tr♥só♥

❮♥ ♥r

Ps♦ t♠♣♦r ♣r♦①♠ó♥ ♠t♥♦♠

♥áss ♣rsst♥ ①t♥♦♥s ♣♦st♣é♠s ♦♥ ♠♣♦rtó♥ ♣r♠♥♥t

♥t♦s

♦♥s♦♥s ♥s ② Prs♣ts

♦♦ ♣♠♦ó♦ ♦♦ ♦♠♣rt♠♥t st♦ást♦

r ♥tr♦ó♥ ♠♦♦ rs ♦ts ♦♦ rs ♦ts

♦rí

♣ít♦

♥tr♦ó♥

❮♥ s♠♥ó♥ ♥ó♥ ♦♥tr

❱♥ó♥ ② rs s♦s ♦♥tt♦s ♣rs♦♥s ② ♥♦

♣rs♦♥s

♦♥t ♥ó♥ ♦♠♦ ♥ rs♦ tr ♦♦

①r♦

Pr♦♣s státs ② ♥á♠s s rs s♦s

Prsst♥ sr♠♣ó♥

♥s ♣rsst♥ ♦♥ stró♥ t♠♣♦s

t♦ strtr r s♦ ♥ ♣rsst♥

strtr tss

s s♦s ♠♥s ♦♥stt②♥ ♠♣♦s ♥♦ts sst♠s ♦♠♣♦sst♦s r♥s ♥tr♠♦s ♥♦s ♥♦s ♥ r ♦♠♣♦rt♠♥t♦s ♠r♥ts ♠s s ♠♣rst♦s ♦♥srr ♦s s♥♦s ♠♥s♠♦s♠r♦só♣♦s ♥♦r♦s st♦ s q t♥t♦ ♣tró♥ ♦♠♦ ♥tr③ s ♥tr♦♥s ♥tr♥♥ts ♥tr ♥ r♥ ♥ú♠r♦ ♥ts s♦ss♠♣ñ♥ ♥ r♦ ♥♠♥t ♥ srr♦♦ ♦♠♣

♥áss ♥ttt♦ ♦s ♥ó♠♥♦s ♣r♦♣ó♥ ♥r♠sr♠♦rs ♦ rs♦s trs ♥tr ♥♦s s♦ tr♦♥♠♥t ♦r♦s ♦s ♠♦s ♠♦♦s ♠♣♦ ♠♦ st♦s ♣♦st♥ ♥ r♥ s♠♣ó♥ ♥ ♠♦♦ ♠t♠át♦ s♦s②r ♥s s tr♦♥sq ♥tr♠♥t ♣rs♥t♥ ♦s sst♠s s♦s

♥tr s tr♦♥s ♥♠♥ts s ♥♥tr rt strtr s②♥t í♥♦s ♥tr st♦s strtr s r♣rs♥trs trés ♥ r ♦♠♣ ♦♥ ss ♥♦♦s ♥t♥ ♦s ♥ts s♦s♠♥trs q ♦s ♥s q ♦s ♦♥t♥ ♣♥ r♣rs♥tr í♥♦s ♦rs♦ ♠rs ♠sts ♣r♦①♠ s♣ ♦ ú♥ r♦ s♠t ♥tr ♦s

♣ít♦ ♥tr♦ó♥

♠s♠♦s ♥ rts r♥st♥s s tr♦♥s ♠♣t♥ rt♠♥t t♥t♦ ♥ ♦ó♥ ♦♠♦ ♥ rst♦ ♥ ♥ó♠♥♦ ♣r♦♣ó♥♥③♦ t♦r♥á♥♦s ♥♣r♦♣♦ s trt♠♥t♦ ♣rtr ♠♦♦s ♠♣♦♠♦

♥③r ♠r♦s ♥tr♠♦ s♦ s t♠♥t ♥ tr♣♦s rs ♥♠♥t♦ s ♥s t♥♦♦ís ♥♦r♠ó♥ ♥áss ♣rt♣ó♥ ♥ rs s♦s ♦♥♥ ❬③r ♥t♦ ❯♥r ③ ❪ ♣tró♥ ♠s ♣♦r t♦♥ír ❬♦♥③③ ♦♥ ❪ st♠ P♦s♦♥♠♥t♦ ♦ P② s t♥♦♦ís ♦♠♥ó♥ ♥á♠r ♠♣♦ r♥♦ s♣♦♥s ♥♦s ♥♦s té♦♥♦s rs ❬ ♦ r♥ ❪t♦♦s ♦s ♦♥tr②♥ s♥trñr s ②s ♠r♦só♣s q ♥ ♦r♠ ♦r♥③ó♥ s s♦s ♠♥s s♠s♠♦ ♥ ♦♠♥③♦ srr♦rs r♥ts ①♣r♠♥t♦s t♥♥ts str ♦ó♥ ♦s ♦♥tt♦s s♦s s♦r ♣♦♦♥s ♣qñs ♥♦s ♥ s♣♦s rr♦s❬ttt♦ s ❪ ♦ s ♥♦r♠ ♦♠♥ st♦s t♦s s♦♥ s♦s ♥tr♦ t♦rí ♥♦♠♥ t ② ís stíst ♣r♦té♥s s ♣r s st♦

♦t♦ st tss s ♥③r ♦s ♣r♦♠s ♥♦s ♦♥ s ♣ú ♦♥ strtr r s♦ ♦♥♦♥ ♥ó♠♥♦ ♣r♦♣ó♥♣rtr q t♥ r s♦r ♦♥♠♥t ♦r♠r♠♦s ♥ ♠♦♦ r♠♥t♦ rs q ♦♥ ♣tró♥ ♦ó♥ ♦sr♦ ♣r ♦s♦♥tt♦s s♦s ♥ st♥t♦s ①♣r♠♥t♦s ♦♥ s ♣r♦♣s státs srs ♦♥strs ♣rtr ♦s ♥s ♠♦s ♥ t♠♣♦

s♠♥ó♥ ♥ó♥ ♦♥tr

♥ó♥ s ♠♦ ♠ás ③ ♣r ♦♠tr ♣r♦♣ó♥ ♥r♠s ♥♦ss ♥ srs ♠♥♦s s♦♦ trás stró♥ ♣♦t ❬P♦t♥ ❪ ó♥ ♣r♠t♦ rró♥ r ② ♦♥tr♦ ♦ ♦trs ♥r♠s ♥♦ss ♦♠♦ sr♠♣ó♥ r♦s ♣♣rs ♣♦♦♠ts ② tét♥♦s ❬♥r r ❪ ❯♥ ♠♣♦♥♦t q ♥t t♦ ♥ó♥ s ♦sr ♥ r ♦♥ s r♣rs♥t ♥♥ sr♠♣ó♥ ♠ ♥♠♥t ♥ ♥♦ ❯♥♦ ♥ts ② s♣és ♠♥stró♥ ♣r♠r ♥ s♠♣♥ sí ♦♠♦ t♠é♥ ♥tr♦ó♥ ♥ r♣ ❱ír ❱♦♥tr sr♠♣ó♥ s ♣♣rs ② r♦ ♥

s♠♥ó♥ ♥ó♥ ♦♥tr

1940 1950 1960 1970 1980 1990 2000 2010 2020Año

r s♦s sr♠♣ó♥ r♣♦rt♦s ♥♠♥t ♥ ♥♦ ❯♥♦ ♥tr ♥ í♥ tr③♦s s sñ♥ ♦s ñ♦s ♦rrs♣♦♥♥ts ♥tr♦ó♥ s ♥s s♠♣ ② r♣ ❱ír ❬P❪ ♥♦r♠ó♥ ♣ú s♣♦♥♥ st♦ P t ♥♥ tt♣♣♦r

♣sr s ♠♦str ♦rtr ♥ ❱ s♠♥♦ ♥ ♥♦s ♣íss srr♦♦s ♦♥ ♠♣ s♣♦♥ ♠s♠♥♦ r r♥ts r♦ts sr♠♣ó♥ ♦♠♦ ♦s ♦s ♥ s ❬❲ ❪ ② ♦s st♦s ❯♥♦s ❬stñ② ❪ st♠ qt s♠♥ó♥ ♥ ♦rtr s ♦r♥ só♥ ♥ st♦ rá♣♠♥t rt♦ q stí ♥ ♦rró♥ ♥tr ♠♥stró♥ ♥ ② s♦s ts♠♦ r♦ ♥ ♥ñ♦s ♣qñ♦s ❬❲ ②♦r ❪st ♦♥tr♦rs ♦♥stt② ♥ ♠♣♦ ♦♥rt♦ q ♣♦♥ ♠♥st♦ ♠♣♦rt♥ s♠♥ó♥ ♦♣♥♦♥s ♥ ♣r♣ó♥ q ♦s ♥♦ssrr♦♥ r ♥ ♥ ❬rss r♦♥ ❪

♥ q♦s s♦s ♦♥ ♥ó♥ s ♦♣♦♥ ♦ ♠♥♦s ♦♥ ①st♥①♥♦♥s ♣♦r r③♦♥s ♦♥♥ ♣♦r ♠♣♦ ♥ ♥♦s st♦s ♦s❯❯ s♦♥ ♦s ♣rs q♥s ♥ rs♣t♦ ♥ó♥ ss ♦s♦ ♦ ♥ ♦♠♦ ♥♦s s♦s s♥♦ ♦♠♦ ♣rt ♥ strtr s♦q ♥ ♥ ss ♦♣♥♦♥s ② s♦r s ③ ♣♥ ♥r s ♣♦r

♦ q r♥t♠♥t ♦♠♥③♦ ♠♦rs s♠♥ó♥ ♦♥t

rs♣t♦ ♥ó♥ ♦♠♦ ♥ ♣r♦s♦ ♠tó♥ s♦ ♥ ♦♥t①t♦ ♠♦♦s ♠♣♦ ♠♦ ❬ ❪ st ♠♦♦ ♦s ♥♦s ♥ s♦♥t ♥t ♥ó♥ ♠t♥♦ ♦♠♣♦rt♠♥t♦ ♠♦ s ♥t♦r♥♦♦♥♠♥t ♦tr♦s ♠♦♦s s♠♥ ♥♦s r♦♥s ♣s r rs♦ t♥t ♦♥t♦ ♥r♠ ② ♦♥trstr♦ ♦♥ ♦s ♣♦sst♦s rs♦s ♥ ♥ ♦♥t①t♦ t♦rí ♦s ❬ ♥♦r♦ ❪

♥③r ♥ ♠♣ ♦rtr ♥ ♣♦r ♥ ♦ s♠♥② ♣r♦ r♦ts ♣é♠♦s ② ♣♦r ♦tr♦ ♣r♦t q♦s ♥♦s q♣♦r ú♥ ♠♦t♦ ♥♦ s ♦♥t ♥♦ ♥ ♣♦♦ ♥rs ♣♦r ♠♣♦ ♥ñ♦s ♠♥♦rs ñ♦ ♥♠♥♦♣r♠♦s ♥♦s ér♦s ♦s ♦♠♣st♦s ♥ t trés t♦ ♥♠♥ r♣♦ ♥ s♥ stút♠♦ s♠ q rs♦ ♦♥t♦ ♥ ♥♦ ss♣t s♠♥②♦♥ ♥♠♥③ó♥ s ♥t♦r♥♦ ♥ st s♥t♦ r♥③ó♥ ♥ r♦♠♥ ♦s ♣íss s♦♦s 95% ♦rtr ♥❱ ♥♦ s♦♦ ♦♥ ♦t♦ ♥③r ♥♠♥ r♣♦ s♥♦ ♣r ♦rr rró♥ ♥t sr♠♣ó♥ ❬❲ ❪

s ♥t q ♥ r♠♥t♦ ♥ ♦rtr ♥ ♠♣ ♥♥r♠♥t♦ ♣r♦ r♦t ♣é♠♦ sr♠♣ó♥ ♦ ♦st♥t♦s r♦ts t♠é♥ s ♥ sst♦ ♥ ♣íss ♦♥ t ♦rtr ♣r♦♠♦♥ ♦♥ ♦♥t ♦♥ ♥ 96% ♦rtr t♦t ❱ ♥ ♥ñ♦s ♠②♦rs ♠ss s♥ ♠r♦ sró ♥ r♦t sr♠♣ó♥ q ♥③ó s♦s r♦s ❬♦♥ ❪ ♥ r♦r ♣rs♦♥s sstr♦♥ ♥ r♥ó♥ ♥ ♥ ♦ ♥♥ ❯❯ ♥ st♦ ♦♥ ♦rtr ♥ ❱ r ♥ s ♥t♦♥s 92% ♥ ♥ñ♦s ♣rs♦rs ② 98%♥ ♥ñ♦s s①t♦ r♦ ❯♥ s sst♥ts r♥ó♥ r ♥ ♠♥♦rt♠r♥ ñ♦s q rrs ♥ ♠só♥ r♦s ♥ ♠♥♦♥ ♦♥tr♦ sr♠♣ó♥ ♦s sst♥ts r♥ó♥ ♣r♦①♠♠♥t 10% t♦t rí♥ ♥♠♥ ♦♥tr sr♠♣ó♥ ♦s s ♦♥trr♦♥ ♥r♠ r♥t s ss s♠♥s ♣♦str♦rs s rstrr♦♥ s♦s t♦ts ♦♥r♠♦s ♦s s 94% ♥♦ st♥ ♥♦s❬Prr ❪ r♦ts s♠♥ts s rstrr♦♥ ♥ ♥ ♦ ♦r♥ ❯❯♥ ❬r♠♥ ❪ ♥ tñ s♣ñ ♥ ❬♦♠í♥③ ❪ ②♥♠♥t ♥ ♦♥ t♠é♥ r♥t ❬♥ ❪ st♦s s s♠♥♦s ② ♠♥♦♥♦s ② ♠ás r♥ts r♦ts ♥ ❯❯ ② s ♥t♦♦s ♦s s r ♥ ♥♦♠♥♦r ♦♠ú♥ ♦s ♥♦s ♥♦♥♦s ♥♦ ♣

s♠♥ó♥ ♥ó♥ ♦♥tr

r♥ str♦s ③r ♥ ♣♦ó♥ s♥♦ q ♣♦r ♦♥trr♦ s ♥♥tr♥r♣♦s ♥ ♦♠♥s ♣♦r ♥ tr

❱♥ó♥ ② rs s♦s ♦♥tt♦s ♣rs♦♥s ②

♥♦♣rs♦♥s

♦r r♣ó♥ ♥♦s ♣♦r rs♦s trs rqr ♣rtrs ♣r♦①♠ó♥ ♣♦ó♥ t♦t♠♥t ♠③ rtríst ♦s ♠♦♦s ♠♣♦ ♠♦ st♦ ♣ ♦rrs r♣rs♥t♥♦ sstrt♦ s♦ trés ♥ r ♦♠♣ ♦♥ ♦s ♥ts s♦s ② ss ♥tr♦♥s ♦rrs♣♦♥♥ ♦s ♥♦♦s ② ♥s r rs♣t♠♥t ♥♦q ♠♣♠♥t♦ ♥ tr♦s r♥ts q ♣r♦♣♦♥♥ ♠♦♦s s♠♥ó♥ ♦♣♥♦♥s♣♦r ♠tó♥ s♦ ♦♥sr♥♦ ♥ ♦♣♥ó♥ ♥r rt ♥ó♥ tr♦ té t ❬té ❪ ♠str q ♦s ♥ts s♦s q♦♠♣rt♥ ♦♣♥ó♥ ♦♥trr ♥ó♥ s r♣♥ ♥ strs ♦ ♠♥s♠♦ ♠tó♥ s♦ st ♠♦♦ t♦ ♥♠♥ r♣♦rst rt♠♥t t♦

r ♠str ♣r♦ r♦t ♣é♠♦ sr♠♣ó♥ ♣rst♥t♦s ♣♦r♥ts ♦rtr ♥ ♦♠♣r♥♦ s♦ q ♥♦r♣♦r ♠♥s♠♦ ♦r♠ó♥ ♦♣♥ó♥ ♦♥ q ♦♥ ♦s ♥♦s ♥♦♥♦s s♦♥ str♦s ③r s♦r r P rs q str③ó♥ ♥♦♥♦s ♥r♠♥t ♥♦t♠♥t ♣r♦ r♦t t♦ qs ♠♥♦ ♣r ♦rtrs ts ♥ ♠r♦ ♥ ❬té ❪ ♠s ♥♦♥♦s ♥♦ s ♦t♥ ♥á♠♠♥t ♣r♦s♦ s♥♦ q s ♠♣st♥ ♠♦♦ ② s ♦♥sr ♦ r♦ ♣r♦s♦ ♦r♠ó♥ ♦♣♥ó♥

tr♦s ♠♦♦s ♣r♦♣st♦s t♠é♥ s♦s ♥ rs s♦s s♠♥ ♥ts r♦♥s q ♥ s ♦♥t rs♣t♦ ♥ó♥ ♥ ♦♥t①t♦ t♦rí ♦s ❬ ❳ r♦ ❪ ♥t ♣♦s ♥♦r♠ó♥ ♦ ♥♥ ♥r♠ ♥ sst♠ ② ♦♥♦ ♦srs♦s srr t♦s rs♦s ♦s ♥ ♦ t♦♠ ♥ só♥r♦♥ ♣rtr rt ♥ó♥ ♣♦ ❯♥ s♣t♦ str t♦♦s ♦s♠♦♦s st qí t♦s s q s♠♥ q ♥r♠ ② ♥♦r♠ó♥s♦♥ tr♥s♠ts trés ♠s♠ r

♥ s♦ ♠♦♦ ♣r♦♣st♦ ♥ ❬ ❪ s ♦♥sr ♥ ♣♦ó♥ ♥ts r♦♥s s♣st♦s s♦r ♥ r ♦♠♣ q ♥ s ♣♦str ♥ó♥ ♦♣t♥♦ ♥ strt ó♣t♠ ♥ ♥ó♥ ♣♦r s♥t♦r♥♦ st ♠♦♦ st♦ ú ♦st♦ ♥ó♥ ♦♥trst

r Pr♦ r♦t sr♠♣ó♥ ♥ ♥ó♥ ♦rtr ♥ ♦♠♣r♥ ♦s s♦s ♦♥ ♠♥s♠♦ s♠♥ó♥ ♦♣♥♦♥s rrs rss ② ♥ s♥ ♠s♠♦ rrs ♥rs r ①trí ❬té ❪

♦ ♦♥ rs♦ ♣r♦ ♥ó♥ ♥ ♥ó♥ s strts ♦♣ts♣♦r s ♥t♦r♥♦ ② ♥♥ ♥r♠ ♥ ♠r♦ st ♠♦♦t♠é♥ ♦♥sr ♥ ♠♥s♠♦ ♠tó♥ s♦ ♠s s rr♦♥♠♣st♦ ♥ ró♥ ♣♦ó♥ ♦♥ ♥ r r r♣ó♥ ♥♦s ♥♦♥♦s ♦sr ♥ ♦s r♦ts ② t♦s ♦ ♠♦♦ ❬ ❪ s♠ ♥ ♣♦ó♥ st♦s q tú♥ r♦♥♠♥t♥ ♥ó♥ s ♦♥♦♠♥t♦ ♦ ♣ró♠♥t ♠③ ♦♥ ♦tr q♠♣♠♥t rr♦♥♠♥t ♠tó♥ s♦ ♣r ♦♣tr ♥ ♦♥t

♥ st tss ♣r♦♣♦♥♠♦s ♥ ♥♦q tr♥t♦ q ♥♦r♣♦r ♦s s♣t♦s ♣r♦♠ st ♠♦♠♥t♦ ♥♦ ♦r♦s s♠♥ó♥ tr ♦♥t ♥ó♥ ② ♥s ♥tr③ ♣rs♦♥ ② ♥♦♣rs♦♥

s♠♥ó♥ ♥ó♥ ♦♥tr

♦♥t ♥ó♥ ♦♠♦ ♥ rs♦ tr

♦♦ ①r♦

s r♦♥s s♦s ♥tr ♥♦s s st♥ ♥♠♥t♠♥t ♣rtr ♥ r♦ ♥ ♣♦r s♠♥③ s ♥r♠♥t r♥t tr♥srs♦ ró♥ ♥ ♣♦r s♠♥③ r ♥♦♠r ♦

♠♦ ♥ s ♥s s♦s ❬③rs Prs♦♥ ❪ ♦s ♠♦♦ssr♣t♦s ♥tr♦r♠♥t s♠♥ ♦♠♦♥ ♥ t♦♦s ♦s í♥♦s s♦s ♠♦♦ q t♦♦s ♦s ♦♥tt♦s rst♥ ♠♥t ♥②♥ts ♠♦♦ qqí ♣r♦♣♦♥♠♦s ♥♦r♣♦r r♦ ♦♠♦ ♥tr ♦s ♥♦s trés ♥ ♣r♦s♦ s♠♥ó♥ tr ♥s♣r♦ ♥ ♣r♦♣st♦ ♣♦r ①r♦❬①r♦ ❪ ♣r str s♠♥ó♥ tr ♥ ♥ s♦ r♣rs♥t trés ♥ r

tr♦ ♦r♥ ①r♦ ♥tr♦ ♥ ♠♦♦ s♦ ♥ ♥ts s♦s s♣st♦s s♦r ♥ r r ♦♥ N = D ×D ♥♦♦s ♥ts♦ s sr♣t♦ ♣rtr ♥ ♥ú♠r♦ rtrísts trs F ∈ N qr♣rs♥t♥ ♣♦r ♠♣♦ ss ♦♣♥♦♥s rs♣t♦ ♣♦ít ♦♥♦♠í ♣♦rtst s ③ ♥ s rtrísts ♣ ♠♥strs sú♥ ♥ ♥ú♠r♦ rs♦s q ∈ N ♣♦r ♦ q t♦♠♥ ♦rs ♥ ♦♥♥t♦ I = 1, 2, 3, ..., q st ♠♦♦ r♦ tr ♥t i strá ♥♦ ♣♦r t♦r ♣♥♥t t♠♣♦ V

i(t) = (V i1 (t), V

i2 (t), ..., V

iF (t)) ♦♥ V

i(t) ∈ IF r♠♥t ①st♥

qF ♣♦ss ♦♥r♦♥s ♣r t♦r Vi(t)♥ ♦♥t①t♦ st ♦r♠s♠♦ r♦ ♦♠♦ ht(i, j) ♥tr ♦s

♥ts i ② j s ♥t ♦♠♦ ♥ú♠r♦ rtrísts trs ♥ s q♠♦s ♦♠♣rt♥ rs♦ ♣♦r ♦ q 0 ≤ ht(i, j) ≤ F P♦r ♠♣♦ s♣♦♥♠♦sV

i(t) = (30, 41, 23, 45, 89) ② Vj(t) = (87, 41, 61, 45, 89) ♥t♦♥s ht(i, j) = 3

str q ♦r♥♠♥t ①r♦ t③ó ♦♠♦ sstrt♦ ♥ r rs♠♣ ♦♥ ♥ ♥ú♠r♦ t♦t ♥ts N = D ×D ♦♥ D = 5 10 ② 100 s♥♦♥♦♥s ♣rós ♦♥t♦r♥♦

♥♠♥t ♥á♠ ♥rá sr♣t ♣♦r s♥t ♣r♦♠♥t♦ ♦rít♠♦

♥ ♥t i ② ♥♦ ss ♥♦s j ♠♦s ♥♦r♠♠♥t ③r

♠tó♥ s♦ ♣♦r ♦♠♦ ♦♥ ♣r♦

P (i → j, t) =ht(i, j)

♥t i ♥trtú ♦♥ j ♥ s♦ ♦♥rtrs ♥tró♥

♥t i ♠trá ♥ rtríst n 1 ≤ n ≤ F ♥t j ♥♦r♠♠♥t ③r ♥ q ♣r♠♥t ♥♦ ♦♥í♥ s r sV in(t− 1) 6= V j

n (t− 1) ♥t♦♥s

V in(t) =

V jn (t− 1) ♦♥ ♣r♦ P (i → j, t)

V in(t− 1) ♦♥ ♣r♦ 1− P (i → j, t) .

♥r♠♥t ♣s♦ t♠♣♦r ♦rt♠♦ t = t + 1 s t♠♣♦rrtrr ♦rt♠♦ s t♥ ♥♦ sst♠ ♥③ qr♦s r ♥♦ ♥♦ ①st♥ tr♥s♦♥s ♣♦ss ♥ s♦ ♦♥trr♦ rt♦r♥ ít♠

s ♦♠♣♦♥♥ts ♦s t♦rs Vi(t) s♦♥ ♥♠♥t s ③r

♥á♠ ♠♦♦ ♦♥ ♥ tr♥só♥ s r qr♦ ♥tr♥ s ♠ttr ② ♦tr ♠♦♥♦tr s ♠ttr s rtr③♣♦r ♣rs♥tr ♥ts s♦s q r♥ ♥ ss t♦rs rtríststrs ♥trs t♥t♦ s ♠♦♥♦tr ♦rrs♣♦♥ ♦♠♦♥tr s r t♦♦s ♦s ♥ts ♦♠♣rt♥ ♠s♠♦ V

i(t) ♣rá♠tr♦ ♦r♥ tr♥só♥ s ♥ ♦♠♦ Smax/N s♥♦ Smax t♠ñ♦ ♠á①♠♦♦♠♥♦ ♦♥r♥ tr ② N ♥t t♦t ♥ts ♥ r s ♣rs♥t ♦♠♣♦rt♠♥t♦ rtríst♦ tr♥só♥ ♥ tér♠♥♦s ♠t♣ rs♦s trs

♥ ❬st♥♦ ❪ ♦s t♦rs ♠str♥ q tr♥só♥ s r ♣r r r s ♦♥t♥ s F = 2 ② ♣r♠r ♦r♥ ♥♦F > 2 P♦r ♦tr ♣rt s rtrísts tr♥só♥ t♠é♥ ♣♥♥ s♣r♦♣s t♦♣♦ós r ❬♠♠ ❪ ♥ t♥t♦ sst♠ q ♣rs♥t♥ tr♥só♥ s r qr♦ ♠♦♦ ①r♦ ú♥ ♣r♠♦s ♥trr♦♥ts t♠♥t st♦s ♥ ♦♥t①t♦ s♦♦♦í

♥ st tss ♣r♦♣♦♥♠♦s ♥ ♠♦♦ s♦ ♥ ①r♦ ♣r ①♣r s♠♥ó♥ ♦♥t ♥ó♥ ♥ ♥ s♦ ♦♥ttrá ♦♠♦ ♥ rtríst ♦♥ ♦♥ rs♦ tr ♥r♦ ♥♦r♦ ♥♦♥♦r q s s♠♥rá ♥ r s♦ s rtrísts trs rst♥ts srá♥ s rs♣♦♥ss srr♦r ♦♠♦ ♥tr ♦s st♦s♣r♦♣♦r♦♥♥♦ ♥ r♦ tr♦♥ ♦♥ ♠♦♦ q tú ♦♠♦t③♦r ♣r♦s♦ s♠♥ó♥ ♦♥t ♥ó♥

♦♠♦ ♠♦s ♠♥♦♥♦ ♥ ♦s ♣árr♦s ♣r♥ts ♠♦♦ ①r♦①♣ sr♠♥t♦ ♦♥s♥s♦ ♥ ♥ r s♦ trés s♠♥ó♥tr ♠♣♥♦ ♣r ♦ ♥ ♠♥s♠♦ ♠tó♥ s♦ ♦s♦tr♦s qí

s♠♥ó♥ ♥ó♥ ♦♥tr

r r♥só♥ s ♦♥s♥s♦s♥s♦ ♣r ♠♦♦ ①r♦ s♦r♥ r r ♦♥ N = 50×50 ② N = 100×100 ♥ ♠♦s s♦s ♦♥ F = 10❬ ❪

s♠♦s ①♣r ♥ó♠♥♦ ♥rs♦ ó♠♦ ♠♦r sr♠♥t♦ strs ♥♦♥♦s ♥ r♥st♥s t ♦rtr ♥ ♥ Pr♦ ♥♦r♣♦r♠♦s ó♥ ♠♦♠♥t♦s ♥♦♥♦rs ♦st♥♦s sr st♦s ② ♦♣♥ó♥ rs♣t♦ ♥ó♥ stá ② ♥♦ ♠ ♣♦r ♣r♦s♦ ♠tó♥ s♦

♦s ♠♦♠♥t♦s ♥t♥ó♥ ♥r♠♥t r♥ r♠♥t♦s ♥tí♦s só♦s ♥q ♠② s♣♦rá♠♥t r♥ ú♥ ♠♠r♦ ♦♠♥ ♥tí ♦♠♦ r ❲ ♥ ② ♠♥♦♥ ♦♥tr♦rs ♥ ❱ ❬❲ ❪ ♥ ♠r♦ s ♣♦♦ r♥t q ♦s♥♦s ♥♦ s♣sts í♥ s ♥t♥♠♥t♦ trés ♠ét♦♦ ♥tí♦ ♠② ♣♦r ♦♥trr♦ t♦♠♥ ss s♦♥s ♥♥♦s ♣♦r ♥♦r♠ó♥q s s s ♥t♦r♥♦ ♣rs♦♥ ♦ trés ♠♦s tr♥t♦s st♦s ♠str♠♥t r♦ ♥ rt♦ ♥ ♣r♠r ♣rs♦♥ q srt♦r♥♦rt♠r♥ ss r ss ♦♥s ♠♦♠♥t♦ t♦♠r só♥ ♥r s ♦ r♥t♠♥t ♣♦ ② st sr ♥ ❯❯❬ss ❪ P♦r t♦♦ ♦ s♠r♠♦s q ♦s ♥ts ♦st♥♦s ♥t♥ó♥♣♥ s♣rr s ♣♦str ♣♦r ♠tó♥ s♦ ♣s ♣♦r ♦♠♦ ♥♣♥♥t♠♥t r♠♥t♦♥s r r♦♥ ♦♠♣♦rt♠♥t♦ ♦s ♥ts s♦s

♦s ♦r♥s♠♦s s ♣ú tú♥ ♠♦♦ ♥t♦♥sts ♦s r♣♦s♥t♥ó♥ Pr ♦ ♥ s♦ ♠♣ñs ♠ss ♥ó♥ q♥②♥ ♠♣ s♣♦♥ ♥s ♥ ♠r♦ ② ♠♦s ①♣t♦q s♣♦♥ ♥s ♥♦ s s♣t♦ ♠ás r♥t ♥q sí ♥sr♦ ♣r s ♥sró♥ ♥ ♦s ♣ss srr♦♦s ♠♦♦ q ♥ st tss ♥♦s♥trr♠♦s ♥ ó♥ ♠♣ñs ♠ss ♥♦r♠ó♥ ♣♦st r ♥ó♥ ♣♦r ♣rt ♦s ♦r♥s♠♦s s ♣ú s ♠♣ñstrá♥ ♦♠♦ ♥ ♠♣♦ ①tr♥♦ q ♥ ♥ ♣♦str ♥ó♥ ♦s♥ts s♦s str♦ ♦t♦ s ♠♦r ♥♠ér♠♥t t♦ ♠t♦r s ♠♣ñs

♦ ♣rt♥♠♦s ♦♥ st st♦ ♣♦rtr rs♣sts ♥ts ♣r♦♠ s♠♥ó♥ ♦♠♣♦rt♠♥t♦ s♦ ♥ ♦ q rs♣t ♦♥t ♥ó♥ str♦ ♦t♦ s ♣♦rtr ♥s rr♠♥ts ♠♦♦ q♠♦t♥ ♥♦s ♥trr♦♥ts ② ♣♦str♦rs ①♣r♠♥t♦s t♥♥ts sr♦s

Pr♦♣s státs ② ♥á♠s s r

s s♦s

♦♥♦r s rtrísts t♦♣♦ós ♣rtrs s rs s♦s s ♥s♣t♦ r♥ r♥ ♣r ♣♦str♦r ♠♦♦ ♥ó♠♥♦s ♣r♦♣ó♥ s♦r s ♦s tr♦s ♥♦♥s ♥ ár r♦♥ ♥t ♥srtrísts st♥ts s rs s♦s ts ♦♠♦ ♣r♦♣ ♠♥♦

♣qñ♦ ❬r♠ ❲tts ❪ str♦♥s r♦ P (k) t♣♦ ②s ♣♦t♥s P (k) ∼ k−γ ♦♥ γ ≥ 2 ❬rs ❪ rt strtr ♦♠♥s ❬♠♥ ♠♥ s ❪ ♥tr ♦trs s ♣r♦♣s rtr③♥ s rs ♦♠♦ ♦t♦s stát♦s rst♦ ♠ó♥ ♥♦♦s ② ♥s trés ♣r♦s♦s ♥á♠♦s ♠r♦só♣♦s s♥trñr srtrísts ♣rtrs st♦s ♣r♦s♦s s♦ ♦s ①♣rt♦s r♥t♦s út♠♦s ñ♦s

♠♦♦ r♠♥t♦ rs ♠ás ①t♥♦ st s♦ ♥ ♣rr♥ ♣r♦♣st♦ ♣♦r rs ② rt ❬rs ❪ ♣r①♣r ♦r♥ s str♦♥s r♦ t♣♦ ② ♣♦t♥s ♣r♥♣♦ ♥♠♥t st q ♣r♦ q ♥ ♥♦♦ t♥ rr ♥♦s ♥s s ♣r♦♣♦r♦♥ ♥t ♥s q ② ♣♦s

st ♠♥s♠♦ s♦ ♣♦ ♦♥ ♥tr♦r ♥ rs♦s ♦♥t①t♦s ❬❨ ♠♦♥ ❪ ♥s♦ t♥ ♥ ♥t♥t í♦ ♥ Prá♦ ♦s ♥t♦s t♦ ♣♦r ♦ t♠é♥ r ♥♦♠r t♦ t♦

Pr♦♣s státs ② ♥á♠s s rs s♦s

♥ rst♦ ♥ s ♦sr♦ ♥ rs s♦s ♠♣írs ♦r♥ t ♠♥s♠♦ ú♥ s♣rt ♦♥tr♦rss ♥ ♦♠♥ ♥tí❬P♣♦♣♦♦s Prr ❪

r♥ ♥t t♦s s♣♦♥s ♣rtr ♥♠♥t♦ s ♥st♥♦♦ís ♥♦r♠ó♥ ♣r♠t♦ r ♦ó♥ ♠r♦só♣ s rs s♦s ♦♥ ♥ rs♦ó♥ t♠♣♦r s♥ ♣r♥ts ❬③r ❪♥ ♣rtr ♥♦s t♦rs ♥ ♠♣♦ s ♦♠♥♦♥s ♣♦r t♦♥í r ♣r str ♦♥tt♦s ♥tr ♥♦s ❬ ♦♥③③ ♦♥ ❪ ♦tr♦s ♥ ♥③♦ s ♣r♦♣s rs ♠♦s ♦♥♥ ♦♠♦ ♦♦ ❬❯♥r ♦♥ ❱s♥t ❪ ♦♠♦ sí t♠é♥ ♦ó♥ ♠♣♥s r s♦rs ♥ ttr❬r♠♥ ❨♥ ♦♥çs ♦♥♦r ❪ ♦ s♦♦ ♥ s♦ rstr♦s ② ♥③♦s ♦s ♦♥tt♦s ♦♥♥ s♥♦ t♠é♥ ♦s ♦♥tt♦s rt♦srr ♥tr ♦s sst♥ts r♥♦♥s s♦s ♣♦r ♠♦ s♥s♦rs ♣r♦①♠ ❬ttt♦ s ❪ ♠♦♦ q ③ st♦s ♥♦st♦s s♦ ♣♦s ♥♦ s♦♦ r ♦ó♥ ♠r♦só♣ r s♥♦t♠é♥ ♥á♠ ♣r♦♣ ♦s ♦♥tt♦s s♦s

♦s ①♣r♠♥t♦s ② sr♣t♦s r♦♥ ♥t ♥á♠ ♦♥tt♦s♥♦ r ♥ ♥ sár st♦ s ♥♦♠♥s rs

rs ♥ t♠♣♦ ❯♥♦ ♦s ♠♦♦s s♥♦s t♥♥ts r♣r♦r ♥á♠ ♦♥tt♦s s♦s s♠ ♥ts ♠♦♦ ♣rtís ts r③♥♦ ♠♥ts ③r ♦♥ ♥ ♣♦t♥ trt♦ ❬tr♥♥ ❪ P♦r ♦tr♣rt ♦ Prr t ❬Prr ❪ ♣r♦♣sr♦♥ ♥ ♠♦♦ rs rs♥ t♠♣♦ ♣r ♥♦♦ s ♣r♦st♦ ♥ rtríst ♥trí♥s♥♦♠♥ t s♦ ♥ ♣s♦ t♠♣♦r ♦s ♥♦♦s s t♥ ♦ ♥♦♣♥♥♦ s t s♦ ② trs ♥♦ st ♥s ♦♥♦tr♦s m ♥♦♦s ♦s ③r ♦ ♣s♦ t♠♣♦r t♦♦s ♦s ♥ss♦♥ ♦rr♦s ② ♣r♦s♦ s r♥ ♥ s♥t ♣s♦ ♦s t♦rs ♠♦strr♦♥ q s stró♥ t s♦ s ♥ ② ♣♦t♥s ♥t♦♥st♠é♥ ♦ rá stró♥ r♦s r ♠ ♥ t♠♣♦

♥ s rs ♦♠♣s ♦♠♥③ó srs ♥tr q♦s q♥③♥ ♣r♦♣s t♦♣♦ós státs ② ♦s q st♥ ♥á♠ ♦s♥s ♦s ♥♥ sr♥ ② s♦♥ ♦rr♦s ♥ s rs rs ♥ t♠♣♦♠♥trs q s s♠♥ ♥trs ♥ r♣rs♥tó♥ stát ♦ stút♠ r♥t♠♥t s rst♦ ♠r ♥ t♠♣♦ ♦s ♥s q♥á♠♠♥t sr♥ ♥ r♣rs♥tó♥ r ♥ t♠♣♦ ♥ ♠r♦ ♠r ♥ t♠♣♦ ♦s ♥s sr♦s ♦s ♠♦♦s rs rs

② t♦s s ♦t♥♥ rs q ♥♦ ♣♦s♥ s rts ♦rr♦♥s ♦s ♠♣♠♥t ♦srs ♥ ♠s rs s♦s státs ♠♦♦ q ♦tr♦ ♦s ♦t♦s st tss s ♣r♦♣♦♥r ♥ ♠♦♦ q ♣r♠t ♥rr rs♦♥ ♥á♠ ♥s ② rsó♥ ♠ r♥t ♥ ♥t♥ t♠♣♦rt♥ s ♣r♦♣s t♦♣♦ós s rs s♦s státs

út♠♦ ♣r♦♠ ♣r♦♣ó♥ ♥ rs s♦s q trtr♠♦s ♥st tss t♠é♥ stá r♦♥♦ ♦♥ sr♠♣ó♥ ♥q ♥ st s♦♦♥ s ♣r♦ ♣rsst♥ sr♠♣ó♥ s ♥ ♥r♠ ♥♦s ①tr♠♠♥t ♦♥t♦s ♦♥tr ♥♠♥t♠♥t trés s ♠r♦♦ts ü q ♦s st♦s ♥t♦s ♠t♥ r ♦ st♦r♥r ② t ♣r♥♣♠♥t ♥ñ♦s ♥ s♦r ♣rí♦♦ t♥t s♣r♦♦♥ ♣♦r ♣r♦①♠♠♥t 8 ís ♥ ♣r♦♠♦ ♦♥ ♥r♠♦ srr♦ ♥r♠ ♣r♦ ú♥ ♥♦ ♦♥t ♠♥trs q ♣rí♦♦ ♥t♦ t♥ ♥ ró♥ ♠ 5 ís ♥ ♦s q ♥r♠♦ s ♦♥t♦s♦❬② ② ♥rs♦♥ ♥ ♦♥♥ ❪ ♥rr ♥♠♥ s ①tr♠♠♥t ♣♦♦ ♣r♦ q ♥ ♣rs♦♥ ♦♥trsr♠♣ó♥ ♦s s ♥ s

♥tr ♦s ñ♦s ② sr♠♣ó♥ ③♦tó r♥ ♣rt ♠♥♦ ♥q①st♥ rstr♦s ♣rtr♠♥t ①st♦s ♥♦ ❯♥♦ ② ❯❯ ♣♦r ♦s♦♥ ♦s s♦s ♠ás ♥③♦s ♥ trtr ♥ r s ♠str ♥t t♦t ♥♦s s♦s r♣♦rt♦s ♥ ♥trr ② s ♥tr ♣r♦s ♠♣ñ ♥♦♥ ♥♠♥③ó♥ ♠ ♥ ♥tr♦s ♥ s♠♥ ♦♥♠♥t r ♣rs♥t ♦s s♦s r♣♦rt♦s♣r r♠♥♠ ♥tr t♠é♥ ♠♦s s♠♥♠♥t ♥trsq ♦s rá♦s r ② ♣rs♥t♥ ♦s rstr♦s ♠♦s ♠♥s♠♥t ♣r s♦ ❨♦r ♥tr ② ♦♣♥♥♥tr rs♣t♠♥t ♦♠♣♦rt♠♥t♦ ♣ró♦ s ♠♥st ♥ s♣tr♦ ♣♦t♥s ♦♠♦ ♣♦s ♠r♦s ♣r ♣rí♦♦ ♥ ② ♥ ♣r ♦s s♦s ♥trrs ② r♠♥♠ r rá♥♦s ♥♦♠♣♦♥♥t rt tr♥ ♣r ♦s s♦s ❨♦r ② ♦♥♣♥♥ r

rrr♥ sr♠♣ó♥ trt ♥ ♣árr♦ ♣r♥t s ♥♥tr♥ ♦♥ ♥ó♠♥♦ ♣rsst♥ ♥③r s srs t♠♣♦rs st♥ts s ♥♦ ❯♥♦ ♠t♠át♦ r rttt ♦sr

r s♦s r♣♦rt♦s sr♠♣ó♥ ♠♦s s♠♥♠♥t ♣r ♥trrs ♥ ② r♠♥♥ ♥ ② ♠♦s♠♥s♠♥t ♣r ❨♦r ♥ ♣rí♦♦ ② ♦♣♥♥♥ ♥ t♦♦s ♦s s♦s s ♦sr ♥ rt ♣r♦ t♠é♥♠♥st ♥ ♦s s♣tr♦s ♣♦t♥ ♣r ♥trrs í♥ só ②r♠♥♥ í♥ tr③♦s s♣tr♦s ♣♦t♥ ♣r sr t♠♣♦r ❨♦r í♥ só ② ♦♣♥♥ í♥ tr③♦s r ①trí ❬♦r ❪

ó q ♥r♠ rst ♥é♠ ♦st♦r ♥ s s r♥s♠♥trs q ♠♦str r♦s ♣rí♦♦s ♠♥♦s s♠♥s ró♥ s♥s♦s r♣♦rt♦s ♥ s s ♠♥♦s ♣♦s st ③♦ t♠é♥ ♣♣rrs ♥ r ♦♥ s ♣rs♥t t♠♣♦ ♠♦ ♥tr r♦ts sr♠♣ó♥ ♥ ♥ó♥ 100/

√N s♥♦ N ♣♦ó♥ s s ♥

r P♦ó♥ s s rs

r ♠♣♦ ♠♦ ♥tr r♦ts sr♠♣ó♥ ♥ ♥ó♥ N−1/2

❬rttt ❪ ♣r s s t

trr ② s ♥s ♥ t r ❬rttt ❪ ♥t♦♥sr♠♦s q ♥r♠ s ♣rsst♥t ♥♦ s ♣r♦ ①t♥ó♥♣♦r ♥ ♣rí♦♦ ♠②♦r s♠♥s s ♥

♦♥ ♦t♦ ①♣r ♥ó♠♥♦ s♥♠♥t♦ ♦ ♦t rttt♥tr♦ ♦♥♣t♦ t♠ñ♦ rít♦ ♦♠♥ NC ♣rtr sr♠♣ó♥ s t♦r♥ ♥é♠♦ ♦st♦r♦ Pr ♦ ♣r♦♣♦♥ ♥ ♠♦♦ ♣é♠♦♦♠♣rt♠♥t st♦ást♦ ② srt♦ t♣♦ ♥♦ ♦♥sr ♦♥ó♥ ①♣st♦ s♠♥♦ q ♦s ú♥♦s st♦s ♣♦ss ♦s ♥ts s♦♥ ss♣t ♥t♦ ♥♠♥ ♦ rrtr♦ ❬rttt ❪ ó♥ ♥ ♠♦♦ st♦ást♦ srt♦ ♣♦r s♦r ♥♦ tr♠♥st s q ♦s ♠♦♦s tr♠♥sts ♦♥t♥♦s ♠t♥ ♣♦s r♦♥s ♥♦s♥t♦s ♥ ♥s ♦s♦♥s t♥♦ ①t♥ó♥ r♦t P♦r ♦tr ♣rt

r tí♦ r♦ 6× 6 t③♦ ♥ s s♠♦♥s st♦ásts rttt ❬rttt ❪

s t♦♥s ♠♦rás t♣♦√N ♣r♦♣s ♦s ♠♦♦s st♦ást♦s

♦♥stt②♥ ♥ ♠♣♦rt♥t ♥t ①t♥ó♥ q t♠♣♦♦ sr ♥tr♠♥t♥ ♦s ♠♦♦s tr♠♥sts ♦♥♠♥t rttt ♦♥sró ♥ strtr♠t♣♦♦♥ ♥♦r♣♦rr ♥ r♦ r♦ ♦♥ ♠r♦♥s st♦s♥t♦s ♥tr s ②♥ts r r

♥ ♠♦♦ rttt sq♠ t♠♣♦r s srt♦ ♦♥ ♥tr♦ ♥ s♠♥ ♦♠♦á♥♦s r♥ ♠str♦ ♦s rstr♦s stór♦s ♠ás ts ♥ó♥ λ ts r♣ró♥ γ ② ♠ró♥ st♦s ♥t♦s ♥tr s ε ♠♦♦ rqr ♥♦r♣♦ró♥ ♣r♠♥♥t ♥♦s ss♣ts trés ♥ ts ♠♦rá ν Pr s♦♣rtr ♥str ❯ rttt st♠ ♥ t♠ñ♦ rít♦ ♦♠♥ ♣r♦①♠♠♥t t♥ts ♦r q st ♠♥t♦s rstr♦s stór♦s ♣r s r♥ts s ♦♥sr♦♥s ♣♦str♦rstr♠♥r♦♥ q t♠ñ♦ rít♦ ♦♠♥ s ♥♥tr ♥tr t♥ts

♦♥ tr♥srrr t♠♣♦ t♥t♦ ♦s ♠♦♦s tr♠♥sts ♦♠♦ ♦s st♦ást♦s ♥r♠♥tr♦♥ ♥ t ♥ sr♣ó♥ ♦♥sr♥♦ ♣♦r♠♣♦ s t♦♥s st♦♥s ♥ ♣tró♥ ♦♥tt♦s s ♦ t♦ ❬♦♥♦♥ ♦r ❪ ♠♦♦ tr♠♥st ♦r③♦♣♦r st♦♥ ♠ás s♥♦ q ♣r♠t ♦t♥r rrr♥ ♣ró ♣

srrs ♦♠♦ds/dt = µ(1− s)− β(t)si;

de/dt = β(t)si− (µ+ σ)e;

di/dt = σe− (µ+ γ)i;

♦♥ β(t) ♦rrs♣♦♥ ts ♦♥tt♦ µ s ts ♥t♠rt st ♠♦♦ s r♠♥t ♦♥ ♥♦s ss♣ts σ s ts tr♥só♥ ①♣st♦ ♥t♦ γ r♣ró♥ ② s + e + i + r = 1 ①♣rs♦ ♥♥ss st♦♥ ♦ t♦ t ts ♦♥tt♦ ♣♦r ♦q s ♣r♦♣♦♥ β(t) = β0(1 + β1 cos 2πt) ♦♥ β0 ② β1 ♦♠♦ ♣rá♠tr♦s rsst ♠♦♦ r♣r♦ s ♦s♦♥s rrr♥ts ♥s ♥s ② tr♥s♦srs ♥ s srs t♠♣♦rs ♣r♦ ♥♦r♣♦rr ♦r③♥t β(t) ♣r♦ ♥♦rr♠♥t♦ ♣r t♠ñ♦ rít♦ ♦♠♥ ❬♦r ❪ rst♥♦ ♦r♥ 5 × 106 t♥ts ♦♦s ♣♦str♦rs ♥♦r♣♦r♥ st♥ó♥ ♦♦rts ♣♦r s ♦s ♥♦s ♦♥ tss ♦♥tt♦ r♥s βij

♥tr s ♦♦rts i ② j ❬♥③ ♦r ❪ q t♠é♥ ♠♣t ♥ ♥ó♠♥♦ ♣rsst♥

♥s ♣rsst♥ ♦♥ stró♥

t♠♣♦s

Pr rs♦r ♥♦♥sst♥ ♥ t♠ñ♦ rít♦ ♦♠♥ ♦s ♠♦♦s q ♥tr♦♥ st♦♥ ② ♦♦rts ♥ ❬♥ ♥ ♦② ❪ s ♣r♦♣♦♥ ♦♥srr str♦♥s t♠♣♦s ♥t♦s ♠ás rsts q s ①♣♦♥♥s s♦s ♦♥ s tss ♦♥st♥ts tr♥só♥ ♦s♠♦♦s tr♦♥s s♦ rs t♦rs ♦♥ stró♥ ①♣♦♥♥♦ ♥ s♠♣ó♥ ♠t♠át ♠ás q ♥ ♦rrs♣♦♥♥ ♦♥ r ❯♥ ♦r♠ ♦rr str♦♥s ♠♥♦s s♣rss s r♠♣③♥♦s ①♣♦♥♥s ♣♦r str♦♥s ♠ás ♥rs t♣♦ ♠♠

P (t) =1

Γ(k)θktk−1e−

s♥♦ k t♦r ♦r♠ ② θ s ❯♥ r ♦♥ stró♥ ♠♠ ♣ ♦t♥rs s♥♦ ♦ q s♠ n rs ♦♥ stró♥ ①♣♦♥♥ Γ(k = 1, θ) ♣♦r rst♦ ♥ r ♦♥ stró♥ Gamma(k = n, θ) ❬② ❪ ♠ét♦♦ tr♦♥ ♣r ♥③r♥ stró♥ ♠♠ ♣r t♠♣♦ ♥t♦ ♥ ♦♥t①t♦ ♠♦♦tr♠♥st ♦♥sst ♥ sr st♦ ♥t♦ ♥ n sst♦s

❬② ② ❪

di1/dt = σe− (µ+ nγ)i1;

di2/dt = nγi1 − (µ+ nγ)i2;

din/dt = nγin−1 − (µ+ nγ)in.

♦♠♥ó♥ s♣ró♥ ♦s ♥♦s ♥ ♦♦rts ♥t♦ ♦♥ ♥tr♦ó♥ str♦♥s t♠♣♦ ♥t♦ ♠ás rsts r ♥ ♠♦r st ♣tró♥ ♣rsst♥ sr♠♣ó♥ ❬♦r ♥ ♥ ❪ ♥ ♠r♦ ♥ st út♠♦ ♠♦♦ s tss ♦♥tt♦ βij ♥trs r♥ts ♦♦rts ♦♥stt②♥ ♦s ♠♥t♦s ♥ ♠tr③ ♠③ ② s♦♥♣rá♠tr♦s st ♠♦♦ ♦s ♣r♦♣♦s t♦rs r♥ ♥ ♥♦r♠ó♥ s♣♠♥tr ♥ tr♦ ♣♦str♦r ❬♦♥♥ ❪ t♦♥ ♦ t ♠♦st

♣♣r♦♣rt ♠①♥ ♠tr① s r② ♠ ♥ rt rtr t♥ s♥

t♦ strtr r s♦ ♥ ♣rsst♥

♦♠♦ q r♦ ♥áss ♣r♥t ♣tró♥ ♣rsst♥ sr♠♣ó♥ rst ♠② s♥s s st♥ts r♥ts q ♥r♠♥t♥ ♥ ♦♠♣ ♠♦♦ ♠♦s st♦ q ♥♦♥r ♣t♥♠♥t ♣r♠ ♠③ ♦♠♦é♥ ♥ ♣♦ó♥ ♥tr♦r ♦♦rts ♦♥tss ♥tró♥ r♥s s t ♠r ♣rsst♥ P♦r ♦s r③♦♥ s♣rr q t♠ñ♦ rít♦ ♦♠♥ rí rs t♦ ssttr ♣ótss ♠③ ♦♠♦é♥ ♣♦r ♥ r s♦ ♥tr♦♥s ♥ ♠r♦ st ♠♥♦ ♥♦ s♦ ♠② ①♣♦r♦ ♦♥ ①♣ó♥ ❬❱rs ❪ ♦♥ s ♥③ ♥♥ ♣r♦♣ ♠♥♦ ♣qñ♦ ♥ ♣rsst♥ sr♠♣ó♥ s♦r ♥ r r ♦♥ r♦♥①ó♥ ♥s

♥ st tss ♥♦s ♣r♦♣♦♥♠♦s ①♣♦rr ♥♥ s ♦rr♦♥s ♦s♥tr ♦s ♥♦s ♥ ♣tró♥ ♣rsst♥ sr♠♣ó♥ s♦r ♥ r♦♠♣

strtr tss

♥ ♣ít♦ r♠♦s ♥ rá♣ ♥tr♦ó♥ t♦rí rs ♦♠♣s ♦♥ ♦t♦ ♣rs♥tr s rs t♦♣♦ós ss q t③r♠♦s ♦ r♦ st tss ♣ít♦ ♥tr♦ ♠♦♦ s♠♥ó♥ ♦♣♥♦♥s ♣r ♥ó♥ ♥ s ♦r♠ s♠♣ ♦♥ r stát ♥ ♣ít♦ ♥r③r♠♦s ♠♦♦ ♣ít♦ ♣r♥t ♥♦r♣♦r♥♦ ♥s ♣rs♦♥s ② ♥♦♣rs♦♥s ♥ ♠♥s♠♦ ♣tó♥ ♥ r s♦ ② ó♥ ♠♣ñs ♥ó♥ ♣ít♦ srr♦ ♠♦♦ ♦r♥ ♥ró♥ rs s♦s ♣rs♥t♦ ♥ st ♥tr♦ó♥ ♥♦ s♦ s rs ♥rs ♠♥t ♠♦♦ ♣ít♦ ♥tr♦r ♥ ♣ít♦ st♠♦s t♦ s ♦rr♦♥s ♦s r♦ ♥ ♣rsst♥ sr♠♣í♦♥ P♦r út♠♦ ♣ít♦ ♦♥t♥ ♥strs ♦♥s♦♥s ♥s ②♣rs♣ts tr♦ tr♦

♣ít♦

r ♥tr♦ó♥ s rs

♦♠♣s

❮♥ s státs ♣r♦♣s t♦♣♦ós

stró♥ r♦

st♥ ♠ ♥tr ♥♦♦s

♦♥t str③ó♥

r♦ ♠♦ ♣r♠r♦s ♥♦s

strtr ♦♠♥s

só♥ t♦♣♦ó rs ♦♠♣s

s t♦rs

s ♥♦ Pqñ♦

s rs s

♥r♠♦s qí s ♣r♦♣s strtrs ♥♠♥ts s rs♦♠♣s r♠♦s ♥ sr♣ó♥ ♥tr♦t♦r s só♥ ♣rtr s rtrísts ♥ st ♣ít♦ ♥r♠♦s ♥s s rtrístst♦♣♦ós s♥s q srá♥ t③s ♦ r♦ ♣rs♥t tss

s státs ♣r♦♣s t♦♣♦ós

st♦ rr♦s♦ ♣r♠ s rs ♦♠♣s ♦ró ♦r ♥tr♦ ♦♠♥ ♥tí r♥t ♦s út♠♦s ñ♦s s♦ ♣s♦ ♥ q♦s ñ♦s ♦s sr③♦s s ♦♥♥trr♦♥ ♥♠♥t♠♥t ♥ sr♣ó♥ ss ♣r♦♣s t♦♣♦ós ♠♦♦ q r♦♥ ♥♠♥t ♥③s ♦♠♦♦t♦s stát♦s s rs ♦♠♣s s st♥♥ s rrs ♦ s

♣ít♦ r ♥tr♦ó♥ s rs ♦♠♣s

st♥ts ♥rstr♦s ♥tr♥t

r ♠♣♦s rs ♦♠♣s r♦♥s s①s ♥tr st♥ts ♥ ♥rs ♥♦rt♠r♥ ❬r♠♥ ❪ 100000rt♦rs ♥tr♥t ② ss ♦♥①♦♥s íss ❬rás ❪

♣tró♥ ♦♥t tr♦é♥♦ ♦♥stt②♥ ♥ s s♥ ♥ r♣rs♥tó♥ ♠t♠át strtr ♥ sst♠ ♦♠♣♦ ♠♦t♦ ♣♦r ♥s♣rt♦ ♠♦ ♥trés ♥ ♦♠♥ ♥tí

❯♥ r ♦ r♦ G(N,L) s ♥ ♦♥♥t♦ N ♥♦♦s ♥♦s trés L

♥s r ♦s ♥♦♦s ♣♥ r♣rs♥tr ♥♦s ♦♠♣t♦rs ♦r♦♥s s♣s ♠♥trs q ♦s ♥s r♣rs♥t♥ ♣♦r ♠♣♦ í♥♦s♣rs♦♥s s♠ts ♦ ♦♥①♦♥s íss ♥tr ♦s ♥♦♦s ❯♥ ♥ lij ♥tr ♦s♥♦♦s i ② j ♣ sr r♦ rs rs ♦ ♥♦r♦ rs ♥♦rs❯♥ ♥ r♦ t♥ ♥ s♥t♦ ♣r♦ ♣♦r ♠♣♦ li→j ♦rrs♣♦♥ ♥ ♥ q ♦♠♥ ♥♦♦ i ♦♥ j P♦r ♦tr ♣rt ♦s ♥s t♠é♥♣♥ t♥r ♥ ♣s♦ s♦♦ q ♥t ♥t♥s í♥♦ ♦ s r♥ ♥ st út♠♦ s♦ r s ♥♦♠♥ r ♣s ♥ st tsstrr♠♦s ♦♥ rs ♥♦rs ② ♥♦♣ss s♦ ♦♥ s r①♣ít♠♥t ♦ ♦♥trr♦

stró♥ r♦

r♦ ki ♦rrs♣♦♥♥t ♥ ♥♦♦ i s ♥ ♦♠♦ ♥t ♣r♠r♦s ♥♦s i stró♥ r♦s P (k) ♥t ♣r♦ s♦♥r ♥ ♥♦♦ r♦ k ② s q③á rtríst t♦♣♦ó ♠ás♥ ♥ só♥ s rs ♦♠♣s Nk ♥t t♦t ♥♦♦s ♦♥ r♦ k ♥ G(N,L) ♥t♦♥s P (k) s ♥ ♦r♠♠♥t ♦♠♦

❬rt ♠♥ ❪

P (k) = lımN→∞

s rs ♦♥ stró♥ r♦ r s rs rs s P (k) ∼ k−ξ ♦♥stt②♥ ♥ sr♣♦ ♠② ♠♣♦rt♥t ♦ q s♦♥s ♥ ♥tr③ ❬rs rt ❪ ❲♦r ❲ ❲❲❲❲ rs ♦♥tt♦s s①s ② s rs ♦♦ró♥ ♥tr ♥tí♦s s♦♥ ♥♦s ♠♣♦s rs rs s s rs ♦♥ stró♥ r♦ t♣♦ P♦ss♦♥ P (k) = µk exp(−µ)/k! s♥♦ µ ♠ stró♥ ♦♥stt②♥ ♦tr♦ s♦ ♣r♠át♦ ♦ q sr♥ ♥tr♠♥t ♠♦♦ rösé♥② ♣r ♥ró♥ rs t♦rs❬rös rös ♠♥ ❪

st♥ ♠ ♥tr ♥♦♦s

♥ st♥ d(i, j) ♥tr ♦s ♥♦♦s i ② j ♦♠♦ ♠í♥♠ ♥t ♥s q s ♥sr♦ r♦rrr ♣r ♦♠♥r♦s ♦ st♥ ♠í♥♠♠ ♥tr sqr ♣r ♥♦♦s i, j ∈ G(N,L) ♥ ♣♦r

N(N − 1)

d(i, j),

♦♥ s s♠ d(i, i) = 0 s s♥♦ r q ♣r ♥ r rr l ∼N1/D s♥♦ D ♠♥só♥ r ♣♦r ♠♣♦ ♣r ♥ r ♠♥s♦♥l ∼ N1/2 ② ♣♦r ♦♥trr♦ ♣ ♠♦strrs q ♣r s♦ ♥ rt♦r rösé♥② s t♥ ❬r♦♥③ ❪

lER =lnN − γ

ln〈k〉 +1

2≈ lnN

ln〈k〉 ,

s♥♦ γ ≃ 0.5772 ♦♥st♥t rsr♦♥ ② 〈k〉 r♦ ♠♦ r ♥♦ ♦♠♦

〈k〉 =∑

kP (k).

①♣rsó♥ ó♥ ♠str q lER s♠♥② s♥s♠♥t ♦♥rs♣t♦ ♦t♥♦ ♣r s♦ rs rrs ♦ s♠♥t ♦rr ♣r

s rs t♦rs ♦♥ stró♥ r♦ P (k) ∼ k−ξ ♣r s q s ♦t♥

lnN s ξ > 3

lnNln lnN

s ξ = 3

ln lnN s 2 < ξ < 3

♣♦r ♦ q ♦♠♣♦rt♠♥t♦ ♥ ♥t♦ l s s♠♥t s♦ s rst♦rs ♥♦ ①♣♦♥♥t ξ > 3 s ♦sr ♥ ró♥ stí♣ ♣r s♦ ξ = 3 ❬♦♥ r♦♥③ ❪ ♠♥trs q s ♦♠♣♦rt♦♠♦ ♦ ♦rt♠♦ N ♥♦ 2 < ξ < 3 ❬♦♥ ❪

r ②s s ♣r C(k) ♦srs ♥ tr♦ rs rs ♥③s ❬s③ ❪ t♦rs ♦s t♦rs s ♥♥tr♥ ♦♥t♦ss tr♦♥ ♥ ♠s♠ ♣í r♦ ♦♥ s ♦♠ s♠á♥t ♦♥t ♦s ♣rs ♦♠ ♥és s s♦♥ s♥ó♥♠♦s r♦ ♦♥ ♦♥r♦ rr♥ ❲str ❲❲❲ r♦ ♦♥ ♦st♦s r♦♣♦s ♥ ❬rt ❪ r ♥tr♥t ♦♠♦ sst♠ tó♥♦♠♦ ♥♦♦ r♣rs♥t ♥ ♦♠♥♦ ♠♥trs q ♦s ♥s ♦♠♥♦♥síss ♥tr ♦s ♠s♠♦s s í♥s ♣♥ts ♥ rá♦ ♦rrs♣♦♥♥ ② ♣♦t♥s ♦♥ ①♣♦♥♥t −1

♦♥t str③ó♥

tr ♠ r♥t♠♥t ♠♣ ♥ rtr③ó♥ t♦♣♦ó s rs ♦♠♣s s ♦♥t str③ó♥ C ♦♥stt② ♥ ♠tr♥t tr♥st r s r ♣r♦ q ①st♥ ♥ ♥tr ♦s ♥♦♦s j ② k ♥♦ ♠♦s ♣♦s♥ ♥ ♥♦ ♦♠ú♥ i ♥ ♦trs♣rs ♦♥t str③ó♥ s ♥rs ♦♥ ♥ rs ♦♥t ♥r q ♥♦r♠ r♥ ♦♥ q ♦s ♠♦s ♠s ♠♦s s♦♥

t♠é♥ ♠s ♠♦s ❬♦ñá ❪

P ♥rs ♦♥t str③ó♥ ♦ ♣r ♥ ♥♦♦ i ♦♠♦ ró♥ ♥s ♥tr ss ♣r♠r♦s ♥♦s j ∈ Nnn(i) rs♣t♦ t♦t ♥s ♣♦ss ♥tr ♦s ♠s♠♦s s r

Ci =∑

j,k∈Nnn(i)

ajkki(ki − 1)

♦♥ ki s r♦ ♥♦♦ i ② ajk s♦♥ ♦s ♠♥t♦s ♥♦♠♥ ♠tr③ ②♥s A s♦ r ts q ajk = 1 s ①st ♥ ♥♥tr ♦s ♥♦♦s j ② k ♦ ajk = 0 ♥ s♦ ♦♥trr♦ ♥t♦♥s ♦r ♠♦ ♦♥t str③ó♥ ♣r t♦ r ♣ ①♣rsrs ♦♠♦

♦♥t str③ó♥ ♠♦ C t♦♠ ♦rs 0 ≤ C ≤ 1 ♥ ♥r 0.1 ≤ C . 1 ♣r s rs ♦r♥ s♦ ♥ s q s ♦sr♦♥ rt t♥♥ ♦s ♦♥tt♦s tr♥st♦s ❬♦ss♥ts ③ ③ ♠ ❪ st út♠ r♥st♥ ♦♥ ♥ ♦r♠ tr♥t ♥r C ♦♠♦ ♣r♦ ♥♦♥trr ♥s í♦s ♥tr ♥♦♦s trá♥♦s ♠♥s♠♦ ♦r♠ó♥ st♦s ♥s í♦s ♥♦♦s r ♥♦♠r ♠♥s♠♦ rr trá♦ ② srá ♦t♦ st♦ ♥ ♣ít♦

Pr s♦ s rs t♦rs C t♦♠ ♦rs ♠② ♣qñ♦s ♦q ♦s ♥s tr♥st♦s ♠♣♥ ♣rs♥ ♦rr♦♥s ♦s ♥♦♦s q s ♥ ①♣t♠♥t rstrs ♣rs♠♥t ♣♦r ♥tr③ t♦r r ♥q st r♠ó♥ s s♦♦ ♦♠♣t♠♥t á ♥♦s ♦♥sr ♥ ♥s♠ r③♦♥s ② q ♣♥ ♣rs♥trs ♦rr♦♥s ♦s ♥ r③♦♥s ss r ❬♦♦♠r ♠♦♥ ❪ P♠♦strrs q C ♣r ♥ r t♦r t♦♠ s♥t ♦r ♦♥st♥t

❬♠♥ rr♥♦ ❪

C =〈k(k − 1)〉2

N〈k〉3 ≃ 〈k〉N

♣♦r ♦ q C s ♦♠♣♦rt s♥♠♥t ♦♠♦ N−1 ♠♦♦ q ♣r rss 〈k〉 ≪ N ♥③ ♦rs ①tr♠♠♥t ♣qñ♦s

tr ♦r♠ str str③ó♥ ♥ r ♦♥sst ♥ ♥r s♣tr♦ ♦♥t str③ó♥ ♠♦ ♥ ♥ó♥ r♦ ♦s ♥♦♦sC(k)

C(k) =1

NP (k)

i∈Deg(k)

♦♥ Deg(k) r♣rs♥t s♦♥♥t♦ ♥♦♦s ♦♥ r♦ k Pr s♦♣rtr s rs t♦rs C(k) rst ♥♣♥♥t k ② t♦♠ ♦r ♦ ♣♦r ♦ ♦st♥t ♥ ♠s rs rs ♥③s s ♦sr♦ ♥ ♦♠♣♦rt♠♥t♦ q rs♣♦♥ ♥ ♥ó♥ s t♣♦❬❱á③q③ s③ ♠ ❪

C(k) ∼ k−β ,

♦♥ ①♣♦♥♥t t♦♠ tí♣♠♥t ♦rs β . 1 r s ♥trs♥t ♥♦tr q st ♣♥♥ ♦♥ k ♥ ♥ ♦r♥③ó♥ rárq ♥ ♦s♥♦♦s ❬s③ ❪ q ♥ ♥ ♥ strtr ♠ó♦s ♦ ♦♠♥s♥tr♦♥t♦s ❬♠♥ s ❪ r ssó♥

r♦ ♠♦ ♣r♠r♦s ♥♦s

r♦ ♠♦ ♦s ♣r♠r♦s ♥♦s ♦s ♥♦♦s r♦ k knn(k) s ♦trrtríst t♦♣♦ó s q ♥ ♦rró♥ r♦s ♥tr ♥♦♦sP♦r ♦ ♦♥stt② ♥ ♠ ♥rt ♣r♦ ♦♥♦♥ P (k′|k) t♥r ♥ ♥ ♥ ♥♦♦ r♦ k′ ♥♦ ♥♦♦ ♦r♥ s r♦k ♦ knn(k) s ♥ ♦♠♦ ❬Pst♦rt♦rrs ❪

knn(k) =∑

k′P (k′|k).

Pr s♦ ♣rtr rs s♥ ♦rró♥ r♦s ♣♦r ♠♣♦ rs t♦rs s♥♦ q P (k′|k) = k′P (k′)/〈k〉 s ♦t♥ knn(k) = 〈k2〉/〈k〉rst♥♦ ♥♣♥♥t k ♠♦♦ q ♥ ♣♥♥ knn(k) ♦♥ k

s ♥ ♥♦r ♦rró♥ r♦s

①st♥ ♦s s♦s ♥ ♠t♦s ❬♠♥ ❪

knn(k) r♥t s q r ♣rs♥t s♦rtt ♥ r♦s st♦s ♦s ♥♦♦s t♥♥ ♦♥trs ♦♥ ♦tr♦s s ♠s♠ s ♥ ♥t♦ r♦ s rr ♦ ♦♥ ♦ t♦ ♦♥ t♦

knn(k) r♥t ♥ st s♦ r♠♦s q ①st ss♦rtt ♥ r

♦s ♣♦r ♦ q ♦s ♥♦♦s r♦ ♦ t♦ t♥rá♥ str ♥s♦♥ ♦s r♦ t♦ ♦

r r♦ ♠♦ ♣r♠r♦s ♥♦s knn(k) ♣r ♦s rs rs ♣r♦tí♥s ♦♥t♠♣♥♦ s♦♦ s ♥tr♦♥s íss trá♥♦s ♦ srt♦rs r♦s ❬s♦ ❪ ♥ st út♠♦ s♦ í♥ só r♣rs♥t ♥ ② ♣♦t♥s knn(k) ∼ k−0.6 ♦♦ró♥ ♥tr♥tí♦s ♠tr ♦♥♥s ♦♥ ♦s ♥s r♣rs♥t♥ ♦♦ró♥♥ ♣♦♥s ♥ ss rs♦♥s ♣s r♦s ② ♥♦ ♣s ír♦s❬rrt ❪

❯♥ ♦r♠ ♥tr s♦rtt s trés ♦♥t s♦r

tt r ♥♦ ♦♠♦ ♦♥t ♦rró♥ Prs♦♥ ♦s r♦s ♦rrs♣♦♥♥ts ♦s ♥♦♦s ♥ ♦s ①tr♠♦s ♥ ♥ ❬♠♥ ♠♥ ❪

r =L−1

i jiki −[

L−1∑

i12(ji + ki)

L−1∑

i12(j2i + k2

i )−[

L−1∑

i12(ji + ki)

r♦ s♦rtts ♣r ♥s rs ♥③s ♥ trtr❬♠♥ ♠♥ ❪

♣♦ N r σr

♦t♦rs ís ♥♦r 52909 0.363 0.002♦t♦rs ♦♦í ♥♦r 1520251 0.127 0.0004

♦s ♦t♦rs t♠át ♥♦r 253339 0.120 0.002t♦rs ♣ís ♥♦r 449913 0.208 0.0002♦♥s st♥ts ♥♦r 573 −0.029 0.037 étr ♥♦r 4941 0.003 0.013

♥♦ós ♥tr♥t ♥♦r 10696 −0.189 0.002❲❲❲ r 269504 −0.067 0.0002♥tr♦♥s ♣r♦tí♥s ♥♦r 2115 −0.156 0.010

♦ós ♠t♦ ♥♦r 765 −0.240 0.007 ♥r♦♥ r 307 −0.226 0.016

s♥♦ ji ② ki ♦s r♦s ♦s ♥♦♦s ♦s ♥ ♦s ①tr♠♦s ♥ i♦♥ i = 1, ..., L tr♠♥t ♦s ♦rs q ♣ t♦♠r ♦♥t s♦rtt s♦♥ −1 ≤ r ≤ 1 ♥ ♣rtr ♣r ♥ r t♦r s♥♦rr♦♥s r♦ s s♥♦ ♥rr q r = 0 ♥trs r > 0 ♥♦rró♥ r♦s ♣♦st ② ♣♦r ♦ s♦rtt ♥ ♠♦ r < 0 stás♦♦ ♦♠♣♦rt♠♥t♦ s♦rtt♦ ♥ r♦s

s rs ♦r♥ s♦ s♥ ♣rs♥tr ♦rró♥ ♣♦st r♦s ②♣♦r ♦ s♦♥ s♦rtts r > 0 P♦r ♦♥trr♦ ♠s s rs t♥♦ós ② ♦ós ts ♦♠♦ ♥tr♥t ❲❲❲ r étr ♥s rs♠tós ② ♥r♦♥s ♣rs♥t♥ ♦rró♥ ♥t r♦s r < 0 rr♦

q ♦rr ♦♥ C(k) ♥ ♠♦s s♦s ♦sr♦s knn(k) t♠é♥rs♣♦♥ ♥ ② ss t♣♦ knn(k) ∼ k−α r r

strtr ♦♠♥s

♦ q ♠s rs ♦♠♣s ♥③s ♥ ♣rát ♣rs♥t♥ ♦rr♦♥s ♦s ♥t♥ss ♦♥ strs ♥♦♦s ♠ás♥s♠♥t ♦♥t♦s ♥tr sí q rst♦ ♦s ♥♦♦s r stút♠ s ♣rs♠♥t ♥ó♥ ♦♠♥ ❬♠♥ ❪ ♣rt♦♥♦ rs ♦♠♣s ♥ ♦♠♥s r♦ ♠ t♥ó♥ r♥t♠♥t ♦ q ♦♥stt② ♥ ♠♦♦ ♦rr ♥ sr♣ó♥ r♥♦ rs♦ r ♦r♥ st té♥ ♣r♠t r ♦♥ s ♥♦r♠s

rs s♣♦♥s ♥ t ♣♦rtr ♥ ♠♦ s q s♠♣ ♣r♦♠ ♦r♥ ó♠♦ ♥♦♥trr ♠♦r ♣rtó♥ ♥ ♦♠♥s ♣r♥ r st s ♣r♥t q ♠♦t♦ ♥ ♣rt ♠♣♦rt♥t ♥stó♥ ♥ ár s rs ♦♠♣s ♥ ♦s út♠♦s 10 ñ♦s

r Prtó♥ ó♣t♠ ♣r ♦s rs tí♣s trtr rt ❩r② ❬❩r② ❪ ♣rtó♥ ó♣t♠ ♦rrs♣♦♥♥t QN = 0.4197 ❬s ❪ Prtó♥ ó♣t♠ QN = 0.546 ❬s ❪♣r r ♦♣ró♥ ♦s ♣rs♦♥s ♦r ♦s srs ❱t♦r♦ sú♥ r♦♣ó♥ ♦♥ ♥t ❬♥t ❪

t♦r ♠ért♦ q ♥ ♣rt♦♥♦ ♥ ♦♠♥s ♥ r s ♥♦♠♥ ♠♦r ♠♥ QN ♠♦r QN ♠ ♥s ♥s ♥tr ♦s ♥♦♦s ♣rt♥♥ts ♥ str ♦ ♦♠♥rt q s ♦t♥rí ♣r s♦ ♥ q ♦s ♥s r♥ ♦♥t♦s♥♦r♠♠♥t ③r ♦♥sr♠♦s ♥ r G(N,L) ♣r q s ♣r♦♣♦♥♥ só♥ ♥ K ♦♠♥s ♥♠♦s eij ♦♠♦ ró♥ ♥s q ♥♥s ♦♠♥s i ② j ♥t♦♥s QN rst

QN =K∑

eii − a2i

s♥♦ ai =∑

j eij st ♠♦♦ eii r♣rs♥t ró♥ ♥s ♣rs♥ts♥ ♦♠♥ i ♠♥trs q a

2i ♦rrs♣♦♥ ró♥ q ♥t♠♥t s

t♥rí ♣r ♠s♠ ♦♠♥ s ♦s ♥s r♥ s♣st♦s ♥♦r♠♠♥t ③r s s♥♦ ♥♦tr q QN ≃ 0 ♣r s rs t♦rs ♠♥trs q

QN = 1 s ♠á①♠♦ ♦r ♣♦s ♣r QN s♥♦ 0.3 < QN < 0.7 ♦s ♦rstí♣♦s ♣r ♠s s rs rs ♥③s ♥ trtr ❬♠♥ ❪r♠♥t ♠♥trs ♠②♦r s ♦r QN ♠♦r srá ♣rtó♥ ♥ ♦♠♥s ♥♦♥tr ♥ r s ♣rs♥t♥ ♥♦s ♠♣♦s ♦♥rt♦s ♣rt♦♥s ó♣t♠s ♣r rs ♣r♠♥t ♥③s ♥ trtr

rs♦s ♦rt♠♦s ♥ s♦ sr♦s ♣r ♥♦♥trr ♣rt♦♥♦ ó♣t♠♦ ♦rt♠♦ r♥♠♥ ♦ r♠♦ó♥ ♥s ❬♠♥ ❪♦♥sst ♥ r ② ♠♥r ♦s ♥s ♠②♦r trá♥st♦ ♦ t♥♥ss♥t♥♥♦ q srá♥ st♦s ♦s q ♦♠♥♥ s ♥ts ♦♠♥s st ♠♦♦ s ♦t♥ ♥ s♦♥♥t♦ P1, P2, ..., PN ♣rt♦♥s ss ♥tr t♦s s ♣♦ss ♣r r G(N,L) s s♦ q q rr♦♠②♦r ♠♦r QN ♦♠♦ ♣rtó♥ ó♣t♠ st s ♦rt♠♦s s♦♦sr♥ ♣r ♥♦♥trr ♣rtó♥ ó♣t♠ ♥tr s q s♦♥ ss ② ♥♦♥sr♠♥t ♣rtó♥ ó♣t♠ ♦ t ♦♠♦ ♠♦s ♠♦str♦ ♥ ♥tr♦ ♣r♦ st tss ❬s ❪ ♥ ♦ tr♦ ♠♦s ♣r♦♣st♦ ♥♦rt♠♦ tr♥t♦ s♦ ♥ té♥ ♦♣t♠③ó♥ ♦ ♥♦♠♥r♦♦ s♠♦ ♣ s♦r t♦r ♠ért♦ QN ♥ ♥♦♥trr ♣rtó♥ ó♣t♠ ♥ s♣♦ t♦s s ♣rt♦♥s ♣♦ss G(N,L) ású♥ ♥ tr♦s r♥ts ♠♦s ♣r♦♣st♦ ♥ ♥ó♥ tr♥t t♦r ♠ért♦ QN ♠ás ♣ró①♠ ♥ó♥ ♥tt ♦♠♥ ♦♠♥③♦ st só♥ ❬s ♦rs♦ ❪

♣rtr s ♥♦♥s ♣rs s ♣♦s r③r ♥ rtr③ó♥t♦♣♦ó s rs ♦♠♣s ♥ st só♥ ♥r♠♦s ♥s s♠s rs ♠ás r♣rs♥tts ♥t♦ ♦♥ ss rtrísts st♥ts

s t♦rs

♥tr♦ sts s st♥ s ♥rs ♣rtr ♦rt♠♦ ♣r♦♣st♦ ♣♦rrös ② é♥② ❬rös ❪ ♥ s♥ ♦♥sst ♥ s♣♦♥r ♥s ③r ♦♥♣r♦ ♦♣ó♥ ♦♥st♥t p s♦r ♥ r ♦♠♣st ♣♦r N ♥♦♦s♥♠♥t s♦s r ♠♣♦ ♥ st ♠♦♦ ♥ í♠t N → ∞ ♠♥t♥♥♦ pN = ♦♥st♥t stró♥ r♦s rs♣♦♥ ♥P♦ss♦♥

P (k) =(pN)k

k!exp (−pN),

♦♥ ♠ 〈k〉 = pN

r t♦r rösé♥② ♦♥ N = 16 ② p = 1/7 ①trí♦ ❬♠♥ ❪

①st♥ ♦trs r♥ts rs t♦rs ♦♥ stró♥ r♦ rtrr P (k) sts s♦♥ ♦t♥s ♠♥t ♥♦♠♥♦ ♠♦♦ ♦♥r♦♥

❬♦♦② ♦♦② ♠♥ ❪ q ♦♥sst ♥ s♥r r♦s ♦s♥♦♦s ♦♥ ♣r♦ ♣♦r P (k) ♣r ♦ str ♦♥tt♦s ③r♥tr ♦s ♠s♠♦s ♦ ♦st♥t ♥ rtríst ♦♠ú♥ t♦s s rs t♦rs s s ♦rró♥ r♦s q s ③ s tr ♥ ♥ C t♠é♥♣qñ♦ t ♦♠♦ s ♠♦str♦ ♣r♠♥t ♥ ó♥ C ∝ N−1tr rtríst ♦♠ú♥ s rs t♦rs s s st♥ ♠ ♥tr♥♦♦s l ∝ lnN

♥ s rt♦ q s rs t♦rs ♥♦ ♦♥stt②♥ ♥r♠♥t ♥ ♥♠♦♦ rs s♦s ♦ t♥♦ós s♦♥ t♠♥t t③s ♦♠♦ ♥♣r♠r ♣r♦①♠ó♥ ♦ q r♥♥ ♣♦s trt♠♥t♦ ♥ít♦ ♣♦r ♠♣♦ trés ♦r♠s♠♦ ♥ó♥ ♥rtr③ ❬♠♥ ❪

①♣r♠♥t♦ s♠♥ ♣só♦♦ s♦ t♥② r♠ ♣♦ ♥ ñ♦ ❬r♠ ❪ ♠♦stró q ♥ ♣r♦♠♦ s ♥sr♦ trsr t♥s♦♦ 6 ♥s ♣r ♦♠♥r ♥ ♥♦ qr ♦♥ ♦tr♦ q ♦ ♦t♦ st ♥ó♠♥♦ ró ♥♦♠r ss r♦s s♣ró♥ ♥♦♠♥ó♥ ♦♥♣t♦ ♥tr♦♦ ♣♦r t♦r ú♥r♦ r②s r♥t② ♥ s♥t♦ ♥♥s ñ♦ ❬r♥t② ❪ st r♥í ♥tr ♥♦♦s

♥ r s ♣tr ♣♦r ♦s ♠♦♦s rs t♦rs ♥ ♠r♦ ♠s s rs s♦s ② t♥♦ós ♥③s ♥ ♣rát t♠é♥ ♣rs♥t♥t♦ C ❬♠♥ ♠♥ ❪

s ♥♦ Pqñ♦

♦ q ♠s s rs ♥③s ♥ trtr ♥ ♣rtr srs s♦s ♣rs♥t♥ ♦♠♦ rtrísts st♥ts st♥ ♣qñ ∼lnN ② t♦ C ≫ 1/N s ♥sr♦ ♣r♦♣♦♥r ♥ ♠♦♦ r♠♥t♦ rs♦♠♣s q ♥tr♦③♥ sts ♦s ♣r♦♣s ♦ ♠♦♦ ♣r♦♣st♦♣♦r ♥♥ ❲tts ② t tr♦t③ ♥ ❬❲tts ❪

r ó♥ ♣r♦s♦ r♦♥①ó♥ ♦♥ ♣r♦ p s♦r r t♣♦ ♥♦ rr ❬❲tts ❪ ♠♥tr p s ♥r♠♥t ♥t t♦s s♠♥②♥♦ l ♣r♦ ♦♥sr♥♦ ♦r t♦ C t♦ ♥r♠♥t♦ p s♦r str♥ C ② s♦r st♥ ♠ L t♠é♥♠♦s ♥♦r♠③♦s rs♣t♦ ss ♦rs p = 0

♠♦♦ rs ♥ q st♥ ♠ ♣qñ s ♥ rtríst ♥tr s rs t♦rs ♠♥trs q t str③ó♥ s♣r♦♣ s rs rrs ♠♦♦ q s rs q ♦♥♥ ♠s rtrísts ♥♦♠♥s rs ♠♥♦ ♣qñ♦ ♥ str ♠t ♠♥♦♥tr s t♦rs ② s rrs s sí ♦♠♦ ❲tts ② tr♦t③ ♣r♦♣sr♦♥ ♥♠♦♦ q ♣rt ♥ r t♣♦ ♥♦ rr ♦♥ t♦ C ♣r♦ t♠é♥t♦ l s♦r q s r③♥ r♦♥①♦♥s ③r ♥s ♦♥ ♣r♦ ps r♦♥①♦♥s s♦♥ s rs♣♦♥ss ♥♦r♣♦rr t♦r r s♠♥②♥♦ l ♦♥sttrs ♥ t♦s s♦rtts s♥ r ♠♣ ♣r♥♣♦ ♥♠♥t ♠♦♦

s rs ♦t♥s ♣rtr ♠♦♦ ❲ttstr♦t③ ♣rs♥t♥ ♦ l♥q ♦♥sr♥♦ t str③ó♥ ♣r ♦rs ♣qñ♦s p ♥ rá♦ r s ♠str ♦ó♥ s♠tá♥ C ② l ♥ ♥ó♥ p♣r ♠♦♦ ❲ttstr♦t③ P ♦srrs q ♣r ♥ ♣r♦ r♦♥①ó♥ p = 0.01 r rst♥♦ ♣rs♥t s rtrísts ♠♥♦♣qñ♦ s rs ♦♦ró♥ ♥tr ♥tí♦s ♦t♦rs ② ♠st♥tr ♠s ♦trs rs s♦s ♣♥ t♦rs ♦♠♦ rs ♠♥♦ ♣qñ♦ ♥ ♠r♦ ♠s s ♣rs♥t♥ s ③ str♦♥s r♦P (k) ♦ ♣s q ♥ ♣r♥♣♦ ♥♦ s♦♥ r♣r♦s ♣rtr ♠♦♦ ❲ttstr♦t③

s rs s

s rs ♦srs ♥ ♣rát ♣rs♥t♥ stró♥ r♦ ♦♥♦ t♣♦ ② ♣♦t♥s P (k) ∼ k−γ r♥t♠♥t ♦♥ 2 ≤ γ ≤ 3 ♣♦r ♦q r♥ ♥♦♠r rs rs s r sr♦rs ♥tr♥t ❲❲❲ r t♦rs ♣ís rts rs ♠tós ②r♥ts rs ♦♦ró♥ ♥tr ♥tí♦s ♦♥stt②♥ ♥♦s ♠♣♦s ♣r♠át♦s st ♦♠♣♦rt♠♥t♦ s s♣r q st ♥ó♠♥♦ ♠r♥ts rst♦ ♥ ♦♠♥ó♥ ♠♥s♠♦s ♠r♦só♣♦s ♦ s♥t♠♥t s♥♦s ② ♥rs ♣r♠r ♠♦♦ r♠♥t♦ rs rs s ♥tr♦♦ ♣♦r rs ② rt ♥ ❬rs ❪ ♣r♦♣♦♥ ♥ts ♥r♠♥t♦ r♦ ♣r♦♣♦r♦♥ r♦ t ♦s ♥♦♦s ♥ ♦trs♣rs q♦s ♥♦♦s ♦♥ ♠②♦r r♦ k t♥rá♥ ♠②♦rs ♥s ♥♦r♣♦rr ♥♦s ♥♦s st ♠♥s♠♦ s ♥♦♠♥ ♥ ♣rr♥ ② s♦♣r♦♣st♦ ♣r♠♥t ♥ ♦tr♦s ♦♥t①t♦s ♣r str str♦♥s t♣♦② ♣♦t♥s ❬❨ ♠♦♥ ❪ P ♠♦strrs q ♣rtr ♥ts ♥ ♣rr♥ ♥ t♣♦ k/2L s ♦t♥

P (k) ∼ k−3.

♠♥s♠♦ ♥ ♣rr♥ ♦♥stt② ♣r♠ ♣t♦ ♣♦r ♦♠♥ ♥tí ♣r ♥rr rs rs s ♥ ♠r♦ ú♥ sst ♦r♥ t ♠♥s♠♦ ❯♥ s ♣ótss ♣r♦♣sts s♦st♥ q ♦r♥ ♥ ♣rr♥ ♥ s rs s♦s s t♥♥ ♦r♠ó♥ trá♥♦s ♥tr ♥♦♦s s r ♠②♦r r♥ ♣ró♥ ♥s tr♥st♦s q t♥♥ rrr trá♥♦s ♠st ❬s♥ ♦♠ ❪ ♥t♠♥t s♦ ♣r♦♣st♦ ♥ ♠♥s♠♦ tr♥t♦ q

st ♣ró♥ ♥ ♣rr♥ ♦ ♥ stró♥ tr♦é♥♣♦r t s♦ ♥tr ♦s ♥♦s ❬Prr ♥ ❪❱♦r♠♦s s♦r sts ♣ótss ♠ás ♥t ♥♦ ♣r♦♣♦♥♠♦s ♥ ♠♦♦ r♠♥t♦ rs q ♦♥ ♦ó♥ t♠♣♦r ♠r♦só♣ ♦♥ s♣r♦♣s ♠r♥ts r ♠ ♥ t♠♣♦

♦♠♦ ② ♠♦s st♦ ♥ ♣ít♦ ♥tr♦t♦r♦ strtr t♦♣♦ó r ♥ ♥ ss rtrísts ♣r♦♣ó♥ Pr s♦ ♣rtr ♥ ♥r♠ ♦♥ ♥á♠ ♣r♦♣á♥♦s s♦r ♥ r r s♦♥ P (k) ∼ k−γ ♣ rs q ♠r ♣♠ rs♣♦♥ λc = 〈k〉/〈k2〉q rst λc = 0 ♥♦ γ ≤ 3 ♦ q 〈k2〉 → ∞ ❬Pst♦rt♦rrs ❪st s t♥ s♦♦ ♥ ♠♣♦ q ♥♦t ♥ó♠♥♦ q ♣rt♥♠♦s ♥t③r♥ st tss ♠♣♦rt♥ st♦ strtr s rs ♦♠♣s② s í♥t♠ ró♥ ♦♥ ♦s ♥ó♠♥♦s ♣r♦♣ó♥

♣ít♦

s♠♥ó♥ ♦♣♥♦♥s ②

♥ó♥ ♠♦♦ r stát

❮♥ ♦♦ r s♦

♦♦ s♠♥ó♥ tr ② ♥ó♥

s♠♥ó♥ ♦♣♥ó♥ rs♣t♦ ♥ó♥

♦♠♥t♦s ♥t♥ó♥

sr♣ó♥ ♦rít♠ ♠♦♦

r♥só♥ ♥♦rs ♥♦♥♦rs

♥áss r♦ts sr♠♣ó♥

♥áss strs

♥♥ s♦r str ♠á①♠♦

t ♦rtr ♥

♦♥s♦♥s ♣r♠♥rs

♥t♦r♥♦ s♦ ♥ ♥ é♥ss sst♠ r♥s ♦s ♥♦sás ú♥ ♠♦s ♦♠♣♦rt♠♥t♦s ♠♥♦s s♦♥ qr♦s ♣♦r ♥ ♣r♦s♦ ♠tó♥ ♥ st ♣ít♦ ♥③♠♦s ♠♣t♦ ♥t♦r♥♦ s♦ ♥ ♣♦strrs♣t♦ ♥ó♥ ♦s ♥♦s trés ♥ ♠♦♦ s♦ ♥♥ts s♦s s♣st♦s ♥ ♥ r ♦♠♣ Pr ♦ ♦♥sr♠♦sq ♦s ♥ts ♥trtú♥ ♣♦r ♠tó♥ ♣s ♣♦r ♥ ♦♥ ♦tr♦s s♥t♦r♥♦ st ♠♦♦ ♥tr♦♠♦s ♦♥♣t♦ ♦♠♦ trés ♥♠♥s♠♦ ♥s♣r♦ ♥ ♠♦♦ ①r♦ ❬①r♦ ❪

♥③♠♦s ♥♥ ♦s r♣♦s ♥t♥ó♥ ♥ s♠♥ó♥ ♣♦str ♥♦♥ó♥ ♦str♠♦s q ♥á♠ s♥ ♠♥s♠♦♣r♦♣st♦ r ♦r♠ó♥ strs st♦s ♥♦ ♥♦s ♣♦r ♦

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

t♥t♦ ss♣ts ♦♥trr ② ♣r♦♣r ♥ ♥r♠ ♥♦s tr♥s♠t♣♦r ♦♥tt♦ ♦♥sr♠♦s sr♠♣ó♥ ♦♠♦ s♦ st♦

♦♠♦ ♠♦str♠♦s ♥ só♥ ♣rs♥t ♣ít♦ ♣r♦ r♦t ♣r ♠♦♦ qí ♣r♦♣st♦ rst ♥♦t♠♥t s♣r♦r s♣r ♣r ♠♦♦s q s♠♥ stró♥ ♥♦r♠ ♦s ♥ts ♥t♦st♥♦ st ♠♦♦ t♦ ♥♠♥ r♣♦

♦♦ r s♦

s rs ♦♠♣s ♦♥stt②♥ ♥ r♣rs♥tó♥ ♥tr♠♦ s♦ ♦♥srr s tr♦♥s ♦s ♦♥tt♦s ♥tr ♦s ♥ts s♦s ♦s♥♦♦s r r♣rs♥t♥ ♦s ♥ts ♠♥trs q ♦s ♥s ♥ ♥t ♦s í♥♦s ♥tr ♦s ♠s♠♦s st♦s í♥♦s ♣♥ ♥r ③♦s ♠rs ♠st r♦♥s ♦rs s♠♣ ♣r♦①♠ s♣ t st ♠♦♦ ♥t s ♥♥tr ♥♠rs♦ ♥ ♥ ♦♥t①t♦ s♦ tr♦é♥♦ q ♦♥rá ♥ s ♦s♠♦só♥

r r stát ♦♥tt♦s ♥tr ♥ts s♦s ♦♥sr♥♦ ♥♦♦ ♥tr r ss ♥s ♦s ♥♦♦s rst♥ts r♦♦s♣♥ tr♥s♠tr t♥t♦ ♦♣♥♦♥s ♦♠♦ ♥ ♥r♠ ♥♦s

♥ ♦♥t①t♦ ♥str♦ ♠♦♦ s♠♠♦s q ♦s ♥ts s♦s r♣rs♥t♥ r♣♦s ♠rs ♦♥ ♦s ♣rs ♥ r r ♥ó♥

♦♦ s♠♥ó♥ tr ② ♥ó♥

s ú♥♦ ♦ ♥ st ♠♣♠♥tó♥ ♦s ♥ts s♦s stá♥ s♣st♦s♥ ♥ r r N = 50 × 50 ♥♦♦s ♦♥ ♥s ♣r♠r♦s ② s♥♦s♥♦s ♠ás r♦ ♥s ♦♥s ♣r ♦♠♣tr ♥ r♦♥ r♦ ♥♦r♠ k = 〈k〉 = 16 r ♥ st ♣ít♦ s♦♦ ♦♥srr♠♦s ♥s ♦♠♦é♥♦s ② stát♦s ♠♥trs q ♥ ♣ró①♠♦ ♥tr♦r♠♦s ♦♥tt♦s ♣rs♦♥s tr♥s♠t♥ t♥t♦ ♦♣♥♦♥s ♦♠♦ ♥r♠s ②♥♦♣rs♦♥s s♦♦ tr♥s♠t♥ ♦♣♥♦♥s sí ♦♠♦ t♠é♥ ♥ ♠♥s♠♦ r♦♥①ó♥ ♣tt ♥s

♦♦ s♠♥ó♥ tr ② ♥

♦♥sr♠♦s ♥ ♦♥ó♥ ♥ s♥ r♥s r♦ts ♣é♠♦s r♥ts sr♠♣ó♥ ♠é♥ ♦♥sr♠♦s ♥ ♣♦ít ♥ó♥ ♦♥tr t s s♦ ♥♦ ❯♥♦ ♦ ♥ ♦♥ s ♣♦s ①♠rs ♠s♠ sr♠♥♦♥ ♦ó♥ ♦♥♥ ♦♠♦ s ♥ ♥♦s st♦s ❯❯ ♥ sts♥r♦ só♥ ♥r ♥♦ ss ♦s r ♥ ♦s ♣rs s♠s♠♦s út♠s r♥s ♣♠s sr♠♣ó♥ ♥ st♦s ♣íss srr♦♦s t♥ é ♣♦r ♦ q ♦s ♣rs ts r♥ rstr♦s ♥s ss ♦♥s♥s ♥ s ♠♠♦r ♥♠t ♥t sts r♥st♥s ♣r♣ó♥ rs♦ ♥♠♥♥t ♦♥t♦ ♥str ♣ótss s q só♥strá ♥ ♠②♦r♠♥t ♥ ♥ ♣r♦s♦ ♠tó♥ s♦ ♣rtr s♣♦strs q ♦s ♥ts s♦rs r♦♥ s ♥t♦r♥♦

♦s ♠♦♦s ♠ás s♥♦s ♦♥sr♥ ♥ ♦♣ó♥ ♥r rs♣t♦ ♥ó♥ q ♥t ♦♣t ♣♦r s♠♣ ♦♣♥ó♥ ♠②♦rtr s ♥t♦r♥♦ ❬té ❪ ♦♦s ♠ás ♦♠♣♦s s♠♥ ♥ts r♦♥s q♠ás ♠♣♠♥tr ♣r♦s♦ ♠tó♥ s♦ ú♥ rs♦ ♦♥t♦ ② ♦s t♦s ♥♦♦s ♥ ♥ ♦♥t①t♦ t♦rí ♦s❬ ❳ r♦ ❪ ♥ ♠♦s s♦s s ♦♥sr q ♥♥ ♦s ♦♥tt♦s s ♦♠♦é♥ ♠♦♦ q t♦♦s ♦s ♥♦s ♥♠t♦s s♦♥♠♥t ♥②♥ts

s♠t ♥ rs♦s trs ét♥♦s ♦ ♥r♦♥s tú ♠♦♦ t③♦r s r♦♥s s♦s r♦ ♥ ♦ s♠t ♥tr ♥♦s s ♥ ♦♥♣t♦ q s♦ ♥tr♦♦ ♥ s ♥s s♦s ♦♥r ♥♦♠r ♦♠♦ ❬③rs Prs♦♥ ❪ ♠♦♦ qí♣r♦♣st♦ ♥♦r♣♦r st r♦ ♦♥ tr♦♥ ♦ ♦♠♦

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

♥tr ♥ts Pr ♦ ♠♣♠♥t♠♦s ♥ r♥t ♦r♥ ♠♦♦ ♥ts ♣r♦♣st♦ ♣♦r ♣♦tó♦♦ ② ♠t♠át♦ ♦rt ①r♦ ♥ ❬①r♦ ❪ q♥ ♦ ó ♦♥ ♦t♦ ①♣r ♦r♥ ♦♥s♥s♦s♦ trés s♠♥ó♥ tr ② ♦r♠ó♥ ♦♠♥♦s ♦♥r♥ tr r

s♠♥ó♥ ♦♣♥ó♥ rs♣t♦ ♥

♦s ♠♦♦s ♠ás ①t♥♦s ♥ t ♦♥sr♥ ♠tó♥ s♦♦♠♦ ♠♥s♠♦ rs♣♦♥s s♠♥ó♥ ♣♦str rs♣t♦ ♥ó♥ ♦s ♦s ♣r♦♣♦♥♥ ♥ ♦♥t①t♦ t♦rí ♦s q♦♠♣r♥ ♥ts r♦♥s ♦♥ ♥♦r♠ó♥ ♦ s ♥t♦r♥♦ ②♦ ♦q♥s ♥ s ♣♦só♥ ó♣t♠ rs♣t♦ ♥ó♥ ♥tr ♦s ♣♦ss ♥r ♦ ♥♦ ♥r ❬ ❳ r♦ ❪tr♠♥t q♦s ♥ts q ♦♣t♥ ♣♦r ♥r r♥ ♥♦ ♥♠♥ ♣r♦ s♠♥ ♣♦r ♠♣♦ ♦st♦ ♦s t♦s s♥r♦s ♥ r ♥♦ ♥r t ♦s ♦st♦s s♦♦s ♦♥ ♥ ♣r♦ ♠♣ rs♦ ♦♥t♦ sr♠♣ó♥ ♥ ♦♥t①t♦ st♦s ♠♦♦s strt ó♣t♠ ♥t ♣♥rá ♥t♦♥s ♦st♦ ♥ó♥ ② rs♦ t♥t ♦♥t♦ ♦ ♦st♥t ♥♦ s ♦♥sr strtr rs♦ t♦rí ♦s ♣♦r sí s♦ ♥♦ s s♥t ♣r str r♣ó♥ ♥♦s ♥♦♥♦s rs♣♦♥s ♥♦s ♦s r♦ts r♥ts ♥♣íss srr♦♦s s ♣♦r ♦ q ♥ ❬ ❪ s rrr ♥ ♦♠♥ó♥ ♥♦s r♦♥s q s♥ ♦♣t♠③r s ♥♦ ♦♥ ♦tr♦s q♥ ♥ ♠♥s♠♦ ♠s s rr♦♥ ♠tó♥ s♦ q ♦♥ ♦r♠ó♥ strs ♥♦♥♦s

str♦ ♥♦q s r♥ ♥♦t♠♥t ♦s ♣r♥ts s♠r q ♦♥t ♥ó♥ s ♣r♦♣ ♦♠♦ ♥ s rtrísts trs♥ ♥ r♥t ♦r♥ ♠♦♦ ①r♦ st ♠♦♦ ♥♦s ♠♦s ♣r♠ ♥ts r♦♥s ♥ sr strts ó♣t♠s ♦♥♦s♦♥ ♥ts rr♦♥s q ♣rt♥ ♠tó♥ s♦

r♦ tr ♥t s♦ ♥ st s♦ ♦♥sr♥♦ ♦♥♥t♦ st♦s s♦rs q ♥tr♥ ♠ ♥ stó♥ ♥rá ♥♦ ♣♦r F +1

rtrísts ♦♥ rtríst ♦♥ r♣rs♥t ♣♦str ♥r ♥ó♥ ♦ ♠t♣ rs♦s trs s♦ ♦♥ ♣♦strrs♣t♦ ♥ó♥ srá q = 2 ♠♥trs q rst♦ s rtrísts

♦♦ s♠♥ó♥ tr ② ♥ó♥

t♥rá♥ ♠t♣ ♦♠♦é♥ q ≥ 2 ♠r♥ st st♥ó♥ ♣♦strrs♣t♦ ♥ó♥ s ♥♦r ♥ ♥á♠ ♣tó♥ ♣♦r ♠tó♥♦♠♦ ♥ rtríst tr ♠ás r♦ ♦♠♦ ♥tr ♦s ♥ts i② j s ♥ ♦♠♦ ♥t rtrísts ♥ s q ♦♥♥ ht(i, j) t♠♣♦ t q ♣r ♠♦♦ ①r♦ s F rtrísts trs①②♥♦ ♣♦str rs♣t♦ ♥ó♥ s ♥♦r♣♦r♥ ♥ ♠♦♦ ♦♥ ♥ srr♦r ♦♠♦ tr♦é♥ ♥tr ♦s ♥ts st ♠♦♦ s♠♥ó♥ s r♥ts ♦♣♥♦♥s ♥ ♣rtr ♥ó♥ s rá♦♥ ♠②♦r ♣r♦ ♥tr q♦s ♥ts ♦♥t♦s q ♣rs♥t♥ t♦r♦ ♦♠♦

♦t♦ s q ♦s ♥ts s♦s ♣t♥ s ♣♦str rs♣t♦ ♥ó♥ ♠t♥♦ q♦s s ♥t♦r♥♦ ♦♥ ♦s q t♥♥ t♦ r♦ ♦♠♦ st♦ s s♠♦s q ♣♦str rs♣t♦ ♥ó♥ s ♠t♦ ♥ ♥ ♣r ♥ ♦trs rtrísts trs Pr ♦ ♥♦r♣♦r♠♦s ♥ ♠r ♦♠♦ κ ♣rtr ♦s ♥ts ♦♠♥③rá♥ ♥trtr ♥ ♥str♦ s♦ ♠♦s κ = 2 ♥q t♠é♥ ♠♦s ♦♥sr♦ κ = 3 ♦t♥♥♦ tt♠♥t é♥t♦s rst♦s ♦♥ κ = 2 t♠♦s ♥tró♥ s♦ ♥tr ♦s ♥ts q s♦♦ ♦♠♣rt♥ ♣♦str ♥r ♥ó♥ ② ♦♥ ♦ s sr ♦ó♥ ♥ qr♦ s♦r♥t ♦♥s♥s♦

♦ ♣r♦ ♥tró♥ P (i → j, t) ♥tr ♦s ♥ts i ② j ♥rá ♣♦r

P (i → j, t) =

0 s ht(i, j) < κht(i,j)F+1

s ht(i, j) ≥ κ

♥♦ ♠♦♦ ♥tr♦♦ ♣♦r ①r♦ ♥♠♦s ♥ t♦r ♦♣♥♦♥sV

i(t) = (V i1 (t), V

i2 (t), ..., V

iF+1(t)) ♦♥ V

i(t) ∈ 0, 1 × IF s♥♦ ♦♠♣♦♥♥t

V iF+1(t) ♦rrs♣♦♥♥t ♣♦str rs♣t♦ ♥ó♥ ♠♦♦ q

V iF+1(t) =

0 s ♥t i s ♥♦r1 s ♥t i s ♥♦♥♦r

♦ ♣r♦s♦ ♣tó♥ ♣♦r ♠tó♥ s♦ t♦r ♦♣♥♦♥sVi(t)

srá sr♣t♦ ♥ ó♥ ♣ít♦

V in(t+ 1) =

V jn (t) ♦♥ ♣r♦ P (i → j, t+ 1)

V in(t) ♦♥ ♣r♦ 1− P (i → j, t+ 1) .

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

♦♠♥t♦s ♥t♥ó♥

❯♥♦ ♥str♦s ♦t♦s s ♥③r ♥♥ ♦s r♣♦s ♥t♥ó♥ ♥ ♦rtr ♥ ❱ st♦s stá♥ r♣rs♥t♦s ♣♦r♥ ♦♥♥t♦ ♥ts Sstub q ♦♣t♥ ♥ ♣♦str ♥♦♥ó♥ ♦r♠ ♦st♥ ♦ ♣♦str rs♣t♦ ♥ó♥ t♦♦s ♦s ♥ts♣rt♥♥ts ♠♦♠♥t♦ ♥t♥ó♥ srá

V iF+1(t) = 1 ∀i ∈ Sstub

♣r t♦♦ t♠♣♦ t st♦s ♥ts s♦♥ s♣st♦s ③r ♥ r s♦ ② ♦q ♦♥stt②♥ ♥ r♣♦ r♠♥t ♠♥♦rtr♦ qí ♥ ♠ás s♠r♠♦sq ♥♦r s♦♦ 1% ♣♦ó♥ t♦t ♦♥srr♠♦s ♥ ú♥ ♦♥ró♥ s♣ ♥ ♥♦♥♦rs ♦st♥♦s ♦♥ ♥ ♣♦r ♦♠♣rrst♥ts r③♦♥s ♠♦♦ ♥♣♥③á♥♦♥♦s sí s stró♥ ♥ ♦ q ♦♥ ♥ 1% ♥ts ♦st♥♦s ú♥ st♠♦s ♦s ♠r ♣r♦ó♥ r ♦♥sr ♣r♦ ♦r♠ó♥ strsr♥s ♥ts ♣rt♥♥ts Sstub s s♣r ② ♥♦ t♥rá ♥♥♥ ♦ó♥ ♠♥s♠♦ s♠♥ó♥

Pr ♠♥r st ♥tr♦ó♥ ♠♦♦ r♠♦s s sr♣ó♥ ♦rít♠ ♦♠♣t s♣♥♦ s ♦♥♦♥s ♥s ② ♦s s♣t♦s ♦♠♣t♦♥s r♥ts ♦ q s♠♦s r♣r♦r stó♥ ♥ rst ♠♣♦ ♦♥s♥s♦ ♥ ♦r ♥ó♥ ♦♠♥③r♠♦s s♠♥♦ ♥ s♥r♦ ♦♥ t♦♦s ♦s ♥ts s♦♥ ♥♦rs V i

F+1(t) = 0 s♦ 1% ♥♦♥♦rs ♦st♥♦s ♣rt♥♥ts Sstub st♦s út♠♦s s♦♥ ♦s♥♦r♠♠♥t ③r ② sts♥ V j

F+1(t) = 1 ∀j ∈ Sstub ♣r t♦♦ t♠♣♦ t

♥ t♦♦ ♦s s♦s ♠♦s ♦ F = 10 Pr ①♣r ♠♦t♦ st ó♥ s s♥t ♦♥ r♠trs s s♦♥s ♦s ♣ró♦s ó♥ r♥t♥ ♥t ♦♥ s♦♥s s r♠♥t ♥ s rsó♥ ♦♥♥ ♠♥trs q ❨♦r ♠s ❯❯ t♠é♥ ♥t ♦♥ s♦♥s rs s② rs ♦ s♠♥t ♦rr ♦♥ Pís s♣ñ s♠♣r rré♥♦♥♦s ss rs♦♥s ♦♥♥ st ♦ r q ♦s t♠s ♥trés ♥rs♦♥ rt♠♥t ss♦s ♣♦ít ♦ ♥tr♥♦♥ ♦♥♦♠í ② ♥♦♦s♣♦rts s♣tá♦s ② rt s♦♥ ♥♦s ♦s ♠♦♦ q F = 10 r♠♥t ♥t t♠s rs♣t♦ ♦s s ♦s ♥ts s♦s

♠♥st♥ ♦♣♥ó♥ ♥♠♥t ♠r ♦♠♦ s stó ♥ κ = 2t ♦♠♦ s sñó ♦♥ ♥tr♦r

♦♥sr♠♦s ♥ r s♦ ♥tr♦r♠♥t sr♣t ♦♠♣st ♣♦r N =

2500 ♥ts r r 50×50 ♦♥ 8 ♥s ♦♥s ♣♦r ♥♦♦♣r ♦♠♣tr 〈k〉 = ki = 16 ∀i ∈ 1, ..., N ♦♠♦ s ♠str ♥ r ② L = 20000 ♥s ♦ ♣♦str rs♣t♦ ♥ó♥ rst♦ srtrísts ♦s ♥ts s♦♥ ♥♠♥t s ♥♦r♠♠♥t ③r ♥1, 2, 3, ..., q ❯♥ ③ ♥s s ♦♥♦♥s ♥s ♠♦♦ s ♦♠♥③♦♥ t③ó♥ sst♠ ♥ ♣s♦ qí s♠♦③♦ ♦♠♦ ♥ t♠♣♦♠♥s♦♥ srt♦ t s♥t ♦rt♠♦

❯♥ ♥t ♥t i s ♦ ♥♦r♠♠♥t ③r ♠♥trs q ♥♦ ss ♣r♠r♦s ♥♦s j s ♦ ♦♠♦ trt t♠é♥ ♥♦r♠♠♥t ③r

♦t♥ ht(i, j) ② ♣r♦ ♥tró♥ P (i → j, t) ♦ s ③r ♥ rtríst n ♥ q i ② j ♥♦ ♦♥♥

n 6= F + 1 s r q rtríst ♥♦ s ♣♦str rs♣t♦ ♥ó♥ ♥t i s ♣r♦s♦ ♣tó♥ sr♣t♦ ♥

n = F + 1 ♥ ♦s tr♥ts

i ∈ Sstub ♥ st s♦ i ♦♥sr s ♣♦str rs♣t♦ ♥ó♥V i

F+1(t+ 1) = V iF+1(t)

i /∈ Sstub ♥ ②♦ s♦ ♥t s ♣r♦s♦ ♣tó♥ sr♣t♦ ♥

♦♥ó♥ t♥ó♥ ♦♥sr♠♦s ♦s ♣♦ss ♦♥♦♥s t♥ó♥

♦rt♠♦ s t♥ ♥♦ ♥③ ♥ ♦♥ró♥ s♦r♥ts r ♥♦ ♥♦ s♦♥ ♣♦ss tr♥s♦♥s ♦♥s

♥ ♦s s♦s ♦♥ ♣r ♦♣♥ó♥ ♥ó♥ ♦♥r ♦s ♥ts Sstub ♣ r r ♣qñs t♦♥s ♥tr♠t♥ts ♥ ♣♦str rs♣t♦ ♥ó♥ ♥ s ♥t♦r♥♦ s ♣♦r♦ q ♠♦s ♦♠♦ ♦♥ó♥ t♥ó♥ ♦♥ ♠♣♦s r③r tr♥s♦♥s ♣r t♦s s rtrísts trs

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

①♣t♥♦ ♣qñs t♦♥s ♥ ♣♦str rs♣t♦ ♥ó♥

t → t+ 1 t♦r♥ ♣s♦

♦♥ rs♣t♦ ♦♥ó♥ st ít♠ ♣ rs s♦ ♥ q ♥ ♥t i ♥♦♥♦r ♦st♥♦ rst r♦♦ ♣♦r ♦tr♦s ♥♦rs♦♥ ♦s q r ♥ t♦♦ s r♦ tr s♦ ♥ ♣♦str r♥t ♥ó♥ rr ♥t♦♥s q ♦s ♥ts ♣rt♥♥ts ♥t♦r♥♦ i

nn(i) ♣♦rá♥ ♠tr ♥tr♠t♥t♠♥t s ♣♦str ♥♦♥ó♥ i ♣r♦ rt♦r♥r ♣♦str ♣r♦♠♥♥t s ♥t♦r♥♦

♦sr q ♥♦s ♦♣ s ró♥ ♥t ♥♦rs ② ♥♦♥♦rs ♥ r s♦ ♥ r s ♣rs♥t ró♥ ♥♦♥♦rs ♣r♦♠ s♦r 1000 r③♦♥s ♦♠♦ ♥ó♥ q

Pr ♦rs ♣qñ♦s q ♦s ♥ts ♥♦♥♦rs ♦st♥♦s ♦r♥ s♠♥r s r♦ ♥③♥♦ ♥ ♠♣♦ ♦♥s♥s♦ ♥♦♥♦r ♥ ♣♦ó♥s♥♦ st st♦ s♦r♥t ♠♦♥♦tr sst♠ ♥ ♠r♦ t♦♠r♦rs ♣ró①♠♦s q ≈ 170 ♦♥s♥s♦ ♥♦♥♦r ♦♠♥③ r s ♥tr♦♥s s♦s rst♥ ♠ás ís strs ♦ ♥r♠♥t♦ ♥ú♠r♦ rs♦s trs ♥♦ r ♦♠♥♦s ♦♥s♥s♦ ♥♦♥♦rr♦r ♦s ♥ts ♦st♥♦s ♦①st♥♦ ♦♥ ♦♠♥♦s ♥tr♦s ♣♦r♥♦rs st ♦ s ♠♥st ♥ r ♦♥ s ♠str ♥ rá♦ ♥s ♦rrs♣♦♥♥t ♣r♦♠♦ s♦r 500 r③♦♥s ♠♦♦ ♥t♥s ♦♦r r♦♦ ♥t ró♥ r③♦♥s ♥ s q ♥t ♥③ó qr♦ ♦♠♦ ♥♦♥♦r

tr♥só♥ s r qr♦ ♣rs♥t ♣♦r sst♠ s rtríst ♠♦♦ ①r♦ ❬♠♠ ❪ s ♦♥s♥s♦ ♥♦♥♦r♦rrs♣♦♥ st♦ s♦r♥t ♠♦♥♦tr ♦♥ t♦♦s ♦s ♥ts ♦♠♣rt♥ ♦s ♠s♠♦s rs♦s trs ♥ ♥t♦ s ♦♥s♥s♦ ♥♦r①♣t♥♦ ♦s ♥ts ♣rt♥♥ts Sstub s ♥ ①st ♦♥s♥s♦ ♥♥t♦ ♦♣♥ó♥ ♥ó♥ ♥♦ ♦rr ♦ ♣r♦♣♦ rs♣t♦ rst♦ srtrísts trs ♣♦r ♦ q s ♦rrs♣♦♥ ♣♦r③ó♥ trt♦t ♥ st út♠♦ s♦ ♦♥s♥s♦ ♥♦r ♣r♦♥ ♦♥ró♥♥ ♣r 99% ♦s ♥ts ♦♠♥③♥ ♣r♦s♦ ♦♥ ♦♣♥ó♥♣r♦♥ó♥

r r♥só♥ ♦♥s♥s♦ ♥♦♥♦r ♥♦r ♥ ♥ó♥ ♠t♣ rs♦s trs q ♠ñ♦ ♥♦r♠③♦ str♠á①♠♦ ♠♦♥♦tr SM/N ♣r s ♣r♠rs F ♦♣♥♦♥s ♥ ♥ó♥ q

♥ r ♠♦str♠♦s t♠ñ♦ ♥♦r♠③♦ ♦rrs♣♦♥♥t ♦♠♥♦ ♠♦♥♦tr ♠②♦r ♣♦ó♥ SM/N ♦♥ SM t♠ñ♦ str♠á①♠♦ ♦♥sr♥♦ s♦♦ s ♣r♠rs F ♦♣♥♦♥s s♦ ♥ó♥ ♥♥ó♥ q P ♦srrs tr♥só♥ ♦♥s♥s♦ ♣♦r③ó♥ trt♦t ♠♥♦♥ ♥ ♣árr♦ ♥tr♦r ♦♥ ♦ s ♥ q ró♥♥ ♥♦♥♦rs s ♥♥tr ♦♥♦♥ ♣♦r ♣♦s ♦r♠ó♥ ♦♠♥♦s ♦♥s♥s♦ tr rtrísts ♠♦♦ ①r♦

♦♠♦ ♠♦s sñ♦ ♥ ♥tr♦ó♥ r ♣ít♦ ♦r♠ó♥ strs ♦ ♦♠♥♦s ♥♦♥♦rs s rs♣♦♥s ♦s r♦ts r♥ts

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

r ♣ ♦r r③♦ ♣rtr 500 r③♦♥s ♠♦♦ s♠♥ó♥ tr ♥t♥s r♦♦ ♥ ró♥ r③♦♥s♥ s q ♥t ♥ stó♥ rstó ♥♦♥♦r ♦s st♦s ♠ás ♦sr♦s♦rrs♣♦♥♥ ♦s ♥ts ♦st♥♦s Sstub

sr♠♣ó♥ ♥ ♣íss ♦♥ ♠♣ ♦rtr ♥ ❱ ♦str♠♦s qíq st♦ s q só♥ ♥♦ ♥r s s♠♥ ♣r♦♠♥♥t♠♥t ♥ r♣♦s q ♦♠♣rt♥ ♥ r♦ tr ♦ ♦st♥t ♦s strs ♥♦♥♦rs ♦①st♥ ♦♥ ♦tr♦s ♥♦s ♥♦s stó♥ ♦①st♥ ♦♠♥♦s ♦♥ st♥ts ♣♦strs s ♦rrs♣♦♥ ♦♥ ♦♠♣♦rt♠♥t♦ ♦sr♦ ♥ ró♥ rít tr♥só♥ ♥ s rs ② s ♣♦r ♦ q ♥ ♦ q rst ♣rs♥t ♣ít♦ r♠♦s q = 170 ♦♥ ♦t♦ str♥♦s ♥ ró♥ rít ♦♥ ♦①st♥ ♠s ♣♦strs ♥ó♥

♥áss r♦ts sr♠♣ó♥

♥ só♥ ♣r♥t s ♦sró ♦①st♥ ♦♠♥♦s ♥♦rs ② ♥♦♥♦rs ♦r♥♦s ♦♠♦ rst♦ ♣r♦s♦ s♠♥ó♥tr ♦r r♠♦s ó♠♦ st♦ ♠♣t ♥ t♠ñ♦ ♠♦ ♥ ♥tr♦t sr♠♣ó♥ ♥ ♣rtr ♥♦s ♥trs ♦♥trstr ♥str♦s rst♦s♦♥ ♣ótss ♥ q ♦♥sst ♥ s♠r stró♥ ③r ♦s ♥♦s ♥♦♥♦s ♥ r s♦ ♠r♠♦s st ♥♦q tr♥t♦♠♦♦

♥áss strs

♣r♦ó♥ st út♠♦ s q ♥tr♠♥t s ♦♥sr ♥ ♦s ♠♦♦s ♠♣♦ ♠♦ ♥q t♠é♥ s s♠rs ♥ ♦s ♠♦♦s rs ♦♠♣s ♠ás s♥♦s ♦♥sst ♥ strr ③r s♦r r ♠s♠ ♥t ♥♦♥♦s ♦t♥ ♠♦♦ s♠♥ó♥ tr s ♠♣♦rt♥t♥♦tr q t♥t♦ ♣r ♠♦♦ s♠♥ó♥ tr ♦♠♦ ♣r ♣r♦ó♥ ♦♥sr♠♦s s ♠s♠s ♣♦s♦♥s ♦s ♥ts Sstub ♦♥ ♦t♦ ♥♦ ♥tr♦r t♦rs ①ó♥♦s q r♥ ♥ ss♦s ♥s♦s

♦♠♦ s sró ♥ ♥tr♦ó♥ sr♠♣ó♥ ♣ r♣rs♥trs ♣♦r♥ ♠♦♦ ♦♠♣rt♠♥t t♣♦ s r ♦s ♥ts s♦♦ ♣♥ ♣rt♥r ♥ s s♥ts t♦rís ss♣ts ①♣st♦s ♥t♦s ♦ rrtr♦s ♦s ts ♠♦♦ ♣♠♦ó♦ t③♦ ♣♥♥♦♥trrs ♥ ♣é♥ s ♠♣♦rt♥t str q s♦♦ ♦s ♥♦s ♥♦♥♦s ♣♥ ♦♥trr ♥r♠ s♠♠♦s q ♥ t♥ ♥♥ 100% P♦r ♦tr ♣rt r♦r♠♦s q ♥t r♣rs♥t ♥♠ ♦♥ ♥ ♦ ♠♦♦ q st út♠♦ s ú♥♦ q ♣ ♦♥trr ♥r♠ s♠♠♦s q ♦s ♣rs ② ♦♥trr♦♥ sr♠♣ó♥ ♥ s ♥♥♦ ♥ s t♦ ♥ s♦ ♥♦s

r♦t ♦♠♥③ ♦♥ ♥ ú♥♦ ♥t♦ ♦ ③r ♥tr ♦s ♥tsss♣ts s r ♦s ♥♦♥♦s tr♦♥ 10000 r③♦♥s ♠♦♦ ♣r♦♣ó♥ sr♠♣ó♥ ♣r ♥ s 500 ♦♥r♦♥s qr♦ ♦t♥s ♠♥s♠♦ s♠♥ó♥ tr ♥ r s ♣rs♥t t♠ñ♦ ♠♦ r♦t ♠ ♥t t♦t ♥t♦s ♥ ♥ó♥ ♣♦ó♥ t♦t ♥♦♥♦s t♥t♦ ♣r ♥str♦♠♦♦ ♦♠♦ ♣r ♣r♦①♠ó♥ ♣r♦ó♥ P rs r♠♥t q♥str♦ ♠♦♦ r ♥ t♠ñ♦ ♠♦ r♦t s♣r♦r ♣r♦ó♥st t♦ s ♠♥♦ ♣r ♣♦♦♥s ♣qñs ♥♦♥♦s

sr♠♣ó♥ ♦♠♦ t♦ ♥r♠ ♥♦s ♣♦r ♦♥tt♦ ♥tr ♠♥♦ss tr♥s♠t ♥♠♥t♠♥t ♥tr ♥♦s ♦♥t♦s s ♣♦r ♦ q rí③ ♦♠♣♦rt♠♥t♦ ♠♦str♦ ♥ r srs ♥ strtr strs ♥♦♥♦s P♦r ♦ ♥ s♥t só♥ str♠♦s stró♥ t♠ñ♦s strs

♥áss strs

tr♠♥t t♠ñ♦ ♠á①♠♦ ♥ r♦t sr♠♣ó♥ strá ♦♥♦♥♦ ♣♦r ♥t ♥ts ♥♦♥♦s q s ♥♥tr♥ ís♠♥t♦♥t♦s ♠♥t ♥ ♥ ♦ ♣♦r ♥ ssó♥ ♦s ♠s♠♦s ♦♥ ♥

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

r ♠ñ♦ ♠♦ r♦t sr♠♣ó♥ s♦r 10000 r③♦♥s ♠♦♦ ♣r♦♣ó♥ ♥ ♥ó♥ ♣♦ó♥ ♥♦♥♦s ♦t♥♦s♠♥ts ♠♦♦ s♠♥ó♥ tr ír♦s r♦♦s ② ♣r ♣r♦①♠ó♥ ♣r♦ó♥ r♦s ③s

t♦ ♥ st♦ s t♠ñ♦ r♦t srá ♦ s♠♦ ♣♦ó♥ str ♥♦♥♦s q ♣rt♥ ♥t♦ ♥ st♦ s ♥r q stró♥ t♠ñ♦s strs ♥ r♦ ♥♠♥t ♣r r ♦r♥ ♦s rst♦s só♥ ♥tr♦r

♥ r s ♠str stró♥ t♠ñ♦s strs ♥♦♥♦s ♣r ♠♦ s 500 r③♦♥s ♣r♦s♦ s♠♥ó♥ tr ♦s rst♦s s♦♥ ♦♠♣r♦s ♦♥ ♦s ♦rrs♣♦♥♥ts ♠♦♦ ♣r♦ó♥ ♦♥sr♥♦ ♥ s♦ ①t♠♥t s ♠s♠s ♣♦♦♥s♥s ♥♦♥♦s Pr s♦ ♠♦♦ ♣r♦ó♥ s tr♦♥1000 r③♦♥s ♣♦r ♥ s 500 ♣♦♦♥s ♥s ♥♦♥♦s ♥str♦ ♠♦♦ ♥áss s r ♣ ♦♥rs q ♦sstrs ♠②♦r t♠ñ♦ s♦♥ ♠♦ ♠ás r♥ts ♣r ♠♦♦ s♠♥ó♥ tr q ♣r ♠♦♦ ♣r♦ó♥ q♥t r♠♦s ♥t♦♥sq ♥str♦ ♠♦♦ s ♣r♦♣♥s♦ str♥ ♥♦♥♦s ♦♠♦ ♣r♦t♦ ♥á♠ ♣r♦s♦ ♠tó♥ s♦ ♣♦r ♦♠♦ ❯♥ ♦r♠ ♥tr s r♥s ♥tr ♠♦s ♠♦♦s s trés á♦ stíst♦zS ♥♦ ♦♠♦

zS(b) =ρ(b)− ρperco(b)

σperco(b)

♥áss strs

r stró♥ t♠ñ♦s strs ♣r ♠♦♦ s♠♥ó♥ tr ír♦s r♦♦s ② ♣r ♣r♦①♠ó♥ ♣r♦ó♥ r♦s③s ♥st stíst♦ z ♦♠♦ ♥ó♥ t♠ñ♦ str ♦sr ♣r♠♥♥ strs r♥ t♠ñ♦ ♥ ♠♦♦ s♠♥ó♥ trrs♣t♦ ♠♦♦ ♥♦ ♣r♦ó♥

s♥♦ ρ(b) ♥s ♦rrs♣♦♥♥t ♥ b ♣♦r ♠♦♦ ♠♥trsq ρperco(b) ② σperco(b) ♦rrs♣♦♥♥ rs♣t♠♥t ♠ ② só♥st♥r ♥s ♣r s r③♦♥s ♣r♦ó♥ ♥ ♠s♠♦ ♥ b♥ ♥st r s ♦sr q zS ♠♥t ♦♥ t♠ñ♦ str♣♦r ♦ s ♦♥② ♥ttt♠♥t t♥♥ str③ó♥ ♣r♥str♦ ♠♦♦ ♦♠♣rr♦ ♦♥ ♣r♦①♠ó♥ ♣r♦ó♥

♥♥ s♦r str ♠á①♠♦

♥ só♥ ♣r♥t ♦sr♠♦s q ♥str♦ ♠♦♦ ♣rs♥t ♠②♦r t♥♥ ♦r♠ó♥ strs q q q s♠ stró♥ t♦r ♦s ♥♦♥♦s ♠♦♦ ♣r♦ó♥ ♦r ♣r♥tr♥♦s s s q s♦srrá♥ r♥s ♥ ♥♥ ♠ sr♠♣ó♥ ♥♦ ést s♣r♦♣ s♦r str ♠á①♠♦ ♣r ♥str♦ ♠♦♦ ② s♦r ♥♦ é♥t♣♦ó♥ ♣r♦ ♦t♥♦ trés ♠♦♦ ♣r♦ó♥ Pr ♦ s♦♥♠♦s r③♦♥s ♥str♦ ♠♦♦ ② ♠♦♦ ♣r♦ó♥ ♦♥ t♠ñ♦ str ♠á①♠♦ ♥ ♠s♠♦ ♥ stró♥ ♦ ♦♠♣r♠♦s ♥t ♠ ♥t♦s s♦r 10000 r③♦♥s ♠♦♦

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

♣r♦♣ó♥ sr♠♣ó♥ ♥ ♥♦ ♦s r ♠str q ♥t ♠ ♥t♦s s♦r ♦s strs ♠á①♠♦s ♦rrs♣♦♥♥ts ♠♦♦ s♠♥ó♥ tr s s♣r♦r ♦t♥ ♠♥t ♣r♦①♠ó♥ ♣r♦ó♥ s♦r strs t♠ñ♦ q♥t t♦ q ♥♠♥t s ♠♥♦ ♣r ♣♦♦♥s ♣qñs ♣♦r ♦ ♠r ♣r♦ó♥

r ♠ñ♦ ♠♦ r♦t sr♠♣ó♥ s♦r 10000 r③♦♥s ♠♦♦ ♣r♦♣ó♥ ♥ ♥ó♥ ♣♦ó♥ ♥♦♥♦s ♦rrs♣♦♥♥ts ♦s strs ♠á①♠♦s ♦t♥♦s ♠♥ts ♠♦♦ s♠♥ó♥tr ír♦s r♦♦s ② ♣r ♣r♦①♠ó♥ ♣r♦ó♥ r♦s ③s

♥ só♥ ♣r♥t ♠♦str♠♦s q ♥t ♥♦♥♦s ♠♦♦ s♠♥ó♥ tr rr♦ ♠②♦r ♥♥ sr♠♣ó♥ q ♦rrs♣♦♥♥t ♣r♦①♠ó♥ ♣r♦ó♥ ❱♥♠♦s st rst♦ ♠②♦r ♣rs♥ strs ♥ ♣r♠r♦ ♦ ♦st♥t ♥ ♣rs♥t só♥♦sr♠♦s q ♠♦♦ s♠♥ó♥ tr t♠é♥ ♣rs♥t ♠②♦r ♥♥ sr♠♣ó♥ q ♣r♦ó♥ ♥♦ s ♦♠♣r♥ strs ♠á①♠♦s t♠ñ♦

sr♣♥ sñ ♥ só♥ ♥tr♦r t♥ s ♦r♥ ♥ s rtrísts t♦♣♦ós ♦s strs ♠á①♠♦s ♥ ♠♦s ♠♦♦s Pr rr♦s stó r♦ ♠♦ 〈k〉 ♦♥t str③ó♥ C st♥ ♠

t ♦rtr ♥

♥tr ♥♦♦s l ② ♠♦r Q s♦r ♦s strs ♠á①♠♦s ♣r♦♥♥ts ♠♦♦ s♠♥ó♥ tr ② ♣r♦①♠ó♥ ♣r♦ó♥

0 500 1000 1500 2000max cluster size

⟨ k⟩

PercolaciónModelo

0 500 1000 1500 2000max cluster size

PercolaciónModelo

0 500 1000 1500 2000max cluster size

PercolaciónModelo

0 500 1000 1500 2000max cluster size

0.550.600.650.700.750.800.850.90

PercolaciónModelo

r Pr♦♣s t♦♣♦ós ♦s strs ♠á①♠♦s ♥ ♥ó♥ st♠ñ♦ r♦ ♠♦ 〈k〉 st♥ ♠ l ♦♥t str③ó♥ ♠♦ C ② ♠♦r Q

♥ r ♣ ♦srrs q r♦ ♠♦ 〈k〉 s ♠②♦r ♣r♦s strs ♠á①♠♦s ♠♦♦ s♠♥ó♥ tr P♦r ♦tr ♣rt st♥ ♠ l ♥tr ♥♦♦s s ♠ás ♣qñ ② C ♠♥t ♠②♦r ♠♥trsq Q s ♠♥t ♠♥♦r ♦t♥♦ ♣r ♠♦♦ ♣r♦ó♥ ♥t♦♥s ♠②♦r 〈k〉 ② st♥ ♠ l ♠ás ♦rt t♥ ♥♥tr♦ ♥tr ♥t♦s② ss♣ts ♣r s♦ ♦s strs ♦t♥♦s ♣rtr ♠♥s♠♦ s♠♥ó♥ tr ♠♥t♥♦ sí ♣r♦ ♦♥t♦

t ♦rtr ♥

♦♠♦ ♠♦s sñ♦ ♣r♠♥t ♦s r♦ts r♥ts sr♠♣ó♥ ♥ ♣íss srr♦♦s s ♥ ♣r♦♦ ♥ ♥ ♦♥t①t♦ t ♦rtr ♣r♦♠♦

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

♥ r ♣ít♦ ♥ r s ♥ q ♥t♦ ♠♥♦rs ♥t ♥♦♥♦s ♠②♦r ♦rtr ♥ ♠②♦rs s♦♥s sr♣♥s ♣r t♠ñ♦ ♠♦ r♦t ♥tr ♥str♦ ♠♦♦ ② qq s♠ stró♥ ③r ♦s ♥♦♥♦s ♥ st só♥ r♠♦só♠♦ st ♦ s ♠♥st ♣r ♥ s♦ ♣rtr q sts ♠r ♦rtr ♥ ❱ 95% r♦♠♥♦ ♣♦r ❬❲ ❪

0 20 40 60 80 100 120tamaño del brote

Pmodel(x≥25)=0.683ModeloPercolación

r st♦r♠ t♠ñ♦s r♦ts sr♠♣ó♥ ♦ ♦♥♦♥s 95% ♦rtr ♥ ♣r ♥str♦ ♠♦♦ s♠♥ó♥ trrrs r♦s ② ♣r ♣r♦①♠ó♥ ♣r♦ó♥ rrs ③s

♦♥sr♠♦s 100 r③♦♥s ♠♦♦ s♠♥ó♥ tr ♦♥ ♦s♣rá♠tr♦s ♣r♠♥t st♦s ♣r s s ♦rtr ♥ ♥ rst ♥tr 94.6% ② 95.4% s r ♥tr 115 ② 135 ♥♦♥♦s ♥ t♦tPr ♣♦ó♥ ♥ ♥♦♥♦s t♠é♥ s stó ♣r♦①♠ó♥ ♣r♦ó♥ ♥ ♠♦s ♠♦♦s ② s♦r ♥ s ♦♥r♦♥s ♥s ♥♦♥♦s s tr♦♥ 10000 r③♦♥s ♦rt♠♦ ♣r♦♣ó♥ sr♠♣ó♥ ♥ r s ♣rs♥t♥ ♦s st♦r♠s ♣r t♠ñ♦ ♦s r♦ts t♥t♦ ♣r ♠♦♦ s♠♥ó♥ tr ♦♠♦ ♣r ♣r♦①♠ó♥ ♣r♦ó♥ r♠♥t ♠♦♦ s♠♥ó♥ tr r r♦ts ♠②♦r t♠ñ♦ q ♦rrs♣♦♥♥t ♣r♦ó♥ ♥♠♦s ♥♣♠ ♦♠♦ ♥ r♦t q r ♠♥♦s 1% ♣♦ó♥ t♦t ♥♥str♦ s♦ N = 2500 stró♥ ♠ r P (x ≥ 25) t♥r

♦♥s♦♥s ♣r♠♥rs

♥ r♦t ♥♦r♥♦ ♠ás 25 ♥ts s ♥ ♦♠♦

P (x ≥ 25) =∑

n≥25

♦♥ p(n) r♣rs♥t ♣r♦ ♦t♥r ♥ r♦t ♦♠♣r♥♥♦ ①t♠♥t n ♥t♦s ♥ s♦ r P (x ≥ 25) = 0.683 ♣r ♠♦♦ s♠♥ó♥ tr ♠♥trs q ♣r♦ rst ♥♣r ♣r♦①♠ó♥ ♣r♦ó♥ ♦♥♠♥t ♥ r ♣rs♥t

r stró♥ t♠ñ♦s strs ♦ t ♦rtr ♥♣r ♠♦♦ s♠♥ó♥ tr ír♦s r♦♦s ② ♣r ♣r♦①♠ó♥ ♣r♦ó♥ r♦s ③s ♥ st út♠♦ s♦ s ♣rs♥t♥ ♦s ♥tr♦s ♦♥♥③ 95%

♠♦s stró♥ t♠ñ♦s strs rstr♥é♥♦♥♦s s♦ ♣rtr t ♦rtr qí st♦ ♥ ♥♦t♠♥t t♦ str③ó♥ ♦♠♦ ♥ ♥ó♠♥♦ q ♠♣t rt♠♥t ♥ ♣r♦ r♦t ás ú♥ ♦ ♠♣t♦ s ♠♥ ♥ ♦♠♣ró♥ ♦♥ ♠♦♦ ♣r♦ó♥ ♥♦ s ♦♥sr t ♦rtr ♥

♦♥s♦♥s ♣r♠♥rs

♥ st ♣ít♦ ♣rs♥t♠♦s ♥ ♠♦♦ ♣r♦♣ó♥ ♦♣♥ó♥ ♥ó♥ s♦r ♥ r ♦♠♣ stát ♥♦ ♥ ♠♦♦ s♠♥ó♥

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦ r

tr ①r♦ ❬①r♦ ❪ ♦ ♠♦♦ ♥♦r♣♦r ♦♠♦ ♥tr♥ts s♦s ♦♠♦ ♥ t♦r ♣s♦ ♥ ♠♥s♠♦ ♠tó♥ s♦ q♦♥ ♦r♠ó♥ ② ♦①st♥ ♦♠♥♦s ♠♦♥♦trs ♦♥ ♣♦strsr♥ts

sr♠♦s q ♠♥s♠♦ s♠♥ó♥ tr s♦r r ♦♥♦♥ ♣♦ó♥ ♥♦♥♦s ② ♦♥ s str③ó♥ ♦ t♦tú ♦♠♦ ♥t♦♥st ♥ó♠♥♦ ♥♠♥ r♣♦ ② s ♠♥ ♥st♦♥s ♠♣ ♦rtr ♥ st♦ ♣r♦♦ ①♣ít♠♥t♣r s♦ 95% ♦rtr sr ♣♦r ♣r ♥ ❱ ♦♥tr sr♠♣ó♥ r♦ ② ♣♣rs

♣ít♦

s♠♥ó♥ ♦♣♥♦♥s ②

♥ó♥ ♠♦♦ ♥r③♦

❮♥ ♦♥tt♦s ♣rs♦♥s ② ♦♣♥ó♥

♥s ♣tt♦s ♠♥s♠♦ r♦♥①♦♥♦

sr♣ó♥ ♦rít♠ ♠♦♦ ♥r③♦

♦r♠ó♥ strs

ó♥ s ♠♣ñs ♥ó♥

♦♦ ♥r③♦ ♦♥ ♠♣♦ ①tr♥♦

rr♦ ♥ q ② φ

ó♥ φ s♦r ♦s ♠♦♦s ♦♥ r stát ② ♣tt

♥♥ φ ♥ t♦♣♦♦í ♦s strs ♠á①♠♦s

♦♦ ♥r③♦ ♦♥ t ♦rtr

♦♥s♦♥s ♣r♠♥rs

♥ st ♣ít♦ ♥tr♦r♠♦s ♠♦rs ♠♦♦ ♣ít♦ ♥tr♦r ❯♥ s srá st♥ó♥ ♥tr ♥s q r♣rs♥t♥ ♦♥tt♦ ♣rs♦♥ ♥tr♥ts q♦s q s♦♦ s♠♦③♥ ♥ ♥tr♠♦ ♦♣♥♦♥s s♥ ♥s ♦♥tt♦ st ♠♦♦ ♦♥sr♠♦s ♥tr♠ó♥ s ♥st♥♦♦ís ② s ♥♥ ♥ ♦r♠ó♥ ♦♣♥♦♥s ♥q s♥ ♣♦s tr♥s♠tr ♥ ♥r♠ ♥♦s

♥♠♥t ♦♥sr♠♦s ó♥ s ♠♣ñs ♥♦r♠ó♥ r ♥ó♥ ♦♠♦ ♥ t♦r t♥♦r ♠♣t♦ ♦s r♣♦s ♥t♥ó♥

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

♦♥tt♦s ♣rs♦♥s ② ♦♣♥ó♥

s ♠út♣s tr♥ts ♦♠♥ó♥ s♥ ♥s ♦♥tt♦ ♣rs♦♥ ♦♥stt②♥ ♥ rtríst st♥t ♥str r ♦s ♥s ♦♠♥ó♥ ♥♦ ♦♥♥♦♥s ts ♦♠♦ ♥s rs s♦s ♦♥♥ ♠♥srí ♥st♥tá♥ ♦ ♠r♦♦♥ ttr ♥tr ♦tr♦s ♦♥stt②♥ ♥ ♥t ♥♦r♠ó♥ ♥s♦s② q ♠♣t rt♠♥t ♥ ♥strs ♦♣♥♦♥s❬r♠♥ ♦♥♦r ♦♥ ♥t♦ ❪ ♥ ♣ít♦ s♠♠♦s ♥ ♦rrs♣♦♥♥ ①t ♥tr ♦s ♥s q ♣r♦♣♥ ♦♣♥♦♥s② ♥r♠s ♥♦ss tr♥s♠ts ♣♦r ♦♥tt♦ rt♦ ♥ ♥str♦ s♦sr♠♣ó♥ ♥ ♠r♦ ♣♦r ♦ ♦ st qí rst ♥t q s♠♥ó♥ ♦♣♥♦♥s ♣ ♣r♦rs s♥ q ♠ ♥ ♦♥tt♦ ♣rs♦♥ q♥ ♠♦ s ♥s♣♥s ♣r ♦♥t♦ sr♠♣ó♥

♦♥ ♦t♦ ♥♦r♣♦rr ♥ st♥ó♥ ♥tr s ♦s ♠♦s ♣r♦♣ó♥ ♣r♠♥t sñs ♣r♦♣♦♥♠♦s ♥ ♠♦♦ ♦r♥ ♥②♥♦♦s t♣♦s ♥s ♥ r♥♦s ♥ ♥str r s♦

♥s ♣rs♦♥s s♦♥ q♦s q ♠♣♥ ♣r♦①♠ ís ♥tr ♥ts ts ♦♠♦ í♥♦s ♠rs ♠sts r♥s r♦♥s ♦rs♥tr ♦tr♦s trés ♦s ♣♥ tr♥s♠trs t♥t♦ ♦♣♥♦♥s ♦♠♦ ♥r♠s ♥♦ss

♥s ♦♣♥ó♥ ♥♦♣rs♦♥s r♣rs♥t♥ r♦♥s st♥ts ♦♠♦ qs ♠s ♣♦r t♥♦♦ís ♥♦r♠ó♥ ♥♦s♠♣♦s ♠sts ♥♦♣rs♦♥s ♥ rs s♦s ♦♥♥ s♦rs ttr ssr♣♦♥s ♦s t

st ♠♦♦ r s♦r q trr♠♦s ♥ st ♣ít♦ srá strtr♠♥t é♥t ♥tr♦ ♥ ♣ít♦ s♦ q ♣rs♥trá ♠♦st♣♦s ♥s ♦s 8 ♣r♠r♦s ♥♦s ♥ ♥♦♦ ♦ i ♦♥sttrá♥ s ♥t♦r♥♦ r♥♦ ♣♦r ♦ ♦s ♥s ♦s srá♥ ♥s ♣rs♦♥s P♦r♦tr ♣rt 8 ♦s s♥♦s ♥♦s i ♦♥sttrá♥ s ♥t♦r♥♦ ♦♣♥ó♥ ♣♦r♦ q i s ♥rá ♦s trés ♥s ♦♣♥ó♥ P♦r s♠♣ s♠r♠♦s q ♦s ♥s ♥♦ s♦♥ r♦♥s ♠♦♦ q t♥t♦ ♥♥ s♦♦♠♦ ♦♥t♦ ♣♥ rs ♥ ♠♦s s♥t♦s ♥st♥t♠♥t r ♠str ♥ r♣rs♥tó♥ ♣tór r q sr♠♦s ♦ r♦ ♣rs♥t ♣ít♦

♠ás tr♦♥ ♥ ♥tr③ ♦s ♥s ♠♦♦ s♠♥ó♥ ♦♣♥♦♥s ♥tr♦♦ ♥ st ♣ít♦ t♠é♥ ♥♦rrá ♥

♥s ♣tt♦s ♠♥s♠♦ r♦♥①♦♥♦

r t③ ♥ ♠♦♦ ♥s ♣rs♦♥s ♥ r♦♦ ② ♥s♥♦♣rs♦♥s ♦♣♥ó♥ ♥ ③

♠♥s♠♦ t♦ ♣tó♥ ♣♦r r♦♥①♦♥♦ rr♥ ♥s s ♣♦r♦ q r♠♦s st♥r ♥tr ♦s sss rs s♦s

stát s♥ ♣♦s r♦♥①♦♥♦ ♥q ♦♥ tr♦♥ ♥s ♣rs♦♥s ② ♥♦♣rs♦♥s

♣tt q q ♥② ♠♥s♠♦ r♦♥①♦♥♦ ♥s

♥s ♣tt♦s ♠♥s♠♦ r♦♥

①♦♥♦

♠♥s♠♦ s♠♥ó♥ tr ♣ít♦ ♦♥sr ♣tó♥ r♦ tr ♥t i r♣rs♥t♦ ♣♦r t♦r V

i(t) trés s ♥tró♥ ♦♥ ♥ ♦♥♥t♦ ♥♦s ♣r♠♥♥ts ♥ ♠r♦ ♥t♦r♥♦ ♦s ♥ts s♦s rs ♥♦ ♣r♠♥ ♥tr♦ ♦♥ tr♥srs♦ t♠♣♦ s♥♦ q t♠é♥ sr ♠♦♦♥s q ♦♠♣ñ♥ ♣r♦s♦ ♣tó♥ ♦♣♥♦♥s ♦♥ ♥ ♥♦r♣♦rr st s♣t♦ ♥ ♥str♦ ♠♦♦♥tr♦r♠♦s ♥ ♠♥s♠♦ r♦♥①ó♥ ♥s ♥ rsó♥ ♥r③

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

♥str♦ ♠♦♦ s♠♥ó♥ tr ♣♦ ♣r♦♣ó♥ ♦♥t ♥ó♥

r♦♥①ó♥ ♥s ♦♥stt② ♥ ♣s♦ ♣tó♥ r ♥strtr sí ♦♠♦ ♦s ♥ts t③♥ s ♦♣♥ó♥ ♥trtr ♦♥ s♥t♦r♥♦ trés r♦♥①♦♥♦ ♣♥ ssttr ♦s ♠♠r♦s s ♥t♦r♥♦♣♦r ♦tr♦s ♦♥ ♦s q ♣rs♥t♥ ♠②♦r r♦ s♠t ♦♠♦ st út♠♦♦♥stt② ♥ ♣r♦s♦ ♣tó♥ t ② ♣♦r ♦ st s rs s s♥♦♠♥ rs ♣tts

str♠♦s ♥ st♥ó♥ ♥tr ♦s ♠♥s♠♦s r♦♥①♦♥♦ ♥s ♦♣♥ó♥ ② ♣rs♦♥s ♦s ♣r♠r♦s s♦♥ ♦áts ♣♦r ♥ó♥ ♥r♠♥t ♥♦ r♣rs♥t♥ ♦♠♣r♦♠s♦ ♣rs♦♥ ② s♦♥ s♦t♠♥t sr♦♥s ♠♦♦ q ♣♥ t③rs ♥ s t♦t ② s♥ rstró♥ ♥ ♥t♦ ♦s ♥s ♣rs♦♥s ♦ q r♣rs♥t♥ ♥tr♦♥s rts ♥tr ♥♦s ♥ sr ♠ás sts q ♦s ♣r♠r♦s s r ♠♥♦s ♣r♦♣♥s♦s r♦♥①♦♥♦ s ♣♦r ♦ q ♥ st s♦ s♦♦ s♦♥r♠♦s ♥ ♣qñró♥ ppc ♥s ♣rs♦♥s ♣tt♦s

♥ ♦♥srr s ♥ts t♦ts ♥s ♣rs♦♥s ② ♦♣♥ó♥♦s r♦♥①♦♥♦s ♥♦ ♣♥ rs qr ♠♦♦ ♥ rt ♦ r♦♥①♦♥♦ r♥ ♥ ♥ lij ♦♥ ♥♦♦s ♥t i ② st♥♦ j srás s♥ts ♣rsr♣♦♥s

lij s ♥ ♦♣♥ó♥ s qr ♥t k r q ♥♦s ♥♦ i r♦ ♦♠♦ ht(i, j) < ht(i, k) ♥t♦♥s r♦♥①♦♥♦ s ♣t ♦♥ ♣r♦ 1 ② ♥ lij s ♦rr♦ ②sstt♦ ♣♦r ♦tr♦ ♥ ♦♣♥ó♥ lik ♥ s♦ ♦♥trr♦ s r③ r♦♥①♦♥♦ ② s ♠♣♠♥t ♠♥s♠♦ ♠tó♥ s♦ sr♣t♦♥ ♣ít♦

lij s ♥ ♣rs♦♥ ♣tt♦ s ♥ ♥t k ♥tr ♦s ♦♥tt♦s ♦♣♥ó♥ ♥t i ht(i, j) < ht(i, k) ♥t♦♥s ♥ ♣rs♦♥♣tt♦ lij s ♦♥rt ♥ ♥ ♦♣♥ó♥ ♠♥trs q ♥ liks rt♦r③ ♦♠♦ ♣rs♦♥ ♣tt♦

str q ♣rsr♣ó♥ r♦♥①♦♥♦ ♣r ♥s ♣rs♦♥s ♣tt♦s t♥ rtríst ♦r ♣r♦♠♦r ♥ ♥ ♦♣♥ó♥ ♥♣rs♦♥ ♦♥t♠♣♥♦ ♥ ♥ ♣r ♥tr ♥ts ♦♥t♦s ♥ r s♦♥r qr ♦tr♦ ♥t r ♥♦r♠♠♥t ③r

sr♣ó♥ ♦rít♠ ♠♦♦ ♥r③♦

sr♣ó♥ ♦rít♠ ♠♦♦ ♥r

③♦

sr♣ó♥ ♦rít♠ ♠♦♦ s s♥♠♥t srr♦ ♥ ♣ít♦ ♦♥ r♦ s ♣rsr♣♦♥s r♦♥①♦♥♦ r♥ ②sñs ♠♥t ♦♥sr♠♦s ♥ r ♥♠♥t r ♦♥ N =

2500 ♥♦♦s ♥♦ ♦s ♦♥ ♥s ♣rs♦♥s ♣r♠r♦s ② s♥♦s♥♦s kp = 8 ♠ás r♦ 8 ♥s ♥♦♣rs♦♥s ♦♣♥ó♥ ♣♦r♥♦♦ kop = 8 t ♦♠♦ s ♠♦stró ♥ r ♦♥♠♥t q ♥ ♣ít♦ ♠♦s F = 10 ♠ás ♦♣♥ó♥ ♥r s♦ ♦♥ ♥ó♥ ♣r ♠♦♦ s♠♥ó♥ tr s♠s♠♦ ♦♥sr♠♦s♥ ♠r ♥tró♥ ♣♦r ♦♠♦ κ = 2 ♠♦♦ q ♦s ♥ts s♦♦♣♦rá♥ ♥trtr s ♣rs♥t♥ ♦♥♥ ♥ ♠♥♦s ♦s ss ♦♣♥♦♥sr

♥ ♦ q rs♣t sr ♥s ♣rs♦♥s s ♥tr t♦t ♥s ♣rs♦♥s ♥ ró♥ ppc = 0.1 ♥s ♣rs♦♥s ♣tt♦s q♣r♠♥rá♥ ♥ ♦♥ó♥ r♥t t♦ ♦ó♥ ♦rt♠♦ st♦ss♦♥ ♦s ú♥♦s ♥s ♣rs♦♥s q ♣♥ sr r♦♥t♦s ♦♠♦ ♣rt ♠♥s♠♦ s♠♥ó♥ ♦♣♥♦♥s s♦r r ♣tt P♦r ♦tr ♣rtt♦♦s ♦s ♥s ♥♦♣rs♦♥s ①s♠♥t ♦♣♥ó♥ s♦♥ ♣tt♦s

♦♥ó♥ ♥ sst♠ s ♠s♠ q ♣r ♠♦♦ ♣ít♦♥tr♦r s ♦♣♥♦♥s s♦♥ s ♥♦r♠♠♥t ③r ♥ 1, 2, 3, ..., q ♣r ♥t ♦♥ ①♣ó♥ ♦♣♥ó♥ ♥ó♥ ♥ st út♠♦ s♦♦s ♥ts ♦♠♥③♥ ♦♥ ♥ ♦♣♥ó♥ ♣r♦♥ó♥ V i

F+1(t) = 0 s♦ 1% ♥♦♥♦rs ♦st♥♦s ♣rt♥♥ts Sstub ② ♣♦str ♦♥tr ♥ó♥ stá ♥♠♥t sr♣ó♥ ♦rít♠ ♠♦♦ ♣r ♣s♦ ♦ó♥ s

❯♥ ♥t ♥t i s ♦ ♥♦r♠♠♥t ③r ♠♥trs q ♥♦ ss ♣r♠r♦s ♥♦s j s ♦ ♦♠♦ ♦t♦ t♠é♥ ♥♦r♠♠♥t ③r

❱r♥ts

stát sr ♥ ít♠

♣tt sr s ♣rsr♣♦♥s ♥s ♥ só♥ s ♦♥rt r♦♥①♦♥♦ ♥ lij sr ♥ ít♠ ♥s♦ ♦♥trr♦ ♥r ♠♥s♠♦ ♠tó♥ s♦ ♥ ít♠

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

♦t♥ ht(i, j) ② ♣r♦ ♥tró♥ P (i → j, t) ♦ s ③r ♥ rtríst n ♥ q i ② j ♥♦ ♦♥♥

n 6= F + 1 s r q rtríst ♥♦ s ♣♦str rs♣t♦ ♥ó♥ ♥t i s ♣r♦s♦ ♣tó♥ sr♣t♦ ♥

n = F + 1 ♥ ♦s tr♥ts

i ∈ Sstub ♥ st s♦ i ♦♥sr s ♣♦str rs♣t♦ ♥ó♥V i

F+1(t+ 1) = V iF+1(t)

i /∈ Sstub ♥ ②♦ s♦ ♥t s ♣r♦s♦ ♣tó♥ sr♣t♦ ♥

♦♥ó♥ t♥ó♥ ♦♥sr♠♦s ♦s ♣♦ss ♦♥♦♥s t♥ó♥

♦rt♠♦ s t♥ ♥♦ ♥③ ♥ ♦♥ró♥ s♦r♥ts r ♥♦ ♥♦ s♦♥ ♣♦ss tr♥s♦♥s ♦♥s

♥ ♦s s♦s ♦♥ ♣r ♦♣♥ó♥ ♥ó♥ ♦♥r ♦s ♥ts Sstub ♣ r r ♣qñs t♦♥s ♥tr♠t♥ts ♥ ♣♦str rs♣t♦ ♥ó♥ ♥ s ♥t♦r♥♦ s ♣♦r♦ q ♠♦s ♦♠♦ ♦♥ó♥ t♥ó♥ ♦♥ ♠♣♦s r③r tr♥s♦♥s ♣r t♦s s rtrísts trs①♣t♥♦ ♣qñs t♦♥s ♥ ♣♦str rs♣t♦ ♥ó♥

t → t+ 1 t♦r♥ ♣s♦

♥ ♥t♦ ♠♦♦ ♣♠♦ó♦ ♣r ♣r♦♣ó♥ sr♠♣ó♥ ♠s♠♦ s sr ♥ ♣é♥ ♦♥t♥ó♥ r♠♦s ♥ sr♣ó♥ ♠s r♥ts ♠♦♦ ♥r③♦ stát ② ♣tt st♥♦ss s♠ts ② r♥s t♥t♦ ♥tr ♦s ♦♠♦ ♦♥ rs♣t♦ ♠♦♦ ♣ít♦ ♥tr♦r

stát t♦s tr♦♥

Pr ♠♦♦ ♥r③♦ ♥tr♦♦ ♥ ♣rs♥t ♣ít♦ s rsstáts s♦♥ qs q ♥♦ ♠t♥ r♦♥①♦♥♦ ♥s ♥q ♦♥

sr♥ s tr♦♥s ♥s trés st♥ó♥ ♥tr ♥s♣rs♦♥s ② ♥s ♥♦♣rs♦♥s ♦♣♥ó♥ ♥ ♣ít♦ ♣r♥t ♠♦s♦♥sr♦ s♦♦ ♥s ♦♠♦é♥♦s ♣s tr♥s♠tr t♥t♦ ♦♣♥♦♥s ♦♠♦♥r♠s tr♠♥t st♥ó♥ ♥tr ♥s ♣rs♦♥s ② ♦♣♥ó♥♥♦ ♣r♦ ♥♥♥ ♠♦ó♥ ♥ ♦ q rs♣t s♠♥ó♥ ♦♥t ♥ó♥ ♥ ♣ít♦ s♠♣r q r ♣r♠♥③ státs♥ r♦♥①♦♥♦ ♦ q r s♦r q s ♣r♦♣♥ s ♦♣♥♦♥s ss♥♠♥t ♠s♠ ♥ ♠r♦ ♣r♥trs á srá t♦ tr♦♥ ♥s ♥ ♣r♦♣ó♥ ♥r♠ s♦r rstát ♦ q ♦r t♥ r ①s♠♥t s♦r ♥ sr ♥s ♣rs♦♥s ♦♥ ♦ ♦t♦ r③♠♦s 10000 s♠♦♥s ♠♦♦ ♣r♦♣ó♥ sr♠♣ó♥ s♦r s rs státs ♦t♥s ♠♥t ♠♦♦♥r③♦ ♥ r ♣rs♥t♠♦s ♦♠♣ró♥ ♥tr ♦s rst♦s ♠♦♦ ♣ít♦ ② qí ♥tr♦♦ ♠♦s ♦rrs♣♦♥♥♦ rstát ♣r t♠ñ♦ ♠♦ r♦t sr♠♣ó♥ ♥ ♥ó♥ ♥tt♦t ♥♦♥♦s

500 1000 1500 2000 2500total unvaccinated

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static

Percolation95% CIModel 2 φ=0.00

500 1000 1500 2000 2500total unvaccinated

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static

Percolation95% CIModel 1 φ=0.00

r ♠ñ♦ ♠♦ r♦t ♥ ♥ó♥ ♣♦ó♥ ♥♦♥♦s♣r r stát ♦♥ ♥s ♦♠♦é♥♦s ② r stát ♦♥ ♥s tr♦é♥♦s ♣rs♦♥s ② ♦♣♥ó♥ ♥ ♠♦s rá♦s s ♣rs♥t♥ ♦s rst♦s♣r ♣r♦①♠ó♥ ♣r♦ó♥ r♦s ③s

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

s sr♣♥s ♦srs ♥ r s ♥ r♥ ♥ r♦ ♠♦ ♥s ♣rs♦♥s ♣r ♦s ♥♦♦s ♠♥trs q ♣r r♦♥ ♥s ♦♠♦é♥♦s t♦♦s ♦s ♥s s♦♥ q♥ts 〈k〉 = 16 ♣r r ♦♥ ♥s tr♦é♥♦s 〈kp〉 = 8 s♥♦ 〈kp〉 r♦ ♠♦ ♦♥sr♥♦①s♠♥t ♥s ♣rs♦♥s s♠♥♦ ♥t ♥s ♣s ♣r♦♣r ♥r♠ ♣r sr ♦♥stt ♣♦r ♥s ♣rs♦♥s s①t♠♥t ♠t q ♣r s♦ r ♦♥ ♥s ♦♠♦é♥♦s r♥ ♥tr ♦s ♠♦♦s s strtr ② ♦s♦♥ ♥ ♠♦ó♥ ♥ t♠ñ♦ ♠r ♣r♦ó♥ rst♥♦ ♠②♦r ♣r s♦ r ♦♥♥s tr♦é♥♦s

♠♦♦ ♦♥só♥ ♣r♠♥r r♠♦s q ♠♦♦ ♠ás rst ♦♥♥s tr♦é♥♦s rst ♦♣t♠st ♦♠♣rr♦ ♦♥ ♣ít♦ ♣r♥t ♠♥♦s ♥ ♦ q rs♣t ♣r♥ sr♠♣ó♥ ♥ ♠r♦ú♥ ♥ st s♥r♦ ♦♣t♠st s♣ró♥ rs♣t♦ ♣r♦①♠ó♥ ♣r♦ó♥ st ♥ ♣ít♦ s s♥t t ♦♠♦ s ♦sr ♥ r ♦♥t♥r♠♦s ♥str♦ ♥áss ♠♦♦ ♥r③♦ ♦r♥♦r♣♦r♥♦ r♥t r ♣tt

♥ st só♥ ♥③r♠♦s ♦s t♦s ♦♥srr r♦♥①♦♥♦ ♥s ♦♠♦ ♥ ♠♥s♠♦ ♣tó♥ ♦♥ r♥t s♠♥ó♥ ♦♣♥♦♥s tr♠♥t strtr r s♦ ♦♦♥rá ♥ t♠♣♦♦ ♦♠♦ ♠♦s ♥♦t♦ ♦ r♦ st♦s ♣r♠r♦s ♣ít♦s ♦s♦♥ s③ ♠♦s ♥ ss ♣r♦♣s ♣r♦♣ó♥ rst♦ s ♥ r♠♥tó♥ ♥tr ♠♥s♠♦ ♣tó♥ ♥s ② ♥á♠ ♠♦♦ ①r♦ ♥t♦♥s ♣r♥tr♥♦s qé ♠♦♦ r♦♥①♦♥♦ ♥strá tr♥só♥ ♥tr ♦♥s♥s♦ ♥♦r ② ♥♦♥♦r ♥ ♥str♦♠♦♦

♥ r ♠♦str♠♦s ró♥ ♥♦♥♦rs ♥ r ♥ ♥ó♥ q ♦r♠ t♦t♠♥t ♥á♦ r ♣ít♦ ♠♥t♦♥sr♠♦s N = 2500 F = 10 〈k〉 = 16 ② 〈kp〉 = 8

tr♥só♥ ♠♦str ♥ r rst ♠♦ ♠ás ♥t q ♦rrs♣♦♥♥t r stát ♦♥ ♦ ♣♦♠♦s ♦♥r q ró♥ rít s♥s♥ ♦ q rst s ♣r s♥t♦♥③r ♥str♦ sst♠ ♥ tr♥só♥P♦r ♦tr ♣rt ♦r rít♦ ♦r s ♥♥tr ♣ró①♠♦ qC = 300 ♠②♦r ♦t♥♦ ♣r r stát qC = 170 tr♦ s♣t♦ str

101 102 103 104

adaptativa

r ró♥ ♥♦♥♦s ♥ ♥ó♥ q ♣r ♠♦♦ ♥r③♦♦♥ r ♣tt

r s q t♥t♦ ♣r r stát ♦♠♦ ♣tt ♠♦♦ ♥r③♦rr♦ ♣r♦①♠♠♥t ♠s♠ ♥t ♥ ♠ ♥♦♥♦s ♣rq = qC qí ♥ ♠ás r♠♦s q = 300 ♣r ♥③r ♦♠♣♦rt♠♥t♦ ♠♦♦ ♥r③♦ ♦ r ♣tt s♦ q s ♥q ①♣ít♠♥t ♦♦♥trr♦

♦r♠ó♥ strs

s♣t♦ ♠ás st♦ ♠♦♦ ♣ít♦ s ♦r♠ó♥ ♦♠♥♦s ♦ strs ♥ts ♥♦♥♦s ♦♥ ♥ r♥ ♠♦ ♠②♦r q ♦t♥ ♥♦ ♠s♠ ♣♦ó♥ ♥♦♥♦s s str ③r♣r♦①♠ó♥ ♣r♦ó♥ ♣r♥tr♥♦s s ♠♦♦ ♦♥ r ♣tt r ♥ ♦♠♣♦rt♠♥t♦ s♠♥t

r ♠str stró♥ t♠ñ♦s strs ♣r ♠♦♦ ♥r③♦ ♦♥ r ♣tt ♦rrs♣♦♥♥t q = 300 ② s♠♣r ♦♥F = 10 s ♠♣♦rt♥t str q ♥ ♦♥t①t♦ ♠♦♦ ♥r③♦s♦♦ st♠♦s strs ♣r sr ♥s ♣rs♦♥s q s♦♥♦s ú♥♦s r♥ts ♣r tr♥s♠só♥ sr♠♣ó♥ ♦♠♦ ♥ s♦ ♣ít♦ ♣r♥t ♦♠♣r♠♦s ♦s rst♦s ♦♥ ♠♦♦ ♣r♦ó♥ ♥ sr♥r s ♦♠♣♦rt♠♥t♦ ♦sr♦ ♦♥stt② ♦ ♥♦ ♥♥t♦ ♦rtt♦ P rs ♥ r q t♠♥t ♠♦♦ s♠♥ó♥ ♦♣♥♦♥s ú♥ ♦♥ ♦r♠ó♥ strs ♦♥ ♠②♦r r

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

r stró♥ t♠ñ♦s strs ♣r ♠♦♦ ♥r③♦♦♥ r stát ② ♠♦♦ ♥r③♦ ♦♥ r ♣tt ♥ ♠♦s s♦ss ♣rs♥t ♦♠♣ró♥ ♦♥ ♣r♦①♠ó♥ ♣r♦ó♥ ② ♠♥t ③s♦r ♥sts

♥ q ♠♦♦ ♣r♦ó♥ ás ú♥ t♥t♦ t♠ñ♦ s♦t♦ ♦♠♦ r♥ ♦s strs ♠á①♠♦s rst ♠②♦r ♣r s♦ ♠♦♦ ♦♥r ♣tt r r ♠♦♦ q ♠♦♦ s♠♥ó♥ ♦♣♥♦♥s ♦♥ r ♣tt ♥♦ s♦♦ ♦♥ ♦r♠ó♥ strs ♥♦♥♦s s♥♦ q s ③ rst ♠ás ♣r♦♣♥s♦ t ♦♠♣♦rt♠♥t♦ str q ♥♦ ♦♠♣r♠♦s ♠♦♦ r ♣tt ♦♥ ♣r♦①♠ó♥ ♣r♦ó♥ ♦♥sr♠♦s r ♥ ♦ ♣r♦s♦ ♣tó♥♣♦r r♦♥①♦♥♦ ♥s P♦r st ♠♦t♦ r s♦r s str②♥ ③r ♦s ♥♦s ♥♦♥♦s ♥ ♠♦♦ ♣r♦ó♥ ② s♦ ♠♦ ♣♦r ♠♦♦ ♥r③♦ ♦♥ r ♣tt s sr♣♥s ♥tr♠♦s srí♥ ú♥ ♠②♦rs ♦♠♣rr ♦♥ ♠♦♦ ♣r♦ó♥ s♦r r♥

♥ ♣ít♦ ♠♦s ♥③♦ t♠ñ♦ ♦s r♦ts s♦r str ♠②♦r t♠ñ♦ SM str ♠á①♠♦ qí r♠♦s ♦ ♣r♦♣♦ ♦♥ ♦s strs♠á①♠♦s ♣r♦♥♥ts ♠♦♦ ♦♥ r stát ② ♣tt ♣r ♠♦♦♥r③♦ ♥ r s ♦sr t♠ñ♦ ♠♦ r♦t sr♠♣ó♥ ♥ ♥ó♥ t♠ñ♦ str ♠á①♠♦ r ♣rs♥t

♦♠♣ró♥ ♥tr ♠♦♦ ♥r③♦ ♣r r stát ② ♣r♦①♠ó♥ ♣r♦ó♥ ♠♥trs q r s q♥t ♣r ♠♦♦ ♥r③♦ ♦♥ r ♣tt ♦♥♥ r♦rr qí q ♣rt♠♦s ♥ ú♥♦♥t♦ ♣rt♥♥t str ♠á①♠♦ ② t♠♦s 10000 r③♦♥s ♦rt♠♦ ♣r♦♣ó♥ sr♠♣ó♥

r ♠ñ♦ r♦t ♥ ♥ó♥ t♠ñ♦ str ♠á①♠♦ ♣r ♠♦♦ ♥r③♦ ♦♥ r stát ② ♠♦♦ ♥r③♦ ♦♥ r♣tt

♦♠♣ró♥ ♥tr s rs ② s ♦sr q r stát ♣rs♥t r♥s s♥ts rs♣t♦ ♣r♦①♠ó♥ ♣r♦ó♥t♦ q ♥♦ s ♣rs♥t ♠♥r t♥ ♠r ♣r s♦ r ♣tt Pr ①♣r st út♠♦ rst♦ s ♥sr♦ rrrr ♥áss strtr ♦s strs ♠á①♠♦s

♥ r ♣rs♥t♠♦s s ♣r♦♣s t♦♣♦ós ♦s strs ♠á①♠♦s ♣r s♦ ♠♦♦ ♥r③♦ ♦♥ r stát r ② ♦♥ r ♣tt r ♥ ♠♦s s♦s ♦♠♣r♠♦s ♦s ♥♦rs t♦♣♦ó♦s ♦♥ ♦s ♦t♥♦s ♣r strs ♠á①♠♦s t♠ñ♦ ♣rtr ♠♦♦ ♣r♦ó♥ P rs r♠♥t q s sr♣♥s♥tr s ♣r♦♣s ♦s strs ♦t♥♦s ♣rtr ♥str♦ ♠♦♦ ② ♦s ♣r♦①♠ó♥ ♣r♦ó♥ s♠♥②♥ ♣r s♦ ♦♥ r ♣tts ♣rs♠♥t ♣♦r st ♦ q s r♥s rs♣t♦ ♠♦♦ ♣r♦ó♥ ♣r ♦s t♠ñ♦s r♦ts r t♠é♥ s♠♥②♥♣r♠♥t rs♣t♦ ♦s ♦sr♦s ♥ r

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

0 500 1000 1500 2000tamaño cluster máx

⟨ k⟩

PercolaciónModelo

0 500 1000 1500 2000 2500 3000tamaño cluster máx

⟨ k⟩

PercolaciónModelo

0.200.250.300.350.40

PercolaciónModelo

0.200.250.300.350.40

PercolaciónModelo

101520253035

PercolaciónModelo

101520253035

PercolaciónModelo

0.650.700.750.800.850.90

PercolaciónModelo

0.650.700.750.800.850.90

PercolaciónModelo

r Pr♦♣s t♦♣♦ós ♦s strs ♠á①♠♦s ♦♦ ♥r③♦♦♥ r stát ② ♦♥ r ♣tt ♥ ♠♦s s♦s s♣rs♥t ♦♠♣ró♥ ♦♥ s ♣r♦♣s ♦s strs ♠á①♠♦s ♣r♦♥♥ts ♣r♦①♠ó♥ ♣r♦ó♥

ó♥ s ♠♣ñs ♥ó♥

s♣♦♥ ♥s s ♥ ♦♥ó♥ ♥sr ♣r ♥③r ♥♦rtr ♠♣ rs ♠♣♠♥tó♥ ♣r♦r♠s ♥ó♥ ♥ ♦ ♣♦r ♣rt ❯ ② ♦tr♦s ♦r♥s♠♦s s ♣ú❬P ❪ st ♠t ♣ ♥③rs P♥ strté♦

ó♥ s ♠♣ñs ♥ó♥

❬P ❪ ♦♥t♠♣ ♥tr ss ♦t♦s ♥ s♣t♦ ♥♠♥t

♦♠♥r ② ♦♠♣r♦♠trs ♦♥strr ♦♥♥③ ② ♠♥ ♣ú ♣r

♥♠♥③ó♥

st s ♣rs♠♥t s♣t♦ ♥ q ♥♦s ♥tr♠♦s ♥ ♣rs♥t só♥ r♠♥t s♣♦♥ ♥s ♥♦ s s♥t s s ♣rs♦♥s s r③♥ r③♦ ♣r♦♥ ♠②♦rtr♠♥t t♦rs trs❬♠♦♥ ♥t❱t♦r ❪ ♦♥♥ ❬Pr ❪ ② ♥ ♣r♣ó♥ rró♥ rs♦ s♦♦ ♦♥ ♥ ❱ P♦r s♣st♦ ♦ q ♥♦s♦tr♦s s♦st♥♠♦s qí s q st ♣r♣ó♥ rs♦ ♥♦ stá ♥ ♥ ♥♦♣♥ó♥ ♥♦r♠ ♣♦r ♣rt ♦s ♥ts s♦rs s♥♦ ♥ ♥ ♠♥s♠♦ s♠♥ó♥ ♦♣♥♦♥s ♣♦r ♠tó♥ s♦

♦♦ ♥r③♦ ♦♥ ♠♣♦ ①tr♥♦

♥ ♦s ♥ts ♦♣t♥ ♥ ♣♦str ♥ó♥ ♦ ♥♥ s ♥t♦r♥♦ ♣ró①♠♦ t♠é♥ s♦♥ ♣r♠s ♠♣ñs ♣r♦♠♦ó♥ ♦s ♥♦s ♥ó♥ s ♦ ♣♦r s t♦rs s♥trs♥♦r♣♦r♠♦s st t♦r ♦♥ ♥str♦ ♠♦♦ ♥ts ♥ ♦r♠ ♥ ♠♣♦ ①tr♥♦ ♦ φ ≥ 0 stá♥ ①♣st♦s t♦♦s ♦s ♥ts ♦♠♣♦ φ t♥ ♣♦r ♥ó♥ ♥tr♦r ♥ ss♦ ♥ ♣r♦ tr♥só♥ ♥♦r ♥♦♥♦r ♥ ♠♥s♠♦ ♠tó♥ s♦

♠♣♠♥tó♥ ó♥ φ ♥ ♥str♦ ♠♦♦ ♥r③♦ s s♥t i ♥ ♥t ♥t ♥♦r V i

F+1(t) = 0 q ♥trtú ♦♥♥ ♥t st♥♦ ♥♦♥♦r j V j

F+1(t) = 1 trés ♠♥s♠♦ ♠tó♥ rtríst ♣r ♠♥s♠♦ ♣tó♥ s ♣♦str ♥ó♥ ♣♦str s t③ sú♥

V iF+1(t+ 1) =

V jF+1(t) ♦♥ ♣r♦ (1− φ)

V iF+1(t) ♦♥ ♣r♦ φ .

st♦ s ♥t ♥♦r ♥t ú♥ ♦♥sr ♥ ♥ ♣rsstr ♥ s♣♦str ♦♥ ♣r♦ φ ♥trtr ♠♥t ♠♥s♠♦ ♠tó♥s♦ ♦♥ ♥ ♥t ♥♦♥♦r trt ♦ st♥♦ ♥ ♥t♦ ♠♣♠♥tó♥ s♦r ♠♦♦ ♥r③♦ ♦♥ r ♣tt ♥t③r q ♠♣♦①tr♥♦ φ ♥♦ ♥tr♥ ♥ ♦r♠ rt ♦♥ ♠♥s♠♦ r♦♥①♦♥♦ ♥s

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

♦♠♦ rst ♥t ó♥ ♥♦ φ = 0 s r♣r ♠♥s♠♦ ♠tó♥ s♦ tr♦♥ P♦r ♦tr ♣rt s ♠♣♦rt♥t strq ♠♣ñ qí ♠♦ s s♦♦ ♥♦r♠ó♥ ♦r ② ♦♥♥t③ó♥ ♥ó♥ ♥ s♥t♦ ♣r♦♣st♦ ♣♦r ó♥ φ ♥♦♣rt♥ ♠♣♦♥r ♥ó♥ ♦♠♣s ♥ t♥ ♥♥ ♥ ♣♦só♥ q♦s q ♥ t♦♠♦ só♥ ♥♦♥r s♥♦ q s♦♦ tú s♦rq♦s ♥ts ♥♦rs ♣r q ♥♦ ♠♦q♥ s ♣♦str ♥ tr♦

rr♦ ♥ q ② φ

♦ ♥♦r♣♦rr ♥ ♠♣♦ ①tr♥♦ φ ♥tr♠♥t ♠♣í ♥str♦s♣♦ ♣rá♠tr♦s s ♣♦r ♦ ♥sr♦ ♠♣♠♥tr ♥ rr♦ t♥t♦ s♦r q♦♠♦ ♣r ♥str♦ ♥♦ ♣rá♠tr♦ φ ♥ r ♣rs♥t♠♦s ♣♦ó♥♠ ♥ts ♥♦♥♦s ♣r ♥str♦ ♠♦♦ ♥r③♦ t♥t♦ ♣rr stát r ♦♠♦ ♣tt r ♥ ♥ó♥ q ② φ ♦♥sr♥♦ ♦ F = 10 ♥ ♠♦s s♦s s st t♦ ♠t♦r φ ♣♦r s♦r ó♥ 1% ♥ts ♦st♥♦s ♥♦♥♦rs ❯♥♠í♥♠ ♥tr♥ó♥ ♠s s♦st♥ ♥ t♠♣♦ ♣rs♥t t♦s ♥♦ts ♥ ♦ó♥ ♣♦str ♥ó♥

r P♦ó♥ ♥♦♥♦s ♣r rr♦ ♥ q ② φ ♦♦ ♥r③♦♦rrs♣♦♥♥t r stát ② r ♣tt

♣rs♥ ♥ ♠♣♦ ①tr♥♦ φ > 0 rstr ♣♦s ♦♥s♥s♦ ♥♦♥♦r s♦t♦ ♦sr♦ ♥ s tr♥s♦♥s ♣rs ♦♥ q < qC r r

ó♥ s ♠♣ñs ♥ó♥

Pr ♦rs ♣qñ♦s q ♣♥ ú♥ ♦r♠rs ♦♠♥♦s ♦♥s♥s♦ s s♥③♥ s♦♦ s ♣r♠rs F ♦♣♥♦♥s s♥ ♠r♦ st♦ ♥♦ ♦rr ♣r s♦ ♦♣♥ó♥ F + 1 rr ♥ó♥ ♦ q ♦rr ♣r q < qC s ♥♦♠♣t♥ ♥tr ♦s ♣♦strs ♥tó♥s s♦r ♥ ♦♠♥♦ ♦♥r♥♥ s ♣r♠rs F ♦♣♥♦♥s

0 200 400 600 800 1000q

⟨ k⟩

0 200 400 600 800 1000q

C0 200 400 600 800 1000

0 200 400 600 800 1000q

r ♦♣♦♦í s srs ♥s ♣rs♦♥s ② ♦♣♥ó♥ ♦♥♠♣♦ ①tr♥♦ φ ♦s ír♦s r♦s r♦♦s ③s ② rs ♦rrs♣♦♥♥ s srs ♥s ♣rs♦♥s ♦♣♥ó♥ ② φ = 0.00 0.01 ② 0.02 rs♣t♠♥t s í♥s ♣♥t♦s ♥♥ ♦s ♦rs ♦rrs♣♦♥♥ts ♣r ssrs ♥s Gpers í♥ ♣♥t♦s r♦♦s ② Gop í♥ ♣♥t♦s ③s

t♠é♥ ♣r♥tr♥♦s ó♠♦ s ♠♦ t♦♣♦♦í r rrrq ♣r φ = 0.00 0.01 ② 0.02 ♦ ♣r♦s♦ r♦♥①♦♥♦ ♥s ♥ s♦ ♠♦♦ ♥r③♦ ♦♥ r ♣tt ♦r♠♦s q r s♥♥tr ♦♥♦r♠ ♣♦r ♥ sr ♦♥tt♦s ♣rs♦♥s Gpers ② ♦tr ♥tr ♣♦r ♦♥tt♦s ①s♠♥t ♦♣♥ó♥ Gop r ♠str r♦ ♠♦ 〈k〉 ♦♥t ♠♥♦ ♠ ℓ ♦♥t str③ó♥ ♠♦ C ② ♠♦r Q t♦♦s ♥ ♥ó♥ q ② ♣r♦♠♦s s♦r

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

100 r③♦♥s ♠♦♦ t♥t♦ ♣r sr ♦♥tt♦s ♣rs♦♥s ♦♠♦♣r ♦rrs♣♦♥♥t ♦♥tt♦s ①s♠♥t ♦♣♥ó♥ ♦r ℓ s♦t♥ s♦r str ♠②♦r t♠ñ♦ ♥ q♦s s♦s ♦♥ sr sr♠♥t ♥t♦ ♠② ♠♣r♦ ♣r s♦ sr ♦♥tt♦s ♣rs♦♥s Gpers ♦♠♦ ♣ ♦srrs ♥ r q ♥♦ t t♦♣♦♦í Gpers ♣r♦ sí ♦ ♦♥ Gop P♦r ♦r♠ó♥ ♠♦♦ 〈k〉 ♣r♠♥♦♥st♥t ♣r ♠s srs s ♥ ♣♦♦ ♠♥♦r ♣r Gop ♦ s r♥s ♦♥ Gpers r ♦s ♦rs ♦r ℓ ♣r♠♥ s ♥tr ♥t ♥r♠♥t♦ q ♠② ♣♦r ♦ s ♦r ♥ ♣r ♠s srsst♦ ♠♣ q ♥t r♦♥①♦♥♦s ♣t♦s s ♥ t♦♦s ♦s s♦ss♥t ♣r s♠♥r ℓ ♦♥ rs♣t♦ ♥ C r ♦♥ q ♣r s♦ Gop st♦ ♠str q ♣r q ♣qñ♦s ♣r♦ ♣tó♥ r♦♥①♦♥♦ s ♥r♠♥t ♥♦ r ♥ t♦♣♦♦í ♦♠♣t ♦♥ ♥sr t♦r ♥ ♠r♦ ♥r♠♥tr q C ♦♠♥③ ♦♥rr ♦r♦rrs♣♦♥♥t ♣r r ♥ ♥♦s ♥ í♥s ♣♥ts ♥ r ♦ ró♥ rást ♥ ♣r♦ ♣tó♥ t♥t♦ ♣s♦s ♣tó♥ ♣s ♦♠♦ t st út♠ ♣♦r r♦♥①♦♥♦ ♥s rá♦ ♣r ♠♦r Q ♥ ♥ó♥ q t♠é♥ sst♥t st♣ótss str q ♦s ♥♦rs t♦♣♦ó♦s Gpers ♥♣r♦♣s ♠♥♦ ♣qñ♦ P♦r ♦tr ♣rt ♥ t♦♦s ♦s s♦s s ♦♥② ♥♣♥♥ t♦♣♦♦í ♥ r rs♣t♦ ♠♣♦ ①tr♥♦ φ②♦s rst♦s s ♣rs♥t♥ ♣r φ = 0.00 sí♠♦♦s r♦♦s φ = 0.01 sí♠♦♦s③s ② φ = 0.02 sí♠♦♦s rs

str ♥á♠ ♦♠♣ st ♠♦♦ ♣r t♦♦ s♣♦ ♣rá♠tr♦s s ♥ ♣r♦s♦ ♠② ①t♥s♦ q s ♥♥tr ♥ ss ♦♠♥③♦s t♦♦s♠♦♦s ♥str♦ ♦t♦ s ♥③r ♦r♠ó♥ strs ♥♦♥♦rs♦ ♥ ♣r♦s♦ s♠♥ó♥ tr ú♥ ♥ stó♥ t ♦rtr ♥ó♥ ② ♥ ró♥ rít ♦♥ ♦s ♦♠♥♦s ♥③♥ ♥qr♦ ♠tst

ó♥ s ♠♣ñs ♥♦r♠ó♥ ♠s ♥ ♠tr s ♣ú♥♦ s♦ ♥ s♣t♦ ♠② ①♣♦r♦ ♥ ♦s ♠♦♦s ♠♣♦ ♠♦ ♦♥ ①♣ó♥ ❬♥♦r♦ ❪ ú♥ ♥str♦ sr ② ♥t♥r st ♦♥stt② ♣r♠r ♦♣♦rt♥ ♥ q s trt t♠ s♦r ♥ ♠♦♦ ♦♥ strtrtr♦é♥ r s♦ ❬s ❪

ó♥ s ♠♣ñs ♥ó♥

101 102 103 104

estática φ=0.00estática φ=0.01estática φ=0.02

101 102 103 104

adaptativaφ=0.00adaptativaφ=0.01adaptativaφ=0.02

r ♣♥♥ ró♥ ♥♦♥♦s ♦♥ q ♣r ♠s r♥ts ♠♦♦ ♥r③♦ s♠♥ó♥ ♦♣♥♦♥s

ó♥ φ s♦r ♦s ♠♦♦s ♦♥ r stát ②

♣r♥tr♥♦s qé r♥s ♥ttts ♦♥rts ①st♥ ♥tr ó♥ ♠t♦r ♠♣♦ ①tr♥♦ φ s♦r ♦s ♠♦♦s r stát ② ♣tt Pr ♦ t♠♦s 500 r③♦♥s t♥t♦ ♠♦♦ s♠♥ó♥ ♦♣♥♦♥s ♦♥ r stát ♦♠♦ ♣tt ♦♥ φ ∈ 0.0, 0.1, 0.2 s ③s♦r ♥ s 500 ♦♥r♦♥s ♥s ♥♦♥♦s t♠♦s10000 r③♦♥s ♠♦♦ ♣r♦♣ó♥ sr♠♣ó♥ rstr♥♦ ♥ s♦ t♠ñ♦ ♥ r♦t Pr ♥tr ó♥ ♠t♦r ♠♣♦①tr♥♦ ♠♦s ró♥ r③♦♥s ♦♥ ♥t ♥t♦st♦ts s♣r ♠r 1% ♣♦ó♥ t♦t ♦ N = 2500 ♦♠r srá 25 ♥ts stró♥ ♠ r ♥ ♥ ♣ít♦ ② ♥♦t ♦♠♦ P (x ≥ 25) s r ♣r♦ t♥r ♥ r♦t q ♥♦r 25 ♦ ♠ás ♥ts

♥ r ♠♦str♠♦s P (x ≥ 25) ♣r φ ∈ 0.0, 0.1, 0.2 t♥t♦ ♣r ♠♦♦ ♦♥ r stát ♦♠♦ ♣tt ♥tr♦ ♦♥♥③ ♥♦♠♣r ♣♥t♦ rst s♣r ♦ r♥ t♠ñ♦ ♠str ❯♥♦♥só♥ q s ①tr ♥♠t♠♥t rá♦ r s q

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

0.0 0.01 0.02campo externo φ

P(x≥2

estáticaadaptativa

r ♦♠♣ró♥ ♥♥ ♠♣♦ ①tr♥♦ ♥ ♦s ♠♦♦s ♦♥r stát ② ♣tt ♣rtr P (x ≥ 25)

t♦ ♠t♦r φ rst ♠②♦r ♥ s♦ ♦rrs♣♦♥♥t r ♣ttPr r ♦s ♦rí♥s st ♥ó♠♥♦ srá ♥sr♦ ♦rr st♦ t♦ φ s♦r strtr tí♣ ♦s strs ♥♦♥♦s rr♦♦s♣♦r ♠s r♥ts ♠♦♦ ♥r③♦

♥♥ φ ♥ t♦♣♦♦í ♦s strs ♠á①

♠♦s

♠♦s st♦ q ♥t ♥ ♥♦♥♦s φ = 0 s q♥tt♥t♦ ♣r ♠♦♦ ♥r③♦ ♦♥ r stát ♦♠♦ ♣tt ♥ q = qC ♥ ♠r♦ ♠♦s r q ♦♥srr φ > 0 ♣r♦ r♦t sr♠♣ó♥ s♠♥② ♠ás r♣t♠♥t ♣r s♦ ♠♦♦ ♦♥ r♣tt st♦ ♥♦ s q r ♣tt ♠ s t♦♣♦♦í ♠♥tr φ ♦ q st♦s ♠♦s ♥♦ s♦♥ s♥t♦s t ♦♠♦ ♠♦str♠♦s ♥ r ♠♦t♦ s r♥s ♦srs s ♦t♥rá ♦♠♣rr t♦♣♦♦í ♦s strs ♠á①♠♦s ♦rrs♣♦♥♥ts sr ♥s ♣rs♦♥s Gpers ♣r ♠♦♦ ♥r③♦ ♦♥ r stát ② ♣tt r ♦♠♣r 〈k〉 ℓ C ② Q ♣r φ = 0.01 s♦ ♦rrs♣♦♥♥t φ = 0.02

s t♦t♠♥t ♥á♦♦ ② r ♠str φ = 0 ♦♠♦ s ♦sr♥ rá♦ ♣r φ = 0.01 ♠♦♦ ♦♥ r ♣tt rr♦ tí♣♠♥t

ó♥ s ♠♣ñs ♥ó♥

0 500 1000 1500 2000 2500tamaño cluster máx

⟨ k⟩

PercolaciónModelo

⟨ k⟩

PercolaciónModelo

0.200.250.300.350.40

PercolaciónModelo

0.200.250.300.350.40

PercolaciónModelo

5101520253035

PercolaciónModelo

5101520253035

PercolaciónModelo

0.650.700.750.800.850.90

PercolaciónModelo

0.650.700.750.800.850.90

PercolaciónModelo

r Pr♦♣s t♦♣♦ós ♦s strs ♠á①♠♦s ♦♥ ♠♣♦ ①tr♥♦φ = 0.01 ♦♦ ♥r③♦ ♦♥ r stát ② ♦♥ r♣tt ♥ ♠♦s s♦s s ♣rs♥t ♦♠♣ró♥ ♦♥ s ♣r♦♣s ♦s strs ♠á①♠♦s ♣r♦♥♥ts ♣r♦①♠ó♥ ♣r♦ó♥

strs ♦♥ r♦ ♠♦ ♠♥♦r q r stát ♦♠♣r♥♦ strs ♠s♠♦ t♠ñ♦ P♦r ♦tr ♣rt st♥ ♠ ♥tr ♥♦♦s ℓ rst ♠♥♦r♣r s♦ ♦rrs♣♦♥♥t r ♣tt ♦♥t str③ó♥ Cs ♠♥t ♠♥♦r ② ♠♦r Q ♥③ ♦rs t♦s ♥ ♠♦s s♦s ♠♦♦ q ♠♥s♠♦ r♦♥①♦♥♦ ♥♦r♣♦r♦ ♥ ♠♦♦ ♦♥ r♣tt r strs ♦♠♣ts ♦♥ rtríst ár♦ t ♦

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

♠♦ ♣ rs ♥ ♠♣♦ r s r♥s strtrs qí

r ♠♣♦s strs ♠á①♠♦s ♥tr♦s ♣♦r ♣r♦①♠♠♥t 200♥♦♦s ♣r ♠♦♦ ♦♥ r stát ② ♦♥ r ♣tt ♠♦s ♣r φ = 0.01 ♦s st♥t♦s ♦♦rs st♥♥ s ♦♠♥s ♦t♥s ♣rtr ♠♦r ♣rtó♥ r♦ ♦♥ ♠ét♦♦ ♦♥ ❬♦♥ ❪ ♥♦ ♥ ♣qt t♦r① P②t♦♥ ♣r ♦♣t♠③ó♥ ♠♦r❬r ❪

♥③s ♥ q ♦s strs ♥♦♥♦rs ♣r♦♥♥ts ♠♦♦ ♦♥r ♣tt rst♥ ♠ás rás q ♦s ♠♦♦ ♦♥ r stát ♥t ó♥ ♠♣♦ ①tr♥♦ φ s♠♥♦ ♥q ♠♦♦ ♥r③♦ ♦♥r ♣tt ♣r♦♠ str③ó♥ ú♥ ♠ás á q ♣r s♦ ♦♥ rstát ♦s strs ♥♦♥♦s ♥③♦s ♠♥t ♣r♠r♦ rst♥♠ás rás q q♦s ♥③♦s trés s♥♦

♦♦ ♥r③♦ ♦♥ t ♦rtr

♥ ♣ít♦ ♥tr♦r ♥③♠♦s ♥♥ ♠♦♦ s♠♥ó♥tr ♥ q♦s s♦s ♦♥ ♥ ♦rtr Pcob ♣ró①♠ ♠t 95%

94.6% ≤ Pcob ≤ 95.4% r♦♠♥ ♣♦r r ♠strq ♠♦♦ ♥r③♦ ♣rs♥t ♥ ♣r♦ r♦t ♥♦t♠♥ts♣r♦r q s ♦t♥ ♣rtr ♣r♦①♠ó♥ ♣r♦ó♥ t♥t♦♣r r stát ♦♠♦ ♥á♠ s♠s♠♦ ♣ ♦srrs t♦ ♠t♦r ♠♣♦ ①tr♥♦ φ s♦r ♣r♦ r♦t t♠ñ♦ ♠②♦r 1% ♣♦ó♥ t♦t P (x ≥ 25)

♦♥s♦♥s ♣r♠♥rs

0 20 40 60 80 100 120tamaño del brote

10-610-510-410-310-210-1100

estática

Pmodel(x≥25)=0.302

AModel φ=0.00Percolation

0 20 40 60 80 100 120tamaño del brote

10-610-510-410-310-210-1100

estática

Pmodel(x≥25)=0.212

CModel φ=0.01Percolation

0 20 40 60 80 100 120tamaño del brote

10-610-510-410-310-210-1100

estática

Pmodel(x≥25)=0.143

EModel φ=0.02Percolation

0 20 40 60 80 100 120tamaño del brote

10-610-510-410-310-210-1100

adaptativa

Pmodel(x≥25)=0.018

BModel φ=0.00Percolation

0 20 40 60 80 100 120tamaño del brote

10-610-510-410-310-210-1100

adaptativa

Pmodel(x≥25)=0.014

DModel φ=0.01Percolation

0 20 40 60 80 100 120tamaño del brote

10-610-510-410-310-210-1100

adaptativa

Pmodel(x≥25)=0.007

FModel φ=0.02Percolation

r st♦r♠ t♠ñ♦s r♦ts sr♠♣ó♥ ♦ ♦♥♦♥s 95% ♦rtr ♥ ♣r ♠♦♦ ♥r③♦ s♠♥ó♥tr rrs r♦s ② ♣r ♣r♦①♠ó♥ ♣r♦ó♥ rrs ③s ♣rs♥t♥ ♦s rst♦s ♣r φ = 0.00 0.01 0.02 t♥t♦ ♣r s♦ ♠♦♦♦♥ r stát ♦♠♦ ♣tt

♦♥s♦♥s ♣r♠♥rs

♥ ♣rs♥t ♣ít♦ ♠♦s ♠♦str♦ q st♥ó♥ rst ♥tr ♥s♣rs♦♥s ② ♥♦♣rs♦♥s ♠♣t ♥ ♥♥ ♠ sr♠♣ó♥ ♠♦r strtr♠♥t sr♦ s♦r s ♣r♦♣ ♦ ♦st♥tú♥ s ♠♥st ♥ó♠♥♦ str③ó♥ ♥♦♥♦rs ♥♦ ♠♥s♠♦ s♠♥ó♥ tr ♦sr♦ ♥ ♣ít♦ ♥tr♦r

♦♥♠♥t ♠♦s ♥♦ r♥t ♦♥ ♣tó♥ strtr r ♣♦r ♠♦ r♦♥①♦♥♦ ♥s ♦♠♦ ♥ r♥t ♠♦♦♥r③♦ ♣r♦♣st♦ ♦str♠♦s q r♥t ♦♥ r ♣tt ♣♦r♥ ♦ ♦r ♦r♠ó♥ strs r♥ t♠ñ♦ ♣r♦ ss ♣r♦♣st♦♣♦ós ♦s t♦r♥♥ strtr♠♥t ♠ás rás q ♦s ♦rrs♣♦♥♥ts

♣ít♦ s♠♥ó♥ ♦♣♥♦♥s ② ♥ó♥ ♠♦♦

♥r③♦

r♥t stát ♠♦♦ P♦r st ♦ ♥♦♥tr♠♦s q ó♥ ♥ ♠♣ñ ♥♦r♠ó♥ ♥ ♦r ♥ó♥ ♥tr♦ trés ♠♣♦ ①tr♥♦ φ ♣r♦ ♥ ♠②♦r s♠♥ó♥ ♥ ♣r♦ r♦ts♦r ♠♦♦ ♥r③♦ ♦♥ r ♣tt

♣ít♦

♥ró♥ rs s♦s ♦♥

s ♠♣ír

❮♥ t s♦ ♦♠♦ ♥r♦r tr♦♥s

strtrs

♦r♠ó♥ ♥ít ♠♦♦

P♦ó♥ ♦♥st♥t γ = 0

P♦ó♥ r♥t

♦rr♦♥s r♦ t♦ ♦r♥

r♦ ♠♦ ♣r♠r♦s ♥♦s

♦♥t str③ó♥

s♦s st♦

♦♠♦ ♠♦s sñ♦ ♥ ♦s ♣ít♦s ♣r♥ts strtr srs s♦s t♥ ♥ ♠♣t♦ rt♦ ♥ ss ♣r♦♣s ♣r♦♣ó♥ s♣♦r ♦ q s ♣r♦♣s t♦♣♦ós státs s rs ♦♠♣s r♦♥♣r♦s♠♥t ♥③s ♥ trtr ♥tí r♥t ♦s ♣r♠r♦s ñ♦s st s♦

♦s sr ♦t♦s stát♦s s rs ♦♠♣s ♥ ♣rtr s ♦r♥s♦ ♣rs♥t♥ ♥ ♦ó♥ ♦♥st♥t q r ró♥ ② ♥t♠♥t stró♥ ♦♥tí♥ ♥s ② ♥♦♦s ♠♥ ♥♦♦s ♥♦s ♣♦r♥s ♦rrs♣♦♥ ♥t♦♥s t♥ s♦♦ ♥ r♣rs♥tó♥ ♠ ♥ t♠♣♦ ♥ ♣r♦s♦ ♥á♠♦ ♦ó♥ r qr r♥ ♥♦

♣ít♦ ♥ró♥ rs s♦s ♦♥ s ♠♣ír

s t♠♣♦ rtríst♦ s ♦♠♣r ♦♥ ♦rrs♣♦♥♥t ♥ó♠♥♦ ②s st ♥ ♥r♠ ♥♦s ♦ ♥ ♦♣♥ó♥ q s ♣r♦♣ s♦r ♠s♠P♦r ♦tr ♣rt r♥t♠♥t ♦♠♥ ♥tí ♣♦ r ♦ó♥♠r♦só♣ r♥s rs ♦♥♥ ♦♥ ♥ rs♦ó♥ t♠♣♦r ② ♦♠♥③ó r③r ①♣r♠♥t♦s ♦♥rt♦s t♥♥ts r ♥á♠ ♦♥tt♦s♥tr ♥♦s ❬s ❪ ♦♥♥ó♥ t♦♦s st♦s ♠♥t♦s ♠♣só st♦ s ♠s rs rs ♥ t♠♣♦ ♦♠♦ ♥ sár ♠♣♦ s rs ♦♠♣s

st♥t♦s ♠♦♦s rs rs ♥ t♠♣♦ ♦♠♣ñ♥ ♣ró♥ ♥♦s sts t♦s ♠♣ír♦s ♥ ♠r♦ ♦s ♠♦♦s s♠♥ t♠♥t ♣ótss t♥♥ts s♠♣r s st♦ ♥ít♦ ♥tr s♣ótss s ♥♥tr ♠♦r ♦s ♦♠rs ♦♠♦ ♠♥♥ts ③r❬tr♥♥ ❪ ♦ ♥ ♠♦r ♦s s♦s ♦♠♦ st♦s s♥ ♠♠♦r ss♦♥tt♦s ♣r♦s ❬Prr ❪ ♦♠♦ ♦♥s♥ ♦ s rs ♠s♥ t♠♣♦ rr♦s ♣♦r ♦s ♠♦♦s ♥♦ ♣rs♥t♥ t str③ó♥ ♥ ♦♠♣♦rt♠♥t♦ s♦rtt♦ ♥ s♠ s ♦rr♦♥s r♦ rtrísts s rs s♦s sts ♥ s rsó♥ stát st♦ ♣♥t ♥ t♥só♥♥tr ♦s r♣rs♥t♦♥s ♠s♠ r q r♣r♦ ♥á♠ ♦s ♦♥tt♦s ② q ♦♦ ♥ s r♣rs♥tó♥ ♠ ♥ t♠♣♦ ♦t♦ ♣rs♥t ♣ít♦ s ♣r♦♣♦♥r ♥ ♠r♦ tór♦ q ♣r♠t rs♦rst ♦♥t♦ ♣♦rt♥♦ ♠♥t♦s ♣r q s rs ♥rs ♦rít♠♠♥t sts♥ s ♦♥♦♥s ♠♣♠♥t ♦srs ♥ s rs s♦s♠s ♥ t♠♣♦ ♦rt♠♦ qí ♥tr♦♦ ♥♦s ♣r♠trá ♥t♦♥s♥rr rs s♦s ♠s ♥ t♠♣♦ ♦♥ s ♠♣ír

t s♦ ♦♠♦ ♥r♦r tr♦

♥s strtrs

s s rs s♦s ♥③s ♥ trtr ♣♦s♥ rts rtrísts ♥rs st♥ts ♣qñ st♥ ♠ ♥tr ♥♦♦s ∝ lnN② t str③ó♥ ♦♠♣rr ♦♥ ♥ r t♦r ♦♥ ♠s♠♦ ♥ú♠r♦ ♥♦♦s N ② ♥s L rt strtr ♦♠♥s ② str♦♥s r♦s P (k) q ♣♥ rr s stró♥ ①♣♦♥♥ ② ♣♦t♥srs rs s ♥ ♥t♦ st út♠ ♣r♦♣ ♠♦s ♠♥♦♥♦♥ ♥tr♦ó♥ ♣ít♦ q ♠♥s♠♦ ♠ás ①t♥♦ ♣r r♠♥t♦ rs rs s s ♥ ♣rr♥ ❬rs ❪ ♦

t s♦ ♦♠♦ ♥r♦r tr♦♥sstrtrs

♠♥s♠♦ stá s♦ s♦♦ ♥ ♦♥t ② rqr r♠♥t♦ ♣♦ó♥ ♣♦r r♦ ♥♦♦s ② ♥s ♣r str ♦r♠ó♥ rsrs s ♦ ♦st♥t ú♥ s st♥ ss ♦rí♥s ♠r♦só♣♦s

♥ ♦♥t①t♦ s rs s♦s rs ♥ t♠♣♦ s ♣r♦♣st♦ q stró♥ r♦s r s ♣ t♥r ♦r♥ ♥ ♥ rtríst♥trí♥s ♦s ♥ts s♦s ♥♦♠♥ t s♦ a ❬Prr ❪♥♦s ♥♦s rst♥ s♦♠♥t ♠ás t♦s q ♦tr♦s ♣♦r ♦ st♥ ♠②♦r ♥t í♥♦s s♦s q ss ♣rs ♦ s stró♥ t s♦ s ♥ ② ♣♦t♥s F (a) ∝ a−γ ♣ ♠♦strrs q stró♥ s r ♣♦r stró♥ r♦s r ♦♥tt♦s♠♦s r♥t ♥ ♥t♥ t♠♣♦r T s r PT (k) ∝ k−γ ♥tsst♦s ♥ ♣r♦♦ q t s♦ ♥♦s q ♣rt♣♥ ♥♣r♦②t♦s ♦♦rt♦s ts ♦♠♦ ❲♣ t ♠ ♥ tér♠♥♦s ♥ú♠r♦ ♣♦rts s ♦rr♦♥ rt♠♥t ♦♥ r♦ q ♦s ♠s♠♦s♥♦s qr♥ ♥ r s♦rs ❬♥ ❪

♠♦♦ ♣r♦♣st♦ ♥ ❬Prr ❪ ♦♥sr ♥ ♦♥♥t♦ N ♥♦♦s ♥♠♥t s♦♥t♦s ♥♦♦ i s s♥ ♥ t s♦ ai r♦ ♦♥ ♥ stró♥ ♣r♦ F (a) ♦r ai ♦rrs♣♦♥♥t♦♥s ts tó♥ ♥♦♦ i ♣♦r ♦ q aiδt s ♣r♦q t♥ ♥♦♦ i trs ♥ ♥tr♦ t♠♣♦ δt ♥♦ ♥ ♥♦♦ st st ♥s ♦♥ ♦tr♦s m ♥♦♦s ♦s ③r st♥ st♦s út♠♦st♦s ♦ ♥♦ ♦ ♣s♦ t♠♣♦r δt t♦♦s ♦s ♥s r s♦♥ ♦rr♦s② ♣r♦s♦ ♦♠♥③ ♥♠♥t st ♠♦♦ ♦s ♥♦♦s tú♥ ♦♠♦ ♥ts ♠r♦♥♦s s♥ ♠♠♦r ss ♦♥tt♦s ♣r♦s ♦t♦ ♣♦r r♠ t♠♣♦r♠♥t rst ♥ r t♦r ♦♥ stró♥ r♦s

PT (k) ∝ F

s♥♦ T t♠♣♦ ♠♦ ♥ r s ♠str♥ ♠♣♦s ♦♥s♥♦s ♥ ❬Prr ❪ ♣r str♦♥s t F (x) ♠♣írs ♦t♥s ♣rtr ss t♦s ♣♦♥s ♥ P②s ttrsP s ♣ís ② ♣♦r út♠♦ ♠♥ss ♥ r ttr P rs q ♥ ♦s trs s♦s ♦♠♣♦rt♠♥t♦ F (x) ♣rs♥t ♥♦♠♣♦rt♠♥t♦ ♦♠♣t ♦♥ ②s ♣♦t♥s

é①t♦ ♠♦♦ ♦ Prr t r ♥t♦♥s ♥ ①♣ó♥♠r♦só♣ ♦r♥ stró♥ r s ♥tr♦ ♣r♠ rs rs ♥ t♠♣♦ ♥ ♠r♦ s rs ♠s ♦t♥s

r str♦♥s ♠s t sr♦s F (x) ♠s♠♣ír♠♥t s♥♦ tr♦ ♥t♥s t♠♣♦rs r♥ts ♥ s♦ ♣♥ ♠str stró♥ t ♠ ♣r r ttr ♣r t ♦s t♦rs ♣ís sú♥ s ② ♣♥ ♣r ♦s t♦rs ♣♦♥s ♥ rst P②s ttrs P ♣♥ ♠str ♥ r♣rs♥tó♥ sq♠át ♠♦♦ ts♦ ♦♥sr♥♦ s♦♦ 13 ♥♦♦s ♠str r ♠ ♥ t♠♣♦♣r 3 ♣s♦s t♠♣♦rs ss♦s

♣rtr st ♠♦♦ r♥ ♦trs rtrísts ♥♠♥ts ♠♥♦♥s ♦♠♥③♦ st só♥ s ♣♦r ♦ q ♥ s♥t só♥ ♥♦s ♦r♠♦s srr♦r ♥ ♠♦♦ tr♥t♦ ♥♦r♦♥♦ q ♦♥sr s ♥ts ♠♦♦ ♥tr♦r ♥q ♥♦r♣♦r♥♦ s rtrísts ♦♥s ♣r♦♣s s rs s♦s rs

Gt(N,L) r♦ ♣♥♥t t♠♣♦ t ♦♠♣st♦ ♣♦r N(t) ♥♦♦s ②L(t) ♥s ♦♠♦ ♥ s♦ s rs ♥s ♣♦r t s♦ t♠é♥ s♠r♠♦s q ♥♦♦ i ♣♦s ♥ t ♥trí♥s ai ♣♦r♥ ♣ F (a) ♥ ♠♦♦ ♦s ♥♦♦s s t♥ ♥ ♣s♦ t♠♣♦r δt♦♥ ♥ ♣r♦ ♣♦r aiδt qí ♣r♦♣♦♥♠♦s ♥ ♠♦♦ r♠♥t♦st♦ást♦ ♥r③♦ ♣r r ♦♥ ai r♣rs♥t ts

♠r♥ ♥♦s ♥s s ♥♦♦ i st ♠♦♦ ♥ r rr♠út♣s ♥s ♥ ♣s♦ t♠♣♦r st♦s rá♥ ♥rá♥♦s ♥♦ ♣♦r ♥♦s♥♦ ♥ ♣r♦s♦ P♦ss♦♥ ♦♥ ts r ♥t♦♥s ♦s ♥s srá♥♥♦r♣♦r♦s ♦♥ ♥ ts ♣♥♥t t♠♣♦ ♣♦r

β(t) =

N(t)∑

sí ♦♠♦ ♦s ♥s s♦♥ r♦s ♠♦♦ t♠é♥ ♠t ♥ ♣r♦s♦ ♠♥ó♥ ♥s ♠♦♦ q ♥ lj s s♥ ♥ ts ♦rr♦ µj ♣r♦♥♥t ♥ ♣ FD(µ) ♥ ♦♥s♥ ts ♦rr♦t♦t ♥ ♥ó♥ t♠♣♦ srá

η(t) =

L(t)∑

♥ ♥str♦ ♠♦♦ ♠t ró♥ ② ♦rr♦ ♥s ♥ ♥ó♥ t♠♣♦ qí ♥♦s ♥trs ♥③r s ♣r♦♣s t♦♣♦ós s rs ♠s ♥ t♠♣♦ AT s ♣♦r ♦ q ♦s ♥s r♦s ♣r♠♥rá♥♥ ♥str sr♣ó♥ ♦♠♦ s r♥ stát♦s ♥ qr s♦ st♠♦sq ♠♦♦ t♠é♥ ♥♦r♣♦r ♥ ♠♥s♠♦ ♦rr♦ ♥srtrr♦

♥á♦♠♥t ♦ q s ♦♥ ♦s ♥s ♣♦ó♥ ♥♦♦s t♠é♥♣ srr ♠♦s ♥ t♠♣♦ s ♣♦r ♦ q ♠t s ③ ♦srí♠♥s ♣r ♦s ♥♦♦s ♣♦ó♥ ♦♥st♥t ♦ ♣♦ó♥ r♥t ♦♥ t♠♣♦ ♥♦♦ r♦ s ♦♥t ♥♠♥t ♥ ♥♦♦ ①st♥t trés ♥ ú♥♦ ♥

♦♠♦ ♠♦s sñ♦ ♣r♦♠ ♠♦♦ ♣r♦♣st♦ ♣♦r Prrt t♥ s ♦r♥ ♥ s♣♦só♥ ♥ts s♥ ♠♠♦r ss ♦♥tt♦s ♣r♦s s ♣rs♠♥t ♣♦r ♦ q s rs ♠s ♥ t♠♣♦ rst♥ s♥♠♥t rs ③r ♦♥ stró♥ r♦s t♣♦② ♣♦t♥s r♠♥t s trt ♥ ♣ótss s♠♣♦r s♥ ♦rrs♣♦♥♥ ♥ ♦♥ ♦sró♥ ♠♣ír ①st ♥ s♥tq ♥t ♦rr♦♥s r♦ t♦ ♦r♥ ♥ rs s♦s rs❬♠♥ ♠♥ ♦ss♥ts ❪ ♥♦r♣♦rr s ♦rr♦

♥s r♦ ♥ ♥str♦ ♠♦♦ ♠♣ s♠r ♠♠♦r ♦s ♦♥tt♦s ♣r♦s♥ ♦s ♥ts Pr ♦ ♠♣♠♥t♠♦s ♥ ♠♥s♠♦ ♦♥①♦♥♦ ♠①t♦q ♣r♦ s♥t ♠♦♦ ♣r r♦ ♥ ♥ lij

♥♦♦ ♥t i ♦♥ ♣r♦ ♣r♦♣♦r♦♥ s t ai

♥♦♦ st♥♦ j ♥ s ♦ r♦ ♦♥ s♥t♣r♦♠♥t♦

♥s♠♦ rr trá♦ ♦♥ ♣r♦ q s ♥s♥♦ ♥♦ i q ♥♦ s ♥♥tr ♣r♠♥t ♥♦ j♦♥ ♥ rrr ♥ trá♥♦ ♣♦r tr♥st

♦♥①♦♥♦ t♦r♦ ♦♥ ♣r♦ 1− q s ♥ ♥♦♦ j ♥♦r♠♠♥t ③r

♠♥s♠♦ rr trá♦ s♦ ♣r♠♥t t③♦ ♦♥ ♦t♦ rr tr♥st ♥ ♠♦♦s r♠♥t♦ rs ❬♦ss♥ts s♥ ♦♠ ❱á③q③ ♦♦♥♥ ♥♦ ❪P♦r ♦tr ♣rt t♥ ♣♦r ♥ r♣r♦r ♠♥s♠♦ ♥ q t♠♥t s ♦sr ♥ s rs ♠st ♦s ♠♦s ♥str♦s ♠♦s ♦♥r♥ t♠é♥ tr♠♥♥ s♥♦ ♥str♦s ♠♦s s ♥ st s♥t♦ ♥ q♥♦s rr♠♦s ♥ts ♦♥ ♠♠♦r ♦ q ♣r q rst ♣♦s str ♥ í♥♦ ♥tr s♥♦s ♥♦s s ♥sr♦ q ♦s ♥ts rr♥t♦♦s ss ♦♥tt♦s ♣r♦s str q ♠♦♦ s♠ ♥tss ♦♥ ♠♠♦r ♥♥t ♥ ♣♦s s♠♣r ♥áss tr♦r

♦r♠ó♥ ♥ít ♠♦♦

♠♦♦ ♣ ♦r♠rs ♠t♠át♠♥t ♦♠♦ ♥ ♣r♦s♦ st♦ást♦ t♠♣♦ ♦♥t♥♦ Ltt∈R≥0

♦♥ ts ♥♦r♣♦ró♥ ♥s β(t)st ♣r♦s♦ t♠♣♦ ♦♥t♥♦ ♣ r♣rs♥trs tr♥t♠♥t ♦r♠♠ás s♠♣ ♦♠♦ ♥ ♣r♦s♦ st♦ást♦ srt♦ ♠♦ Lℓℓ∈N ♥ tér♠♥♦s ♥ú♠r♦ ♥s r♦s ℓ ♥ st út♠♦ s♦ s s♠♣ ♥♦t♠♥t ♦r♠ó♥ ♠t♠át ② s ♣♦str♦r ♥áss ♣r♦s♦ srt♦♠♦ s♠ ♥ ts ♥r♠♥t♦ ♣♦ó♥ ♥s 1♦ q ♣♦r ♣s♦ ♦ó♥ s ♦♥ ♥ ♥ st ♠♦♦ stss rst♥ts ♠♦♦ strá♥ rrs ts ♥tr

♥ ♦rr ♦r♠ó♥ ♠ás ♥r ♠♦♦ ♦♥sr♠♦s ♥ts r♠♥t♦ ♣♦ó♥ ♥♦♦s γ P♦r ♦ t♥t♦ ♥ú♠r♦ t♦t

♥s ♥rá ♦ ♣♦rL(ℓ) = L0 + (1 + γ)ℓ,

♦♥ γℓ ♣r♦♥ ♦s ♥s s♦♦s ♦♥ ♦s ♥♦♦s r♦s ② L0 s ♥t ♥ ♥s ♥á♦♠♥t ♣♦ó♥ ♥♦♦s s ♥r♠♥t♦♠♦

N(ℓ) = N0 + γℓ,

s♥♦ N0 ♥t ♥ ♥♦♦s ♥ r

♥♠♦s Pk|a(ℓ, ℓ0) ♦♠♦ ♣r♦ q ♥ ♥♦♦ ♦♥ t a

♥tr♦♦ ♥ ♣s♦ ℓ0 t♥ r♦ k ♣r ♥ ♣s♦ ♣♦str♦r ℓ > ℓ0 ♦♥♠♥t ♥♠♦s Nk|a(ℓ) ♦♠♦ ♥t ♠ ♥♦♦s ♦♥ r♦ k ♥trq♦s q ♣♦s♥ ts t a ② ①♣rsó♥ ♥ tér♠♥♦s Pk|a(ℓ, ℓ0)

Nk|a(ℓ) =ℓ∑

ℓ0=1

Pk|a(ℓ, ℓ0) .

♥ ♣é♥ ♠♦strr♠♦s q Nk|a(ℓ) ♣r♦♣♦r ♠♦♦ rs♦ts ♣r♦♣st♦ ♥ ❬♦♥ ❪

♦ó♥ t♠♣♦r Nk|a(ℓ) ♣ ♦r♠rs ♠♥t ♥ sr♣ó♥ ♦♥t♥ trés s♥t sst♠ ♦♥s r♥s ♦♣s❬r♣s② r♣s② ❪

dℓ=q(

Θ(k − 1, a, ℓ)Nk−1|a −Θ(k, a, ℓ)Nk|a

+1− q

N(ℓ)

〈a〉 + 1

Nk−1|a − Nk|a

N(ℓ)

Nk−1|a − Nk|a

+ γδk1 ,

♦♥ ♣r♠r tér♠♥♦ ♠♠r♦ r♦ rs ó♥ ♦rrs♣♦♥ ♦♥tró♥ ♠♥s♠♦ rr trá♦ s♥♦ Θ(k, a, ℓ)

♥ú♦ s♥♦ tér♠♥♦ rs ó♥ ♦rrs♣♦♥ ♦♥tró♥ ♦s ♥s r♦s ③r st út♠♦ ♣ s♦♠♣♦♥rs♥ ♦♥tró♥ ♥♦♦ ♥t ♦ ♦♥ ♣r♦ ♣r♦♣♦r♦♥ st

1− q

N(ℓ)

〈a〉

Nk−1|a − Nk|a

s♠ ♣♦rt ♦rrs♣♦♥♥t ♥♦♦ ♦t♦ ♦ ♥♦r♠♠♥t ③r

1− q

N(ℓ)

Nk−1|a − Nk|a

♦♥t♥♥♦ ♦♥ sr♣ó♥ ó♥ út♠♦ tér♠♥♦ rs ♦rrs♣♦♥ ♦♥tró♥ ①tr♠♦ rst♥t ú♥♦ ♥ q ♥ ♥t♦ ♥♦♦ t a ♥tr♦♦ sst♠ str q ♦ ①tr♠♦ ♥ s ♦♥t ♥♦r♠♠♥t ③r qr ♥♦♦ ♣rt♥♥t r ♥♠♥t út♠♦ tér♠♥♦ rs γδk1 r♣rs♥t ♣♦rt ♣♦ó♥ N1|a(ℓ) ♦s ♥♦♦s r♦s ♦♥ t a ② r♦ ♥ k = 1

♥ ♥t♦ ♥ú♦ ♠♥s♠♦ Θ(k, a, ℓ) ♥ ♥♦♦♠♦

Θ(k, a, ℓ) =a

〈a〉N(ℓ)+

2L(ℓ).

❯♥♦ ♦s ①tr♠♦s ♥ r♦ ♠♥t ♠♥s♠♦ s♦♥t♦ ♥ ♥♦♦ ♥t ♦ ③r ♣r♦♣♦r♦♥♠♥t s t a♥♦ ♦r♥ ♣r♠r tér♠♥♦ rs ♥trs q s♥♦tér♠♥♦ rs ♣r♦♥ ♦♥①ó♥ ①tr♠♦ rst♥t ♥ r♦ ♥ s♥♦ ♥♦ ♥♦♦ ♥t st út♠♦ tér♠♥♦ r♣rs♥t ♥♣rr♥ q sr ♥tr♠♥t ♣rtr ♠♥s♠♦ rst♦ qs ♦rrs♣♦♥ ♦♥ ♦s ♣r♠♥t r♣♦rt♦s ♥ ❬♦♠ ❱á③q③ ❪

r♣♥♦ tér♠♥♦s ♣ rsrrs ♦♠♦

dℓ= Nk−1|aΦγ,q(k − 1, a, ℓ)− Nk|aΦγ,q(k, a, ℓ) + γδk1

♦♥ Φγ,q(k, a, ℓ) s ♥ú♦ ♥r③♦ ♦♥①ó♥ ♦ ♣♦r

Φγ,q(k, a, ℓ) =1

N(ℓ)

〈a〉 + 1− q + γ

2L(ℓ)q .

♦ ①♣rsó♥ ♦rrs♣♦♥♥t ♥ú♠r♦ ♠♦ ♥♦♦s ♦♥ r♦ k♥♦t Nk(ℓ) ♦♠♣r♥♥♦ ♣♦rt t♦s s tss t ♣♦ss♦♥ ♣ F (a) q ♥ ♦♠♦

Nk(ℓ) =

F (a)Nk|a(ℓ)da

s♥♦ Ω ♦♠♥♦ ♥ó♥ F (a) P♦r út♠♦ ♣♦♠♦s ♥r stró♥ r♦s ♣r ♥ ♥t ♥s r♦s ℓ ♦♠♦

Pℓ(k) = Nk(ℓ)/N(ℓ) .

♠♦♦ qí sr♣t♦ ♣rs♥t ♥ ♠♣ rst ♣r♠t♥♦

♠♦r t♥t♦ stró♥ ts ♦♠♦ ♦s rí♠♥s r♠♥t♦ s ♣♦♦♥s ♥♦♦s ② ♥s s ♣♦r ♦ q ♥ s ss♦♥s s♥ts str♠♦s s s♦♦♥s ♣r ♣♦ó♥ ♥♦♦s ♦♥st♥tγ = 0 ② r♥t γ > 0 tr♥t♠♥t str♠♦s ♦s ♣ t ♣r♠áts t ♦♥st♥t F (a) = δ(a− a0) ② t t♣♦ ② ♣♦t♥s F (a) ∝ a−ξ q r♣r♦ ♠♥t ♣tró♥ t ♦s ♥ts ♣r ♥s rs s♦s ♥③s ♥ trtr❬Prr ♥ ❪

P♦ó♥ ♦♥st♥t γ = 0

♥ st s♦ ♣rtr r♠♥t♦ r s♦♦ t♥ r trés r♦ ♥s s♦r ♥ ♣♦ó♥ ♦♥st♥t ♥♦♦s stt②♥♦ γ = 0

♥ ② ♦ r♠♣③r ♥ ♠s♠ ♦rrs♣♦♥♥t ①♣rsó♥♣r Φ0,q(k, a, ℓ) s ♦t♥

dℓ=Nk−1|a

〈a〉 + 1− q

+(k − 1)q

2L(ℓ)

〈a〉 + 1− q

2L(ℓ)

♦♥ s s♠ N(ℓ) = N0 ∀ℓ ≥ 0 ② ♣♦ó♥ ♥s r ♦♠♦ L(ℓ) =

L0 + ℓ ♥ sts r♥st♥s r♦ k ♣rs♥t ♥ rstró♥ ♥tr ♣♦r k ≤ (N0 − 1) s♠s♠♦ s♦ó♥ s♥tót ♦♣t ♦r♠tr Nk|a = N0δk,(N0−1) ∀a ≥ 0 ♦ ♦st♥t ♥ st s♦ ♥♦s ♥trs ♥③r s♦ó♥ ♥♦tr tr♥st♦r st♦ s q ré♠♥ ♣♦ó♥♦♥st♥t ♥♦♦s ♥r♠♥t s ♥♥tr s♦♦ ♦♥ ♥áss rss♦s s♦r ♥ ♥t♥ t♠♣♦r ♦s s♥t♠♥t ♣qñ ♣r q ♣ótss rst

s ♣♦s rs♦r ♣♦r ♠♦ ♥ sq♠ trt♦ ♦♠♦ s

Nk|a(ℓ) =

Nk|a(0) +

∫ ℓ

Nk−1|a(ℓ′)

Πk|a(ℓ′)Φ0,q(k − 1, a, ℓ′) dℓ′

Πk|a(ℓ)

♦♥ Πk|a(ℓ) = A (L0 + ℓ)−kq/2 exp (− (a/〈a〉+ 1− q) ℓ/N0) s s♦ó♥ ó♥ r♥

dΠk|a(ℓ)

dℓ= −Πk|a(ℓ)Φ0,q(k, a, ℓ).

P♦r ♦tr ♣rt s ♣♦s ♥♦♥trr s♦♦♥s ♥íts rrs ♥ ♥♦s s♦s ♣rtrs P♦r ♠♣♦ s q = 0 ♣ rs á♠♥tq s♦ó♥ rst

Nk|a(ℓ) = N0 × Pois (k;λ = ℓ(a+ 〈a〉)/(〈a〉N0)) ,

s♥♦ Pois(k;λ) = (λk/k!) exp(−λ) stró♥ P♦ss♦♥ ♦♥ ♠ λ♠♣③♥♦ ♥ s ♦t♥

Pℓ(k) =Nk(ℓ)

ℓk − 1

r♣r♥♦ sí s♦ó♥ ♠♦♦ t s♦ ❬Prr ❪

str♦ ♦t♦ s ♥♦♥trr s♦♦♥s ♣r♦①♠s ♦ ♦♥♦♥s ①tr♠s q ♥♦s ♣r♦♣♦r♦♥♥ ♥ ♥tó♥ ♦♠♣♦rt♠♥t♦ ①t♦ Nk|a(ℓ) ② ♦ stró♥ r♦s s♦ Pℓ(k) ♦♥ t ♥s♠r♠♦s ♥♠♥t ♦♥ó♥ ①tr♠

〈a〉 + 1− q

≫ qk

2(L0 + ℓ),

♥ rt s♦ó♥ ♣r♦①♠ rst

Nk|a(ℓ) ∼1

a/〈a〉+ 1− q

a/〈a〉+1−qN0

s r ♥ P♦ss♦♥ ♦♥ ♠ λ = ℓ(a/〈a〉 + 1 − q)/N0 s♦ó♥ rst ♥tt ♦ q ♦♥ó♥ ♠♣ s♣rr ó♥ tér♠♥♦ ♥ ♣rr♥ rs♣t♦ ♦rrs♣♦♥♥t ♥ ③r♣s♦ ♣♦r t ♦s ♥♦♦s

tr♥t♠♥t ♦♥ó♥ ♦♣st

〈a〉 + 1− q

≪ qk

2(L0 + ℓ),

♦♥ ♦tr s♦ó♥ ♣r♦①♠ ♥ st s♦ ♣♦r

Nk|a(ℓ) ∼ (L0 + ℓ)−q/2

L0 + ℓ

)−q/2]k−1

≈ (L0 + ℓ)−q/2 e−(k−1)

L0+ℓL0

)−q/2

q ♠str ♠♥t♦ ①♣♦♥♥ ♥♣♥♥t t a ♥ s♦ ♦♥ó♥ ♠♥s♠♦ ♥ ♣rr♥ s q s ♠♣♦♥P♦r ♦ s♦ó♥ ♦♥ ♠♥t♦ ①♣♦♥♥ s ♦♥sst♥t ♦♥ s♣r♣r ♥ ♠♥s♠♦ ♥ ♣rr♥ ♦♥ ♣♦ó♥ ♥♦♦s ♦♥st♥t❬rs ❪

①♣rsó♥ Nk(ℓ) s ♦t♥ ♣rtr ♣r ♥ s♣ t F (a) s♦♥s ♥ ♥str♦ s♦ ♦♥st♥t F (a) = δ(a−a0)♦ ② ♣♦t♥s F (a) ∝ a−ξ

20 40 60 80 100k

10−6

10−5

10−4

10−3

10−2

10−1

40 60 80 100k

10−10

10−9

10−8

10−7

101 102 103 104

10−7

10−6

10−5

10−4

10−3

10−2

10−1

(b)q=0.1

r stró♥ r♦ Pℓ(k) = Nk(ℓ)/N0 ♣r ré♠♥ ♣♦ó♥♦♥st♥t stró♥ t F (a) = δ(a − a0) r ♠ ♥ t♠♣♦ ♦♥ N = 105 ② 〈k〉 = 20 stró♥ t F (a) ∝a−1.5 r ♠ ♦♥ N = 105 ② 〈k〉 = 200 ♥ ♠♦s s♦s ♦s sí♠♦♦s♦rrs♣♦♥♥ ♦s ♣r♦♠♦s s♦r 100 s♠♦♥s ♥♠érs ♣r q = 0.1ír♦s ♥r♦s q = 0.3 r♦s r♦♦s ② q = 0.9 trá♥♦s rs ♦♥ ♦t♦ s♠♣r s ♦s rá♦s ♣r st♥t♦s ♦rs q ♥s♦ s♣③♦s rt♠♥t ♥ t♦♦s ♦s s♦s s í♥s sós ♦rrs♣♦♥♥ s♦ó♥ ♥♠ér ♦ ♥tr ♥ ♦♥ ♦t♦ ♦t♥r Pℓ(k) = Nk(ℓ)/N(ℓ) ♥st ♥ t ♣r ♦rs r♥s k ♥ ♦♠♣rr ♦♥ ♠♥t♦ ①♣♦♥♥ s♦ó♥ ♣r♦①♠ ♣♦r í♥ tr③♦s

♥s②♠♦s s♠♦♥s ♦♠♣t♦♥s ♦rt♠♦ ♥r♥♦ rs ♦♥ ♥ ♣♦ó♥ ♦♥st♥t N = 105 ♥♦♦s ♣r s ♦s ♣ t F (a) ♦♥srs ♠tá♥♠♥t r③♠♦s ♥tró♥ ♥♠ér s ♦♥s ② ♦♥ ♥ ♦♠♣rr s str♦♥s r♦sPℓ(k) = Nk(ℓ)/N(ℓ) ♦♥ s ♦rrs♣♦♥♥ts ♣r s rs s s♠♦♥s r ♠str ♠② ♥ r♦ ♥tr s s♦♦♥s

♥íts Pℓ(k) ② s ♦t♥s s s♠♦♥s ♦rt♠♦ ♣rF (a) = δ(a−a0) r ② F (a) ∝ a−ξ r ♦♥ st♥t♦s ♦rs q

P♦ó♥ r♥t

s♦ tr♥t♦ ♥③♦ ♥ ssó♥ ♥tr♦r ♦rrs♣♦♥ s♠r♣♦ó♥ r♥t s r ♦♥srr γ > 0 ♥ ♦ r♣rtér♠♥♦s ♣ rsrrs ♦♠♦

dℓ=Nk−1|a

N(ℓ)

〈a〉 + 1− q + γ

+(k − 1)q

2L(ℓ)

− Nk|a

N(ℓ)

〈a〉 + 1− q + γ

2L(ℓ)

+ γδk1

♦♥ ♦r N(ℓ) = N0 + γℓ ② L(ℓ) = L0 + (1 + γ)ℓ ♦♥trr♦ ♦ q♦rrí ♣r s♦ ♦♥ ♣♦ó♥ ♦♥st♥t γ = 0 ♦r s ♣♦s ♦t♥rs♦♦♥s s♥tóts ♥♦trs ♥♦ ❬r♣s② ❪ ♣♦♥rs q st s s♦♦♥s s♠♥ ♦r♠ ♥r Nk|a(ℓ) = nk|aℓs♥♦ nk|a ♥ ♥ó♥ s♦♥♦ ♣♥♥t r♦ k ② ts t a ♥ í♠t s♥tót♦ ℓ → ∞ ♣ rs♦rs trés ♥ ♣r♦♠♥t♦ trt♦ rr♦♥♦ s♥t s♦ó♥ ♥r

nk|a = n1|a

k−1∏

2(γ + 1)(a/〈a〉+ 1− q + γ) + qγ j

2(γ + 1) (a/〈a〉+ 1− q + 2γ) + qγ (j+ 1)

s♥♦ n1|a s♦ó♥ ♦rrs♣♦♥♥t ♦r k = 1 ♣♦r

n1|a =2γ2(γ + 1)

2(γ + 1) (a/〈a〉+ 1− q + 2γ) + qγ.

ás á ①♣rsó♥ rr♦s♠♥t ①t ♣r nk|a ♣rt♥♠♦s qrr♥tó♥ ♣rtr ♥áss s ♦♠♣♦rt♠♥t♦ ♥ s♥r♦s ♣r♦①♠♦s q ♣r ♣♦ó♥ ♦♥st♥t ♦♠♥③r♠♦s ♦♥sr♥♦ stó♥①tr♠ ♥ q ♣ s♣rrs tér♠♥♦ ♥ ♣rr♥ ♥

〈a〉 + 1− q + γ

〈k〉ℓ ≫ qk

s♥♦ 〈k〉ℓ = 2L(ℓ)/N(ℓ) r♦ ♠♦ ♦ ℓ ♥s r♦s ♦ ♣s♦s ♦rt♠♦

♥tr♦♥♦ ♣r♦①♠ó♥ ♥ ② sstt②♥♦ Nk|a(ℓ) =

nk|aℓ ♦t♥♠♦s s♦ó♥ s♥tót ♦rrs♣♦♥♥t st s♥r♦ ♣r♦①♠♦

nk|a ∼(

a/〈a〉+ 1− q + γ

)−(k−1)

q ♠str ♥ r♦ ♠♥t♦ ①♣♦♥♥

❱♠♦s ♦r ♣r♦①♠ó♥ tr♥t

〈a〉 + 1− q + γ

〈k〉ℓ ≪ qk .

♦ ♦♥ó♥ t♦♠ s♥t ♦r♠ s♥tót

nk|a ∼ k−1−2(γ+1)

♠♦str♥♦ ♥ ♦♠♣♦rt♠♥t♦ t♣♦ ② ♣♦t♥s ♦♥ ①♣♦♥♥t α =

1 + 2(γ+1)q

q rst α > 3 ♥♦ 0 < q ≤ 1 ② γ > 0 ❯♥ ♦♥s♥♠♣♦rt♥t s q nk|a rst ♥♣♥♥t a ♠♦♦ q Nk(ℓ) r é♥t ♣r♦♣ ♦♣t♥♦ ♠s♠♦ ①♣♦♥♥t ♥ s ♦r♠s♥tót

Nk ∼ k−1−2(γ+1)

♥ q ♦♠♣♦rt♠♥t♦ s♥tót♦ t♣♦ ② ♣♦t♥s♠♦str♦ ♣♦r Nk(ℓ) rst ♥♣♥♥t ♣ t F (a) ② s s♦st♥ s♦♦ ♣rtr ♠♥s♠♦ rr trá♦ ♠♥t ♦♠♥ó♥ ♥ tér♠♥♦ ♥ ♣rr♥ ♦♥ r♠♥t♦ ♣♦♦♥ r stró♥ r♦s t♣♦ ② ♣♦t♥s t ♦♠♦ s st ♥❬rs ❪

♥ r s ♦sr ♠② ♥ r♦ ♥tr ♦s rst♦s ss♠♦♥s ♥♠érs ② s♦ó♥ ♥ít ♣r P (k) ♦F (a) = δ(a − a0) ♥ í♥ tr③♦s s ♣rs♥t♥ s♣r♣sts s s♦♦♥s♣r♦①♠s q ♣r♦♥♥ s ② r ♣rs♥t é♥t♦♠♣ró♥ ♥q ♥ st s♦ ♦♥sr♥♦ F (a) ∝ a−ξ ♦♥ ξ = 1.5

♥ s rs ② ♣ rs q ♦♠♣♦rt♠♥t♦ P (k)

♣r F (a) ∝ a−ξ s ♦♠♥♦ ♣♦r st út♠ ② ♣♦t♥s ♥ ró♥♥tr♠ ♦s ♦s s♦s ①tr♠♦s ♥③♦s

50 100 150 200 250 300k

10−7

10−6

10−5

10−4

10−3

10−2

10−1

102P(k)

(a)q=0.1

100 101 102 103

10−7

10−6

10−5

10−4

10−3

10−2

10−1

(b)(b)(b)q=0.8

r str♦♥s r♦ Pℓ(k) = Nk(ℓ)/N(ℓ) ♣r ré♠♥ ♦♥ ♣♦ó♥ r♥t ② F (a) = δ(a − a0) ♣r rs ♠s ♥ t♠♣♦ ♦♥♣♦ó♥ ♥N = 105 ② 〈k〉 = 20 ♦s sí♠♦♦s ♦rrs♣♦♥♥ ♣r♦♠♦s s♦r100 s♠♦♥s ♥♠érs ♣r q ∈ 0.1, 0.2, 0.3 ② q ∈ 0.8, 0.9, 1.0s í♥s sós ♦rrs♣♦♥♥ s s♦♦♥s ♥♠érs ♣r♦♣♠♥t ♥♦r♠③ ♣r ♦t♥r P (k) s t♥♥s s s♦♦♥ss♥tóts s s ② r♦♥ rs ♠♥t í♥s tr③♦s♠♥t ♦s rá♦s ♦rrs♣♦♥♥ts r♥ts ♦rs q ♥ s♦ ♦♥♥♥t♠♥t s♣③♦s ♣r tr s s③ó♥

♦rr♦♥s r♦ t♦ ♦r♥

♥ st só♥ ♥③r♠♦s s rtrísts s ♦rr♦♥s r♦r♣rs♥ts ♣rtr ♦s ♥♦rs r♦ ♠♦ ♦s ♣r♠r♦s ♥♦sknn(k) ② ♦♥t str③ó♥ ♠♦ C(k) ♠♦s ♥ ♥ó♥ r♦ k ♦♠♦ ♠♦s ♠♥♦♥♦ s ♦rr♦♥s r♦ t♦ ♦r♥♦♥stt②♥ ♥ s♥tr s rs ♦r♥ s♦ s ♣♦r ♦ q ♦t♦r♠♦s ♣rtr t♥ó♥ st s♣t♦ ♦ ♦st♥t ♦r♠ó♥ ♥ít♥ st♦s s♦s s t♦r♥ rá♣♠♥t ♥trt ♣♦r ♦ q s♦♦ ♦rr♠♦s ♥trt♠♥t♦ ♥♠ér♦ ♣rtr s♠♦♥s ♥str♦ ♠♦♦

r♦ ♠♦ ♣r♠r♦s ♥♦s

❯♥ ♠ó♥ ♥rt ♦rró♥ r♦s s♥♦ ♦r♥ ♣♦t♥rs ♣rtr á♦ r♦ ♠♦ ♦s ♥♦♦s r♦ k knn(k)

♦rr♦♥s r♦ t♦ ♦r♥

100 101 102 103 104

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100101

ξ = 1.5

(b)q=0.8

102 103 104 105 106

10−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1

100 101 102 103 104

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

(a)(a)(a)

r stró♥ r♦ Pℓ(k) = N(ℓ)k/N(ℓ) ♣r ré♠♥ ♣♦ó♥ r♥t ♦♥ F (a) ∝ a−1.5 rs ♦♥ ♣♦ó♥ ♥ N = 105 ② 〈k〉 = 200♦s sí♠♦♦s ♦rrs♣♦♥♥ ♣r♦♠♦s s♦r 100 s♠♦♥s ♥♠érs ♣r q ∈ 0.1, 0.2, 0.3 ② q ∈ 0.8, 0.9, 1.0 ás í♥s sós ♦rrs♣♦♥♥ s s♦♦♥s ♥♠érs ♣r♦♠s ♠♥t ② ♣r♦♣♠♥t ♥♦r♠③ ♣r ♦t♥r P (k) ♦♥ ♦t♦ s♠♣r s ♦s rá♦s ♣r st♥t♦s ♦rs q ♥ s♦ s♣③♦s rt♠♥t ♥ t♦♦s ♦s s♦s ás í♥s sós ♦rrs♣♦♥♥ s s♦♦♥s♥♠érs ♣r♦♣♠♥t ♥♦r♠③ ♣r ♦t♥r P (k) st♥♥s s s♦♦♥s s♥tóts s s ② r♦♥ rs♠♥t í♥s tr③♦s ♥st ♥ t ♣r ♦rs r♥s k ♥ ♦♠♣rr ♦♥ ♠♥t♦ t♣♦ ② ♣♦t♥s s♦ó♥ s♥tót♣r♦①♠ ♣♦r í♥ tr③♦s

♦r♠♠♥t ♥ ♦♠♦ ❬Pst♦rt♦rrs ❪

knn(k) =∑

k′P (k′|k) ,

♦♥ P (k′|k) s ♣r♦ ♦♥♦♥ q t♥ ♥ ♥♦♦ r♦ k str ♦♥t♦ ♦tr♦ ♦♥ r♦ k′ st ♦r♠ ♣♥♥ knn ♦♥k ♥t ①st♥ ♦rró♥ r♦r♦ ♥ ♣rtr ♥ q♦s s♦s ♣r ♦s s knn r r ♦♥ k r♠♦s q r ♣rs♥ts♦rtt s♦rtt r♦ ❬♠♥ ♠♥ ❪ s♦rtt r♦ s ♣rs♠♥t ♦♠♣♦rt♠♥t♦ q ♦sr♠♦s ♣r ré♠♥ ♣♦ó♥ r♥t t♥t♦ ♦♥ F (a) ♦♥st♥t ♦♠♦ ② ♣♦t♥st ♦♠♦ s ♦sr ♥ r ② P♦r ♦♥trr♦ ♠♦♦ ♦♥ ♣♦ó♥ ♦♥st♥t ② F (a) = δ(a− a0) ♣rs♥t ♥ rá♦ s♦♥st♥t

10 20 30 40 50 60 70 80k

knn(k)

(a)q=0.00

q=0.30

q=0.50

q=0.70

q=0.80

q=0.90

100 101 102 103

knn(k)

10 20 30 40 50 60 70 80k

knn(k)

100 101 102 103

knn(k)

r r♦ ♠♦ ♣r♠r♦s ♥♦s ♥ ♥♦♦ ♦♥ r♦ k knn(k) ♣r♣r s rs ♦t♥s ♠♥ts ♠♦♦ ♦♥ ♣♦ó♥ ♥ N = 105

② 〈k〉 = 20 ♥ ② ♠♥trs q N = 104 ② 〈k〉 = 20 ♣r ② ♦ssí♠♦♦s ♦rrs♣♦♥♥ ♦s ♣r♦♠♦s s♦r 100 s♠♦♥s ♥♠érs ♦s s♥ts ♦♥♦♥s P♦ó♥ ♦♥st♥t ② ♣ t F (a) =δ(a − a0) ♣♦ó♥ r♥t ② ♣ t F (a) = δ(a − a0) ♣♦ó♥ ♦♥st♥t ② ♣ t F (a) ∝ a−1.5 ♣♦ó♥ r♥t ②♣ t F (a) ∝ a−1.5

knn(k) r st ♦♠♣♦rt♠♥t♦ s ♦♠♣t ♦♥ ♦rrs♣♦♥♥t rs t♦rs rösé♥② ♣r s s knn(k′) = 〈k〉 ∀k′♥q ♥ r ♠♦strr♠♦s q t ♦rrs♣♦♥♥ s s♦♦ ♣r Pr s♦ tr♥t♦ ♦♥ F (a) ∝ a−1.5 rá♦ knn(k) s t♠é♥ ♣♥♦ ♥q♠♥t s♦rtt♦ r

Pr s♦ ♣rtr ♦♥ q = 0 ② γ = 0 ♣ ♠♦strrs r ♣é♥ ♣r ♥ ró♥ ♦r♠ q s sts s♥t ② s q

♦rr♦♥s r♦ t♦ ♦r♥

10−2 10−1 100

(2〈a〉/〈k〉)k10−4

10−3

10−2

10−1

(2〈a〉/〈k〉)(

knn(k)−

1)−2〈a〉

(a) 〈k〉=100〈k〉=200〈k〉=400

10−2 10−1 10010−4

10−3

10−2

10−1

q = 0.5

10−3 10−2 10−1 100

(2〈a〉/〈k〉)k10−4

10−3

10−2

10−1

(2〈a〉/〈k〉)(

knn(k)−

1)−2〈a〉

q=0.00

q=0.10

q=0.20

q=0.30

q=0.50

q=0.90

r r♦ ♠♦ ♣r♠r♦s ♥♦s ♥ ♥♦♦ r♦ k rs♦♣r rs ♠s ♥ t♠♣♦ ♦t♥s ♠♥ts ♠♦♦ ♦♥♣ t F (a) ∝ a−1.5 ② ♣♦ó♥ ♦♥st♥t N = 104 ♦s sí♠♦♦s♦rrs♣♦♥♥ ♣r♦♠♦ s♦r 100 s♠♦♥s ♥♠érs ♦♣s♦ ss♦♦♥s ♣r q = 0 ② 〈k〉 = 100 200 ② 400 ♥st ♦rrs♣♦♥ q = 0.5 í♥ tr③♦s r♣rs♥t ♣ró♥ ♥ó♥ s knn(k) rs♦ ♣r st♥t♦s ♦rs q ≥ 0

♥♦r knn(k)

2〈a〉〈k〉

knn(k)− 1)

− 2〈a〉 ≃ (〈a2〉 − 〈a〉2)(

2〈a〉〈k〉 k

♥ r s ♠str ♦♣s♦ ♣r♦ ♣♦r ② s ♣r r♥ts ♦rs 〈k〉 ♦♥ q = γ = 0 s ♠② í r♥ ①♣rsó♥ s♠r ♣r q > 0 ♦ s ♦rr♦♥s♦s ♥s ♣♦r ♠♥s♠♦ ♦ ♦st♥t ♥ r ♠♦sr♦ (2〈a〉/〈k〉)

knn(k)− 1)

− 2〈a〉 ♦♠♦ ♥ó♥ 2〈a〉k/〈k〉 ♣r q > 0♠♦str♥♦ q ①str ♥ ♥ó♥ s ♦♠♦ sr ♥st r ♣r q = 0.5 st rá ♣rs♥tr ♥ ♣♥♥ ♦♥ q ♦r♠(2〈a〉/〈k〉)

knn(k)− 1)

− 2〈a〉 ≃ F qscal (2〈a〉k/〈k〉)

♦♥t str③ó♥

tr♦ ♥♦r ①st♥ ♦rr♦♥s r♦ s ♦♥t str③ó♥ ♠♦ C(k) ♥♦ ♥ ♦♠♦

C(k) =1

i∈Deg(k)

♦♥ Deg(k) s ♦♥♥t♦ ♥♦♦s ♦♥ r♦ k ♦♥ Nk s r♥ ② Ci s ♦♥t str③ó♥ ♦ ♥♦ ♥

♦♥t str③ó♥ s ♥♥tr s♦♦ ♦♥ r♥ ♣ró♥ ♥s tr♥st♦s ♦ st♦ ♥ ♦r♠ ♦♠étr ♦♠♦ ♣r♦ ♦rr♥ trs ♥♦♦s ♥tr♦♥t♦s trá♥♦s ♥ rs rs ♦r♥ s♦ s♠♥t ①♥ ♥ ② s ♣r C(k)

C(k) ∼ k−β ,

②♦ ①♣♦♥♥t sts β . 1 ♣r ♠s s rs ♥③s ♥ trtr ❬♠ ❪ st ② s s sts ♣♦r s rs ♠s ♥ t♠♣♦ ♥rs ♠♥t ♠♦♦ t ♦♠♦ s ♠str ♥ r st ♦♠♣♦rt♠♥t♦ ♠str q s rs ♥rs ♠♥t ♦♥♣♦ó♥ ② ♣ t ♦♥st♥ts ♥♦ s ♦♠♣♦rt♥ ♦♠♦ rs t♦rs ♦♠♦ srí r ♦ q ♥ ♦ s♦ rí♥ ♣rs♥tr C(k) ♦♥st♥t ♦rrs♣♦♥♥t rs t♦rs ② ①♣rsó♥ ①t s❬♦r♦♦ts ❪

C(k) =(〈k2〉 − 〈k〉)2

N〈k〉3 .

P♦r út♠♦ ♥ r ♠♦str♠♦s ♦♥t str③ó♥ ♣r♦♠♦ r ♥♦ ♦♠♦

Ci =∑

P (k)C(k),

♥ ♥ó♥ ♣r♦ q ♦sr ♥ ♦♠♣♦rt♠♥t♦ r♥t C ♦♥ q t ♦♠♦ rst s♣r ♣♦r ♥ó♥

s♦s st♦

100 101 102

10−4

10−3

10−2

10−1

q=0.20

q=0.30

q=0.50

q=0.80

q=0.90

100 101 102 103

10−4

10−3

10−2

10−1

100 101 102

10−4

10−3

10−2

10−1

100 101 102 103

10−4

10−3

10−2

10−1

r ♦♥t str③ó♥ ♠♦ C(k) ♦♠♦ ♥ó♥ r♦ k♣r rs ♠s ♥ t♠♣♦ ♦t♥s ♠♥t ♠♦♦ ♦♥♣♦ó♥ ♥ N = 105 ② 〈k〉 = 20 ♥ ② ♠♥trs q N = 104 ②〈k〉 = 20 ♣r ② ♦s sí♠♦♦s ♦rrs♣♦♥♥ ♦s ♣r♦♠♦s s♦r 100s♠♦♥s ♥♠érs ♦ s s♥ts ♦♥♦♥s P♦ó♥ ♦♥st♥t② ♣ t F (a) = δ(a− a0) ♣♦ó♥ r♥t ② ♣ tF (a) = δ(a− a0) ♣♦ó♥ ♦♥st♥t ② ♣ t F (a) ∝ a−1.5 ♣♦ó♥ r♥t ② ♣ t F (a) ∝ a−1.5 ♠str♥ ♦♠♦ rr♥í♥s tr③♦s ♦♥ ♣♥♥t −1 ♦rrs♣♦♥♥ts C(k) ∼ k−1

s♦s st♦

♥ st só♥ str♠♦s ♦s ♠♣♦s rs s♦s ♦♥ ♥ tr ♥ ♦♠♣ró♥ ♦♥ ♦s rst♦s ♠♦♦ ♦♠♣ró♥ srás♠tt ♦ q ♥♦ ♣rt♥♠♦s r♣r♦r ♦♥ ①tt ♦s rst♦s ♥ r s♦ ♥ ♣rtr ♣r ♦ ♥strí♠♦s ♥♦r♣♦rr ♠ás ♣rá

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0q

without growth

with growth

0.1 0.3 0.5 0.7 0.9q

r ♦♥t str③ó♥ ♣r♦♠♦ C ♣r r ♥ ♥ó♥ ♣r♦ q ♥ ♦♦s ♦s rst♦s ♦rrs♣♦♥♥ ♣r♦♠♦ss♦r 100 r③♦♥s ♦♥ N = 104 〈k〉 = 20 ② F (a) ∝ a−1.5 ♥strst♦s ♣r ♠♦♦ ♦♥ ♦s ♣rá♠tr♦s ♣r♦s ♣r♦ ♦♥ t♦♥st♥t F (a) = δ(a− a0)

♠tr♦s ② ♠♥s♠♦s s♣í♦s ♠♦♦ s♥♦ s♦♦ ①trr ss rtrísts♥♠♥ts

♣r♠r s♦ q ♥③r♠♦s ♦rrs♣♦♥ ♥ st♦ ①♣r♠♥t ♦♥tt♦s s♦s ♣♦r ♣r♦①♠ r③♦ ♣♦r ♦♦ró♥ ♦♦Pttr♥stt♣s♦♦♣ttr♥s♦r ♦♥ ♦t♦ rstrr ♦s ♦♥tt♦ss♦s ♥tr ♦s sst♥ts ♥ r♥ó♥ s♦ s s♣♦♥♥ ♠s♦rs r♦r♥ ♥ t♦rs♦ ♥♦s ♠t♥♦ ♣s♦s ♦rrs♣♦♥♥ts st♦ ♦♥ rs♦ó♥ t♠♣♦r s♥♦s ② ♥ ♥ ∼ 1♠❬ttt♦ ❪ st ♠♦♦ ①♣r♠♥t♦ ♥♦ s♦♦ rstr ♦s ♥♥tr♦ss♥♦ t♠é♥ s ró♥ ❯♥♦ st♦s ①♣r♠♥t♦s r③♦ r♥t ♦♥r♥ ②♣rt①t r③ ♥ r♥ t ♥ s rstrr♦♥ ♦s ♦♥tt♦s s♦s r♥t ♦s ♦s ís ♦♥r♥ ♥♦s♦tr♦strr♠♦s s♦♦ ♦♥ ♦s t♦s ♣r♦♥♥ts ♣r♠r í r ♥t♦♥ N = 113 ♥♦♦s ② L = 2196 ♥s r♦♥s ❬s ❪

s♥♦ s♦ q ♥③r♠♦s s ♥ sr♦ r s♦ ♦♦ ♦rrs♣♦♥♥♦ N = 63731 sr♦s ♦♥ Ncc = 63392 ≈ 0.995 × N

♥tr♥♦ ♠á①♠ ♦♠♣♦♥♥t ♦♥t ♣rt♥♥ts r♥s ♦s♥ ❯❯ ♥♦s ♣♦r L = 817090 ♥s q r♣rs♥t♥ r♦♥s ♠st r♦♥s sr♦ r♣♦rt♦

s♦s st♦

♥ ❬❱s♥t ❪ ♦♠♦ ♥ s♦ ♥tr♦r st r♦ t♠é♥ ♣♦rt ♥♦r♠ó♥ t♠♣♦r trés ♦♠♥③♦ ♠st ♦♥trr♦ ♦ q ♦rr ♦♥ r ① r♠♥t♦ ♣♦ó♥ ♥ ♥♦♥t①t♦ s♥ rstr♦♥s s♣s ♥trs

20 40 60 80 10010−4

10−3

10−2

20 40 60 80 100

GSG model

10110−1

101 102

GSG model

knn(k)

101 102

GSG model

r stró♥ r♦ P (k) ♦♥t str③ó♥ ♠♦ C(k)② r♦ ♠♦ ♣r♠r♦s ♥♦s knn(k) ♣r r ♠♥ t♠♣♦ ② ♣r ♠♦♦ ♦♥ ♣♦ó♥ ♦♥st♥tN = 113 〈k〉 = 38.9 ♣r♦ q = 0.8 ② ♣ t t♣♦ ② ♣♦♥t♥s ♦♥ ①♣♦♥♥t ξ = 0.79

♥ r s ♠str P (k) C(k) ② knn(k) ♣r r ♥ ♥t♦ stró♥ r♦s P (k) ♣rs♥t ♥ ♦♠♣♦rt♠♥t♦ P♦ss♦♥♥♦♣r k ♣qñ♦ ♦♥ ♥ ♠♥t♦ ♣r♦①♠♠♥t ①♣♦♥♥ ♦♠♣t♦♥ ♦t♥♦ ♣r ♥str♦ ♠♦♦ ♦♥ ♣♦ó♥ ♦♥st♥t r r P♦r ♦♥trr♦ r ♠str ♥ P (k) ♦♥ ♥ ♦♠♣♦rt♠♥t♦ t♣♦ P (k) ∼ k−α ♦♥ α ≈ 3.4 ♣r k r♥s r r st út♠♦ ♦♠♣♦rt♠♥t♦ t♠é♥ rst ♦♠♣t ♦♥ ♦sr♦ ♣r ♣r♦ ♥ st s♦ ♦♥ ♣♦ó♥ ♥♦♦s r♥t ♠♦♦ t♥t♦

C(k) ♦♠♦ knn(k) ♣r s rs ② ①♥ tt♠♥t ♠s♠♦♦♠♣♦rt♠♥t♦ q ♥str♦ ♠♦♦ ♣r ♣♦ó♥ ♦♥st♥t ② r♥trs♣t♠♥t

100 101 102 10310−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100 101 102 103

GSG model

101 102 103

10−2

10−1

101 102 103

GSG model

100 101 102 103

knn(k)

100 101 102 103

GSG model

r stró♥ r♦ P (k) ♦♥t str③ó♥ ♠♦ C(k)② r♦ ♠♦ ♣r♠r♦s ♥♦s knn(k) ♣r r ♠ ♥ t♠♣♦ ② ♣r ♠♦♦ ♣r ♣♦ó♥ r♥t ♦♥N = 63731 〈k〉 = 25.6 ♣r♦ ♥s q = 0.8 ② ♣ t t♣♦ ② ♣♦♥t♥s ♦♥ ①♣♦♥♥t ξ = 1.5

Pr ♣rsr ♦♠♣ró♥ ♣árr♦ ♣r♦ r③♠♦s ♥ s♠ó♥ ♠♦♦ ♦♥ ♣♦ó♥ ♦♥st♥t N = 113 L = 2196 k〉 = 38.9♦♥ ♥ ♣ t ♦r♠ F (a) ∼ a−ξ ② ♥♦ ♣r♦ rr trá♦ q = 0.8 ② ó♥ srá st ♥ r st ♠♦♦ ú♥♦ ♣rá♠tr♦ st ♥str♦ ♠♦♦ rst ①♣♦♥♥t ξ F (a)st♥♦ ξ ♣♦r ♠ét♦♦ ♠á①♠ r♦s♠t s♥♦ ♠♦♦ ♦♥ ♣♦ó♥ ♦♥st♥t s♦r ♦s t♦s ♦rrs♣♦♥♥ts P (k) ♣r r ♦t♠♦s rá♦ r st s r③ó ♦♥sr♥♦ξ ∈ [0.5, 3] ♦♥ ♥ ♣s♦ r δξ = 0.01−0.1 ♦s rá♦s s rs ② ♥♦ r♦♥ st♦s s♥♦ q ♦rrs♣♦♥♥ ♦s ♣rá♠tr♦s ♣r♠♥t♦s ② ξ ♦t♥♦ st P (k) ♦♠♣♦rt♠♥t♦ ♠♦♦ rst♥ t♦♦s ♦s s♦s tt♠♥t s♠♥t ♦sr♦ ♣r r r

s♦s st♦

♥ ♥t♦ r ♣r♦♠♦s é♥t ♦r♠ ♥q ♥ st s♦st♥♦ trés ♠♦♦ ♦♥ ♣♦ó♥ r♥t ♣rt♥♦ ♥♣♦ó♥ ♥N0 = 100 ♦♥ ♥ ts r♠♥t♦ ♥♦♦s λ = N/L s♥♦N = 63731 ② L = 817090 ♥t ♥♦♦s ② ♥s ♥s ♥str r♥r rs♣t♠♥t r rs ②

♦ q t♥t♦ r ♦♠♦ ♥t♥ ♦♥ t♦s s ♦ó♥t♠♣♦r ♦♥ ♥ rs♦ó♥ s ♣♦s ♥③r s ♠♥s♠♦ t♥ ♦♥♦ r ♥ ♦s s♦s Pr ♦ s lt = (i, j)t ♥ q ♠r t♠♣♦t ♥♥♦ ♦s ♥♦♦s i = lt(1) ② j = lt(2) P♦r ♦tr ♣rt s d(lt, t) st♥♥tr ♦s ♥♦♦s ♥ ♥st♥t ♥♠t♠♥t ♣r♦ ♣ró♥ lt st ♠♦♦ q♦s ♥s lt ♣r ♦s s d(lt, t) = 2 ♠♦s ♥str♥st♦s s♦♥ ♣r♦t♦ ♥ ♠♥s♠♦ qr s s ♦r♥

♦ ♥♠♦s Nd(T ) ♦♠♦ ♥ú♠r♦ ♠♦ ♥s ♦♥ d(lt, t) = d

♣r t < T s♠s♠♦ ♥♠♦s d(lt, t) = 0 ♥♦ ♥♦ ♦s ♥♦♦s lt(1)

♦ lt(2) s ♥♦ ♥ r ② d(lt, t) = ∞ ♥♦ ♥♦ ①st ú♥ ♠♥♦ qr st♥ q ♥ ♦s ♥♦♦s ♣r♦ ♦♥ró♥ lt ♦q ♥♦ s ♣r♠t♥ ♠út♣s ♥s ♥tr ♥♦♦s ♦♥sr♠♦s Nd=1(T ) = 0♦ ♦ ♥tr♦r stró♥ st♥s ♣ ♥rs ♦r♠♠♥t ♦♠♦

PT (d) = lımL→∞

Nd(T )

s♥♦ L =∑

i∈N0Ni(T ) ♥ú♠r♦ t♦t ♥s

stró♥ st♥s PT (d) ♣♥ ♦r♥ t♠♣♦r ♦s ♥s ltt≤T ② s ♣♦r st ♠♦t♦ q ♣r♦ ♥♦r♠ó♥ ♦s r ♠♥s♠♦ r♠♥t♦ r ♦♥ ♦t♦ rr s s q ①st♥ ♠♥s♠♦ q ♣r ♣ró♥ ♥s tr♥st♦s d = 2 ♣r ♦ss♦s qí ♥③♦s ♦♠♣r♠♦s stró♥ PT (d) ♣r ♦r♥ t♠♣♦rr ♦s ♥s l1, l2, ..., lT ♦♥ stró♥ ♣r♦♠♦ 〈PT (d)〉rand s♦r100 ♣r♠t♦♥s t♦rs lσtt=1,...,T ♥ ♣r♠tó♥ t♦r ♦♥σt ∈ Perm(T ) ♣rt♥ ♦♥♥t♦ ♣r♠t♦♥s T í♥s Perm(T ) ♥t♦♥s lσtt=1,...,T ♥r ①t♠♥t ♠s♠ r ♠ t♠♣♦T ♣r♦ ♦♥ ♥ stró♥ P σ

r ♠str PT (d) ♣r s rs r ② r ♥t♦ ♦♥ 〈PT (d)〉rand ♥ s♦ ♥ ♠♦s s♦s s ♦sr

2 3 4 5 6d

HT random order

2 3 4 5 6d

FB random order

r stró♥ ♣r♦ st♥s PT (d) ír♦s ③s② ss ♦rrs♣♦♥♥ts ♦rs ♣r♦♠♦ s♦r 100 ♣r♠t♦♥s t♦rs ssó♥ r ♥s ♦r♥ ♥ t♠♣♦ r♦s r♦♦s ♣r r ♦♥tt♦s rr ② sr♦ ♠sts ♦♥♥ ♦srr♦rs st♥r rst♥ ♠ás ♣qñ♦s q ♦s sí♠♦♦s

♥ r♥s s♥ts ♥tr PT (d) ② 〈PT (d)〉rand ♥ ♣rtr PT (d) <

〈PT (d)〉rand ♣r d > 2 ♠♥trs q PT (2) > 〈PT (2)〉rand ♣r ♠s rs ② rt sr♣♥ ♣r d = 2 ♣ ♥trs ♣rtr á♦ zs♦r ♥♦ ♦♠♦

z =PT (2)− 〈PT (2)〉rand

σrand

♥♦ ♣♦r rst♦ zHT = 32 ② zFG = 185 P♦r ♦tr ♣rt ♦s ♥str♥st♦s rst♥ r♠♥t ♣r♦♠♥♥ts ♥ ♠♦s s♦s ♥③♦s ♦♥PT (2) = 0.91(2) ♣r ② PT (2) = 0.77(4) ♣r st♦s ♦ ♥ ♥t ♥ rt t♦ ♠♠♦r ♥ ♠♥s♠♦ r♠♥t♦ r q r ♥ ♣r♦♠♥♦ ♥s ♦ tr♥st♦s

♣ít♦

Prsst♥ sr♠♣ó♥ s♦r

rs s♦s

❮♥ ♠♦♦

r s♦

ss tr♥só♥

Ps♦ t♠♣♦r

♣r♦①♠ó♥ ♠t♥♦♠

♥áss ♣rsst♥

①t♥♦♥s ♣♦st♣é♠s ♦♥ ♠♣♦rtó♥ ♣r♠♥♥t

♥t♦s

♠♦s s♦ ♣s♦ ♠t♠át♦ ♥és r rttt ♠♦stróq ♥ r ♣r♥ sr♠♣ó♥ rst ♥é♠♦ ♦st♦r♦ ♥ sr♥s ♦♥ ♣♦ó♥ > 250000 ♠♥trs q ♣rs♥t ①t♥♦♥s ②rr♦ts ♥ s ♣qñs s sí ♦♠♦ ♥ó ♥ t♠ñ♦ rít♦ ♦♠♥

250000 − 500000 ♣♦r ♥♠ s ♦srrí ♣rsst♥ sr♠♣ó♥ ♥ ♣♦♦♥s ss ❬rttt rttt ❪ ♦ ♦st♥t♠♦♦s ♠ás s♦st♦s ♠♦strr♦♥ q ♠r ♣r♦♣st♦ ♣♦r rttt srt♠♥t ♣♥♥t s str♦♥s ♦s t♠♣♦s rtríst♦s tr♥só♥ ♥tr ♦s st♥t♦s st♦s ❬♥ ♦② ❪

P♦r ♦tr ♣rt t♠é♥ s ♠♦stró q ♣♥ s tr♦♥s ♥ s r♥s ♦♥tt♦s ♥tr ♦s st♥t♦s r♣♦s tr♦s ② t♦rs st♦♥s ♥♦s ♦♥ ♦ t♦ ❬♦r ♦r ❪♥ ♠r♦ s ♦ ♠② ♣♦ t♥ó♥ ♠♣t♦ s tr♦♥s ♦♥tt♦s ♥tr ♥♦s r♥t♠♥t sr♣t♦s ♠♥t ♥ r♦♠♣ ♥tr s sss ①♣♦♥s ♣♦♠♦s tr tr♦ tt ♥ ② ♦♦r♦rs ❬♥ ❪ r ♠♦♦ ♣r♦①♠ó♥ ♣rs

♣ít♦ Prsst♥ sr♠♣ó♥ s♦r rs s♦s

♦♥sst♥t ♥ ♣r♦①♠r ♥♦♥s ♦rró♥ 3 r♣♦s ♦♠♦ ♣r♦t♦s ♦rr♦♥s 2 r♣♦s ♣♦ ♣r♦♣ó♥ ♥ ♥r♠ ♥♦s s♦r ♥ r ♦♠♣ P♦r ♦tr ♣rt t♠é♥ str tr♦ ❱rs ② ♦♦r♦rs ❬❱rs ❪ ♦♥ ♦s t♦rs st♥ ♦s t♦s ♣r♦♣ ♠♥♦ ♣qñ♦ ♥ ♦s r♦ts rrr♥ts sr♠♣ó♥ trés st♦s ♥♦qs s ①♣♦r♦ ♠♣t♦ stró♥ r♦s② ♦ s♠♦ s ♦rr♦♥s ♦s r♣♦s ♥tr ♦s r♦s ♦s ♥♦♦s ♥♠r♦ sú♥ ♥str♦ sr ② ♥t♥r st ♠♦♠♥t♦ s s♦♥♦ s ①st♥ ♦rr♦♥s ♦s ♦r♥ s♣r♦r ♥tr ♦s ♥♦♦s ♦♠♦ ssr♣ts ♣♦r ♦♥t str③ó♥ C r t♥♥ ú♥ ♠♣t♦♥ ♣tró♥ ♣rsst♥ sr♠♣ó♥

♥ st ♣ít♦ ♣rs♥t♠♦s ♥ ♠♦♦ st♦ást♦ ② s♦ ♥ ♥♦ ♣r str ♣r♦♣ó♥ sr♠♣ó♥ ♥ t♣ ♣r♥ó♥ ♣rtr ♠♦str♠♦s q ♣rsst♥ s♠♥② ♥r♠♥tr C strtr ♦rr♦♥s ♦s r srá ♦♥tr♦ ♣rtr ♣r♦ ♥s tr♥st♦s q ♥tr♦ ♥ ♠♦♦ ♥r♦r rs s♦s ♣ít♦

♠♦♦

❯♥ ♠♦♦ ♦♠♣rt♠♥t s♣t①♣st♦♥t♦rtr♦ t♣♦ st♦ást♦ rst ♠ás ♣r♦♣♦ ♣r srr ♣rsst♥ sr♠♣ó♥ ❯t③♠♦s ♣r ♦ s♠♦♥s ♥♠érs ss ♥ ♥♦ ♦♥sr♥♦ s rs ♥rs ♥ ♣ít♦ ♣r♥t

♥t r ♣ ♦♣tr ♥♦ ♦s tr♦ st♦s ♣♦ss ss♣t ①♣st♦ ♥t♦ ♦ rrtr♦ ❯♥ ♥t ss♣t s♦♦ ♣♦♥trr sr♠♣ó♥ ♥♦ ss ♥♦s rt♦s ♥t♦s ♣r rstr ①♣st♦ ♦s ♥ts ①♣st♦s ♥♦ ♦♥t♥ ♥r♠ st t♥t♦②♥ qr♦ r r ♥sr ♣r sr ♦♥sr♦s ♥t♦s ♥♠♥t ♦s ♥t♦s ♣s♥ st♦ rrtr♦ ♦♥ qr♥ ♥♠♥♣r♠♥♥t ♦♥♠♥t ♦♥sr♠♦s ♥ ts ♠♦rt µ ② ♦tr ♥t ν ♠♦♦ q ♠s rst♥ s ♦♥sr♥♦ sí ♣♦ó♥ r ♦♥st♥t ♥só♥ t♦rs ♠♦rá♦s ♥ ♣r♦♠ rsts♥ ♣r r♥♦r ♣♦ó♥ ss♣ts

♠♦♦ st♦ást♦ s♦ ♥ ♥♦ q ♠♦s ♣r str ♣rsst♥ sr♠♣ó♥ s ♥ ♥ r♦ ♠♣♦ srt♦ st♦ s st♦ sst♠ t♠♣♦ t+δt s♦♦ ♣♥ st♦

♠♦♦

t♠♣♦ t ó♥ ♠ás r♥t ♥ ♦s ♠♦♦s ♠♣♦ ♠♦ s trr♦♥ ♥ ♥ r♦ ♠♣♦ ♦♥t♥♦ ♥ ♠r♦ ó♥ rqr ♥ t♠♣♦ ó♠♣t♦ ♣r♦t♦ ♣r s ♣r♦♠q ♦r♠♦s qí ♥♦r♥♦ ♣♦♦♥s ♦r♥ N ∼ 106 ♥♦s ♥♦ ♦♥ s ts ♦♥t♦ ♥ ♣♥♥t ♥t♦r♥♦ st ♠♣♠♥tó♥ rí r t♥ts tss tr♥só♥ ♦♠♦ ♥♦s ② ♥ ♣♦ó♥ st ♠♦♦ s♦ó♥ ①t ♥ ♣rtr ♦rt♠♦ s♣ ❬s♣ ❪ rst ①tr♠♠♥t ♥t ♣r ♠♦♦s s♦s♥ ♥♦ ♦♠♦ qí ①♣st♦

r s♦

r s♦ ♠♣ ♥ s♦ s ♥r ♠♥t ♦rt♠♦ ♣ít♦ ♥tr♦r ♣r ♣♦ó♥ ♦♥st♥t ♦s ♥♦♦s ② ♥s r♣rs♥t♥ ♦s ♥♦s ② ss í♥♦s s♦s ♣s tr♥s♠tr ♥ ♥r♠♥♦s rs♣t♠♥t ♥ ♣rtr s♠♦s ♥rr rs ♦♥ r♥t♥ ♦rr♦♥s ♦s rs ♣♦r ♣r♦ ♥ tr♥st♦q ♦♥ ♥ ♥③r s ♠♣t♦ ♥ ♣rsst♥ sr♠♣ó♥ s♦r rr♠♦s ♦♥ rs státs r♦ ♠♦ 〈k〉 ② ♣♦♦♥s ♦♠♣r♥s♥tr 100000 ≤ N ≤ 3000000 ♥♦s

♦ str♠♦s tr♦♥s ♥tr ♦s ♥♦s ♠ás á s ♣r♦♥♥ts r ♦♥tt♦s ♦♥ ♦t♦ r♥srr ♥str♦ ♥áss t♦ s♦ stás út♠s ♥ ♣rtr ♥♦ ♦♥srr♠♦s tr♦♥s ♥tr r♣♦s tr♦s ② ss r♥ts r♥s ♦♥tt♦ ♠ástrr♠♦s ♦♥ stró♥ ♥♦r♠ t ♣r ♦s ♥ts s♦s♠♣♦♦ ♥♦r♣♦rr♠♦s t♦r st♦♥ s♦♦ ♥r♠♥t♦ r♥ ♦♥tt♦s ♥tr ♥ñ♦s ♣qñ♦s r♥t ♦ t♦

ss tr♥só♥

s ♣r♦s tr♥só♥ ♣♦r ♥♦ ♥tr ♦s st♥t♦s st♦s s♦♥s♥♠♥t s sr♣ts ♥ ♣é♥ s♦ r♦ ♥ ts ♠♦rt µ = 4.2 × 10−5 ♦♠♣t ♦♥ ♥ ①♣tt 65 ñ♦s♦♥sr♥♦ ♦♠♦ rr♥ ♦rrs♣♦♥♥t ♥trr ♥ é 1950 r♥t s ♠②♦rs ♣♠s sr♠♣ó♥ ♥ r ♣r♥ó♥ ❬tt♣♦♥s♦❪ P♦r ♦ ①♣t♦ ♥tr♦r♠♥t ts ♥♠♥t♦ s♦♥sr ♠rt ν = µ s st♥ts ♣r♦s tr♥só♥♣r ♥ ♥♦ i r♥t ♥ ♥tr♦ t♠♣♦r δt s rs♠♥ ♥

♥ ♣rtr ♣r♦ ♦♥t♦ ♣r ♥ ♥♦♦♥♦ i ss♣t♦ ♥ ♥ú♠r♦ z(i) ♥♦s ♥t♦s s ♥ ♦♠♦

P i(S → E) = 1− (1− δt c/〈k〉)z(i)

s♥♦ c/〈k〉 ts ♣r♦♠♦ ♦♥t♦ ♣♦r ♥ ♣r ♥ r ♦♥ r♦♠♦ 〈k〉 ♦r ♣rá♠tr♦ c s ♥ c = 2.8 ♠♦♦ q ♥ú♠r♦r♣r♦t♦ ás♦ rst ♦♠♣t ♦♥ ♦♥♦♦ ♣r sr♠♣ó♥ 12 ≤R0 ≤ 18 ❬♥rs♦♥ ❪

r♦ Pr♦s tr♥só♥ ♣♦r ♥♦s ♣r ♠♦♦ ♣♠♦ó♦

r♥só♥ Pr♦S → E 1− (1− δt c/〈k〉)z(i)E → I σδtI → R γδtS,E, I, R → ♠rt µδt→ S ♥♠♥t♦ S µδt

r sq♠ s ♣r♦s tr♥só♥ ♣♦r ♥♦ ♣r ♠♦♦

str q ♦s ♥♦s ♦s qr s s st♦ s♦♥ sstt♦s ♣♦r ré♥ ♥♦s ss♣ts s♠s♠♦ ♦♥srr♠♦s q ♣r♦ ♠rt ♦ t♥r s♥♥ s ♥♣♥♥t st♦ ♣♠♦ó♦ ♦s ♥♦s ♦ s♦♦ ♦♥ ♥str♦ ♠♦♦ ♣r sr♠♣ó♥s r♣rs♥t♦ ♥ r

♠ás s ② ♠♥♦♥s tss ♥♠♥t♦ ② ♠rt rst♥tstss ♠♣s ♦ r♦ st ♣ít♦ s ♦t♥♥ ♣rtr ♦s t♠♣♦s

♠♦♦

♣r♠♥♥ ♥ st♦ ①♣st♦ TE = 8 ís σ = 0.125í ♥t♦TI = 5 ís γ = 0.2í

s ♠♣♦rt♥t str q s str♦♥s t♠♣♦s s♣r ♣r ♦sst♦s ② ♦♥t♠♣s ♥ st ♣ít♦ srá♥ ①♣♦♥♥s rtríst strt♠♥t r♦♥ ♦♥ tss ♦♥st♥ts ♠♦♦ ♠♦sst♦ rt ♣♥♥ ♥tr s str♦♥s t♠♣♦ s♣r ②s ♣r♦♣s ♣rsst♥ ♥ ♣ít♦ ❬♥ ♥ ♦② ♦♥♥ ❪ ♥ ♠r♦ qí ♣rt♥♠♦s ♠♦strr s♦♦ t♦ s ♣r♦♣s t♦♣♦ós r ♣rsr♥♦ rst♦ s rs ♣r♦♠ ♥ s ♦r♠ó♥ ♠ás s♥

Ps♦ t♠♣♦r

ó♥ ♣s♦ t♠♣♦r δt rqr rt♦s ♦s P♦r ♥ ♦ ss q r♥t δt ♦rr♥ ♠♦s ♥t♦s st♥t s♣ ♥ s♠♥r t♠♣♦ ó♠♣t♦ ♦♥ rs♣t♦ ♦rt♠♦ s♣ q ♠♦ ♥ ♣r♦s♦ P♦ss♦♥ ♦♥ ♥ ú♥♦ ♥t♦ ♣♦r ♣s♦ ♦ ♦st♥t s δt s♠s♦ r♥ ♣♦rí♥ t♥r r ♠út♣s tr♥s♦♥s ♣r ♥ ♠s♠♦ ♥♦ ♥♦ rstrs ♣♦r sq♠ ♣r♦♣st♦ ♥ ♠♦ s ♣s♦ t♠♣♦rrstr ♠② ♣qñ♦ s s♣rrí t♠♣♦ ó♠♣t♦ ♥ ♣s♦s r♥t♦s s ♥♦ s rstrrí ♥♥ú♥ ♥t♦ rst♥♦ ♥ ♦rt♠♦ ú♥ ♠ás ♥t♦q s♣ ♥♠♥t ♣s♦ t♠♣♦r stsr

δt < ♠♥TE, TI,

♥ tr s tr♥s♦♥s ♠út♣s ♣r♦ ♦ s♥t♠♥t r♥ ♦♠♦♣r ♦♠♣r♥r ♠♦s ♥t♦s ♦♥t♠♣♥♦ ♦s s♣t♦s ♠♦s δt =0.5 í ♣r t♦s s s♠♦♥s r③s

♦ q ♣r♦ ♦♥t♦ ♥♦♦ qr ♥ ♦r♣rtr ♣♥♥♦ s ♥t♦r♥♦ t③ó♥ ♦s ss♣ts r③rs ♥ ♥ ♥t♦♥s ♥ ♣s♦ t♠♣♦r δt ♥ ♥♦♦ ♣ ♦♥trs ♦s ♥t♦s s ♥t♦r♥♦ ② ♣sr st♦ ♣t♥r s♥♥ ♦ ♠♦rr ♠s ♦♥ ♣r♦ µδt ♦ ♣ ♦♥t♥r ♥s st♦ ♦r♥ ♥ ♦♥trst t③ó♥ s♥ró♥ r③ s♦r ♦s♥♦♦s ② ♥♦ rqr trt♠♥t♦ ♥ ♦ q s tss tr♥só♥♥tr st♦s st♦s s♦♥ ♦♥st♥ts ♣r ♥♦ s ♣♦r ♦ q ♥ st♦ss♦s t③ó♥ ♣ r③rs ♥ ♣♦♦♥s ♦♥ ♦t♦ rr t♠♣♦ ó♠♣t♦

♣r♦①♠ó♥ ♠t♥♦♠

Pr ♠♣r ♥♦q ♥ ♣♦♦♥s ♦♥sr♠♦s s♦ ♦s ♥♦♦s ♥ st♦ t♠♣♦ t ♥ ♣s♦ t♠♣♦r ♦s ♥♦♦s ♣♥r③r tr♥só♥ st♦ ♦♥ ♥ ♣r♦ pI = σδt ♣♥ ♠♦rrs♠tá♥♠♥t ♥♦ r ♥ ré♥ ♥♦ ♦♥ ♣r♦ pd = µδt ♦♣♥ ♣r♠♥r ♥ st♦ ♦♥ ♣r♦ ps = 1− (σ+µ)δt t♦s ♣♦r♥♦ st♦ s ♦s ♥♦♦s ♥ sr r♣rt♦s ♥tr 3 st♦s ♣♦ss ♥♦ rtr③♦ ♣♦r ♥ ♣r♦ ♦rr♥ ♦♥st♥t ♦r♠ ♦♥rtr♦ s trés stró♥ ♠t♥♦♠ MX1,X2,X3 ♦♥ stró♥ ♣r♦ pM ♣♦r

pM(n1, n2, n3) =n!

n1!n2!n3!pn1I pn2

d pn3s

s♥♦ ♣r st s♦ n = n1+n2+n3 ♥ú♠r♦ t♦t ♥♦♦s ♣r t♠♣♦ ♦♥sr♦ ② nii=1,2,3 s ♥ts ♥♦♦s q rst♥ ♥t♦s♦s ♥rs rs♣t♠♥t ♣r ♦ ♣s♦ t♠♣♦r

Pr ♠♣♠♥tr stró♥ ♠t♥♦♠ ♠♦s s♦ s s♦♠♣♦só♥ s♥ ♥ str♦♥s ♥♦♠s BXi

(n, p)

MX1,...,Xk= BX1 (n, p1)⊗BX2

n− n1,p2

1− p1

⊗⊗BXk

n−k−1∑

1−∑k−1

i=1 pi

P♦r s♣st♦ ①♣rsó♥ ♥♦ ♠♣ ♥ ♠♦♦ ♥♦ q stró♥ ♠t♥♦♠ ♣ s♦♠♣♦♥rs ♦♠♦ ♣r♦t♦ ♥♦♠s ♦q s rs t♦rs Xi ♥♦ s♦♥ ♥♣♥♥ts st ♦ s r r♠♥t ♥ r♣rs♥tó♥ s♥ ♣♦r r③ó♥ Xi rst♣♥♥t s ♦rrs♣♦♥♥ts X1, ..., Xi−1

♥áss ♣rsst♥

str♦ ♦t♦ s ♥③r ♣rsst♥ sr♠♣ó♥ s♦r ♥ r s♦ s ♣♥♥ ♦♥ ♣♦ó♥ t♦t ② ♥ ♣rtr ♠♣t♦ s♦rr♦♥s ♦s ♦r♥ s♣r♦r ♥tr ♦s ♥♦♦s s♠♥t ♥♦ ♦♥t♠♣s ♥ s ♣r♦①♠♦♥s ♠♣♦ ♠♦ ♥s♦ ♥ qs ♠♣♦♠♦ tr♦é♥♦ q ♥♦r♣♦r♥ strtr r ♥ s ♦r♠ó♥ Pr♦ ♦♥sr♠♦s s♠♦♥s ♥♠érs ♠♥t s rs ♥rs ♣rtr ♠♦♦ ♦♥ ♣♦ó♥ ♦♥st♥t N ∈ [105, 3× 106] r♦ ♠♦

♥áss ♣rsst♥

〈k〉 = 40 ② ♣r♦ ♥s tr♥st♦s q

♥ t = 0 s ♥tr♦♥ I(0) = 10−4 × N ♥t♦s ♦s ♥♦r♠♠♥t ③r ♥tr ♣♦ó♥ t♦t N rst♦ ♦s ♥♦s s ♦♥sr♥ss♣ts S(0) = N − I(0) ♣♦ó♥ str t♦t♠♥t s stró♥ ♥ ♥t♦s ♥ ♠♦s s♦s rrí ♥ ♥ ♣r♦♥t ①t♥ó♥ ♣♦r ♦ s ♥sr♦ ♦♥srr ♥ tr♥st♦r♦ ♥ r♥t s ♥tr♦♥ ♥t♦s ♠♣♦rt♦s ♥ ♥str♦ s♦ s♥♦ tr♦ ♦② ❬♦② ❪ ♠♦s ♥ ts ♦♥st♥t ♥t♦s ♠♣♦rt♦sλimport = 5ñ♦ ró♥ tr♥st♦r♦ ♥ s ó ♥ 30 ñ♦s ♣rí♦♦q rst s♥t ♣r str ♦s♦♥s st♦ásts rrr♥ts rr

5000 10000 15000 20000Tiempo (días)

r Pr♥ sr♠♣ó♥ ♣r ♥ s♠ó♥ ♠♦♦ s♦r ♥r ♦♥ N = 800000 〈k〉 = 40 ② q = 0.7 í♥ ♣♥t ③ ♥ t♠♣♦♣r ♠♥ tr♥st♦r♦ ♥ ♠♣♦ ♠♦str♦ ♦rrs♣♦♥ ♥ r③ó♥ ♣rsst♥t

♦ tr♥st♦r♦ s ♦♦♥r s♠ó♥ s♥ ♥tr♦ó♥ ♥t♦s ①tr♥♦s r♥t 30 ñ♦s ♦♥s r ♠str ♥♠♣♦ tí♣♦ s ♦s♦♥s st♦ásts ♦srs ♣r ♣r♥ sr♠♣ó♥ ♥ s s♠♦♥s ♥♠érs ♥str♦ ♠♦♦ r♠♦s q sr♠♣ó♥ rst ♣rsst♥t ♥ ♥ r③ó♥ ♣rtr ♠♦♦ ♥♦s ♦♥② s♥♦ ♣rí♦♦ 30 ñ♦s s♥ ①t♥ó♥ ♥t sr♠♣ó♥

Pr ♣♦ó♥ N ∈ [105, 3× 106] t♠♦s 100 r③♦♥s ♠♦♦ ♣r♦♣ó♥ ♥t♦♥s ♥♠♦s ró♥ r③♦♥s ①t♥ts Pextin ♥ts ♠♥ó♥ s♥♦ ♣rí♦♦ 30 ñ♦s♦♠♦ ♥♦r ♣r♦ ♣rsst♥ sr♠♣ó♥ ♥ r

Pextin =r③♦♥s ①t♥ts

♥ú♠r♦ t♦t r③♦♥s

0 500000 1000000 1500000 2000000 2500000 3000000N

r ró♥ r③♦♥s ①t♥ts Pextin ♣r ♠♦♦ sr♣t♦ ♥ t①t♦ ♣r♥♣ ♦♠♣r♥ r③♦♥s s♦r rs ♦♥ q = 0.9 ②q = 0.0 ♦s ♥tr♦s ♦♥♥③ 95% ♦♥sr♦s s♦♥ ♥♦♠s

♦♠♦ ♠♥♦♥♠♦s ♦♠♥③♦ ♣ít♦ s ♥ rst♦ ♦♥♦♦ q♣r ♣♦♦♥s ♦ s♥t♠♥t r♥s ts ♥♠♥t♦s ② ♠rts µrst s♥t ♣r r♥♦r ♣♦ó♥ ss♣ts ♣r♠t♥♦ sí q sr♠♣ó♥ s t♦r♥ rrr♥t ♥é♠♦ st s ♣rs♠♥t ♦♠♣♦rt♠♥t♦♦sr♦ ♥ r ♦♥ ♣rs♥t♠♦s Pextin ♥ ♥ó♥ N t♥t♦♣r rs ♥rs ♦♥ q = 0.7 ♦♠♦ ♣r rs ♦♥ q = 0.0 s♥♦ stsút♠s s♥♠♥t rs t♦rs r ♠str q Pextin ♣rs♥tr♥s s♥ts ♥tr ♠♦s s♦s ♥③♦s s rs ♥rs ♦♥q = 0.0 rst♥ ♠ás ♣rsst♥ts ♥t sr♠♣ó♥ q qs ♦♥ q = 0.9sts út♠s ♣rs♥t♥ rts ♦rr♦♥s r♦ ♥s ♣♦r ♦s ♥str♥st♦s q ♥ ♣rtr r♥ ♥ ♥ ♦♥t str③ó♥ ♦C ♦♠♦ ♠♦str♠♦s ♥ ♣ít♦

sr♠♦s ♥ r q t♠ñ♦ rít♦ ♦♠♥ s ♥♥tr ♥ ♥tr♦ [5× 105, 106] ♣r q = 0 ② ♥ [106, 2× 106] ♣r q = 0.9

♥áss ♣rsst♥

♠♦s ①♥ ♦s ♦rs ♦sr♦s ♣r s srs t♠♣♦rs ♥♥♥ ♥♦ ❯♥♦ r♥t s és 1940 ② 1950 st♦ s s♣r ♦q ♥ ♥str♦ ♠♦♦ s♠♣♦ ♥♦ ♠♦s ♦♥sr♦ t♦rs st♦♥s♥♦s ♣♦r ♦ t♦ ♥ tr♦♥s í♥♦s ♥tr st♥t♦s r♣♦s tr♦s ❬♦r ♥ ♥ ❪ ♥tr ♦tr♦s r♥♠♥t♦s ♠♦♦ t♦♦s ♠♦♦s rst♦ qí ♣rs♥t♦ s s♠♠♥t ♥trs♥t ♦ q ♠str ♥ s♣t♦ s♠r♦ ♥ ♦s ♠♦♦s ♣r♦s ♥♥ ♥ ♣rsst♥ s ♦rr♦♥s ♠♣sts ♣♦r strtr ♦♥tt♦s r♦ ♠♦ 〈k〉 ♣r♦ ♦♥t♦ ♣♦r ♦♥tt♦ ♥tr ♥♦s S I ② s r♥ts tss tr♥só♥ ♥tr st♦s s ♥ ♠♥t♥♦♥r♥ts s r ♠♦s ♦♥sr♦ t♦♦s ♦s ♣rá♠tr♦s q t♠♥ts ♦♥sr♥ r♥ts ♥ ♦s ♠♦♦s ♣♠♦ó♦s ♠♣♦ ♠♦ st♦s st ♦ q ♠♦s ♦s ♣rá♠tr♦s ♥ s♦ st♦s ♥♣♥♥t♠♥t ♣rtr ♦♥sr♦♥s ♦ós ② ♥♦ ♦♠♦ ♣rá♠tr♦s st ♥ ♠♦♦ ♣rtr ♦♠♦ ♦rr ♣r ♥♦q ♠t♣♦♦♥♦♥ r♣♦s tr♦s ♣r♦♣st♦ ♥ ❬♦r ♥ ♦♥♥ ❪

♠ ♣rsst♥ trés Pextin ♥♦ s ♠ás ♥tr ♥♦s ♥③♥ srs t♠♣♦rs rs s♣t♦ ♠ás rít♦ s q s ♣♦♦♥s s st♥ts s ♥♦ s ♥♥tr♥ ss ss ♥s s ♣♦r ♦ qs srs t♠♣♦rs rs ♥♦ ♣rs♥t♥ ①t♥♦♥s ♥ts s♥♦ ♣rí♦♦s s♥ ♥t♦s q s ♣r♦♦♥♥ st q ♥♦s ♠♣♦rt♦s ♥rs♥ ♣r♦♦♥♦ ♥ rr♦t ♥ s♥t só♥ ♥r♠♦s ♥ ♠tr♥t ♣rsst♥ q ♦♥t♠♣ st ♦

①t♥♦♥s ♣♦st♣é♠s ♦♥ ♠♣♦rtó♥ ♣r

♠♥♥t ♥t♦s

s srs t♠♣♦rs ♣r ♦s rstr♦s stór♦s ♥♥ sr♠♣ó♥♣rs♥t♥ r♦ts ♣é♠♦s s♦s ♣rí♦♦s ♦ ♥ ♥♥♣♥♥♦ t♠ñ♦ ♣♦ó♥ Pr ♣♦♦♥s r♥s ♦s ♣rí♦♦s♣♦st♣é♠♦s ♣rs♥t♥ ♥♠♥t♠♥t ♥q ♥♦ ♥ ♥♥♥ ♠♦ ♥ s s ♠ás ♣qñs ♦s r♦ts s♦♥ s♦s ♣rí♦♦s♣r♦♦♥♦s ♦♥ s♥ ♥t♦s ♥ ♦s rstr♦s ♥t♦♥s ♥ ♦r♠ ♥tr ♣rsst♥ sr♠♣ó♥ s trés ró♥ ①t♥♦♥s♣♦str♦rs r♦ts ♣é♠♦s q ♥♦♠♥r♠♦s Epost

Epost =①t♥♦♥s ♣♦st♣é♠s♦t r♦ts ♣é♠♦s

♥ó♥ Epost ♥st ♥ rtr♦ ♣r str á♥♦ ② ♥r♦t ♣é♠♦ ② á♥♦ ♥ ①t♥ó♥ ♣♦st♣é♠ ♥ ♥str♦ s♦ ♦♣t♠♦s ♣♦r sr rtr♦ t③♦ ♥ ❬♦♥♥ ❪ ♦♥sr♥♦ ♥ ①t♥ó♥♣♦st♣é♠ ♥♦ tr♥srr♥ trs s♠♥s ♣♦str♦rs ♥ r♦t s♥♥t♦s

♥ ♦ q rs♣t ♦s r♦ts ♣é♠♦s ♥ ❬rttt ❪ rttt ♦♣tó♣♦r ♦♥srr ♥ ♠r r♦t ♣é♠♦ 1 ♥♦ s♦ ♥ó♥ ♣♦r 4000 t♥ts ♣r ♥ ♥tr♦ t♠♣♦r ♥ s♠♥ ♥ ♠r♦ ♣r♣♦♦♥s ♣qñs st rtr♦ rst ♣♦♦ rstrt♦ sr t♦♥s st♦ásts ♠♦rás ♦♠♦ r♦ts ♣é♠♦s s ♣♦r ♦ q rtttt♦ q ♠♣♠♥tr ♥ ♠♦ó♥ ♦ ♣r ♣♦♦♥s N . 105 t♠♦♦ q s ♥s♣ó♥ s s srs t♠♣♦rs ♦♥r ♦♥ só♥ rtr♦ ♦t♦ ♥ s q ♠r ♦ s s♥t♣r q ♦s ♣♦s ♦s r♣♦rts s♠♥s ♥♥ ♥ s srs t♠♣♦rsrs ♦ s♠s s♥ s♦s ♦♠♦ r♦ts ♣é♠♦s ② q s ♣qñst♦♥s s♥ srts ♠♥t sr♠♦s rtr♦ ♠ás s♥s♠♣st♦ ♥ ❬♦♥♥ ❪ ♦♥ r♦♥ ♥ ♠r 1 ♥♦ ♥t♦♣♦r 2000 t♥ts ♥ ♥ ♣rí♦♦ ♥ s♠♥

s ♥sr♦ ♦♥srr ♥ ♦ ♦♥st♥t ♥t♦s ♠♣♦rt♦s ♣r qEpost ♣ ♥rs ♠♦♦ t q ♦ st ♥♦ ♥♦q s♠r♠♦sλimport = 5ñ♦ ♦♥st♥t r♥t t♠♣♦ t♦t s♠ó♥ ♥ t♦♦s ♦ss♦s ♦♠♦ ♥ts ♥ s♠ó♥ s tr♥srrr ♥ tr♥st♦r♦ 30

ñ♦s ② ♦ s st♠ Epost s♦r ♦s 30 ñ♦s s♥ts ♥ r ♠♦str♠♦s Epost ♥ ♥ó♥ t♠ñ♦ ♣♦ó♥ ♣r ♦s ♠s♠♦s ♣rá♠tr♦s♣♠♦ó♦s ♥tr♦r♠♥t r♣♦rt♦s ♠♥t ♦♠♣r♠♦s ♦s s♦s♦rrs♣♦♥♥ts rs ♥rs ♣rtr ♠♦♦ ♣ít♦ ♦♥ q = 0.0

② q = 0.9 r ♥ q s♦ ♦rrs♣♦♥♥t rs ♦♥ q = 0.0

♣rs♥t ♠♥♦r ♣r♦♣♦ró♥ ①t♥♦♥s ♣♦st♣é♠s q ♦rrs♣♦♥♥t q = 0.9 ♥ t♦t r♦ ♦♥ ♦s rst♦s só♥ ♥tr♦r

s ♦♠♣r♦♥s r③s st ♦r ♦♥trst♥ ♦s rst♦s ♣r rs ♥rs ♦♥ st♥t ♣r♦ ♥s tr♥st♦s ♦ rr trá♦q P♦r ♦ ♠♦s s r♥s ♦srs ①st♥ ♦rr♦♥s t♦ ♦r♥ ♥tr ♦s r♦s ♦s ♥♦♦s ♥ ♠r♦ ♠♦s st♦ ♥ ♣ít♦ ♣r♥t q ♠♦r q t♠é♥ s ♦sr♥ r♥s ♥ttts♥ s str♦♥s r♦ s rs ♦t♥s ♥t♦♥s ♣r♥tr♥♦s s s r♥s ♦srs ♥ ♣rsst♥ s♦♥ r♠♥t s ♦♥t str③ó♥ ♦ ♦ s ♥ ♠♦ t♥♥ ♦r♥ ♥ ♠♦

♥áss ♣rsst♥

500000 1000000 1500000 2000000N

r ró♥ ①t♥♦♥s ♣♦st♣é♠s Epost ♣r ♠♦♦ sr♣t♦ ♥ t①t♦ ♣r♥♣ ♦♠♣r♥ r③♦♥s s♦r rs ♦♥ q = 0.9② q = 0.0 ♦s ♥tr♦s ♦♥♥③ 95% ♠♦str♦s ♦rrs♣♦♥♥ 1.96s rr♦r st♥r ♣r ♣r♦♠♦ s♦r 100 s♠♦♥s 30 ñ♦s ♣♦str♦rs tr♥st♦r♦ ♦tr♦s 30 ñ♦s ró♥

ó♥ stró♥ r♦ Pr r st stó♥ ♥ r ♦♠♣r♠♦s ♠♥t Epost ♥ ♥ó♥ N ♦♥sr♥♦ ♣♦r ♥ ♦ srs ♦r♥s ♦♥ q = 0.9 ② ♣♦r ♦tr♦ rs t♦rs ♥rs ♦♥ é♥tstró♥ r♦s ♣r♦ ♦♥strs ♠♥t ♠♦♦ ♦♥r♦♥♣r♦♣st♦ ♥ ❬♦♦② ❪ st út♠♦ s♦ ♦rrs♣♦♥ ♥rr rs té♥t♠♥t t♦rs ♣r s s ♦s r♦s ♦s ♥♦♦s s♦♥ ①t♠♥t ♦s♦t♥♦s ♥ s rs ♦r♥s ♣r♦ ♦s ♥s s♦♥ ♦♥t♦s ③r

♥ r ♣ ♦srrs q t♠♥t s sr♣♥s ♥♥t♦ Epost ♣r♦♥♥ ♠②♦rtr♠♥t t♦ ♦♥t str③ó♥♦ ♦st♥t t♠é♥ s ♦sr q ró♥ ①t♥♦♥s ♣♦st♣é♠s♣r ♠♦♦ ♦♥r♦♥ ♦♥ stró♥ r♦s ♦rrs♣♦♥♥t q = 0.9 s ú♥ ♠♥♦r q ♦t♥ ♣r rs ♥rs ♦♥ q = 0.0 ♥ r st ♦ ♣♦rí rs t♦ stró♥ r♦♦♥ ♠♥t♦ ①♣♦♥♥ ♠ás ♣r♦♦♥♦ ♣r q = 0.9 r r ♣ít♦ ② ♦rrs♣♦♥♥t ♣rs♥ ♥♦♦s ♦♥ r♦ ♠á①♠♦ s♣r♦r ♦s ♦t♥♦s ♣r q = 0.0

500000 1000000 1500000 2000000N

q=0.9 modelo configuracionalq=0.9

r Epost ♦♠♦ ♥ó♥ t♠ñ♦ ♣♦ó♥ ♣r rs ♥rs♠♥t ♠♦♦ ♦♥r♦♥ ♦♠♣r ró♥ ①t♥♦♥s ♣♦st♣é♠s ♦♥sr♥♦ stró♥ r♦ s rs ♦♥ q = 0.9 ♣r ♠♦♦ ♥tr♦♦ ♥ ♣ít♦ ♦s ♥tr♦s ♦♥♥③ 95% ♠♦str♦s ♦rrs♣♦♥♥ 1.96 s rr♦r st♥r ♣r ♣r♦♠♦ s♦r 100s♠♦♥s 30 ñ♦s ♣♦str♦rs tr♥st♦r♦ ♦tr♦s 30 ñ♦s ró♥

♣ít♦

♦♥s♦♥s ♥s ② Prs♣ts

♥ st tss st♠♦s ♦s ♠♣♦s ♦♥rt♦s ♦♥ ♦ ♦♥srr♥ strtr r ♥ r♠♥t ♥ rst♦ ♥ ♥ó♠♥♦ ♣r♦♣ó♥ ♥③♦ ♣r♠r♦ ♦s ♦rrs♣♦♥ ♣r♦♣ó♥ ♣♦str rt ♥ó♥ ♣r s♦ ♣rtr ♥ ❱♥ ♣ít♦ ♥tr♦♠♦s ♥ ♠♦♦ s♠♥ó♥ tr s♦r ♥r stát s♥ q ♦♥ str③ó♥ ♥ts ♥♦♥♦rs♦♠♦ rst♦ ♥ ♠♥s♠♦ ♠tó♥ s♦ ♠♦ ♣♦r ♦♠♦ r♥ ❬té ❪ ♥str♦ ♥♦q ♥♦ ♥tr♦ ♥ ♥t ♥♦♥♦rs ♥ r s♥♦ q st♦s sr♥ ♥tr♠♥t ♦♠♦ ♦♥s♥ ♠♥s♠♦ ♦r♠ó♥ ♦♣♥♦♥s ♥tr♥♦ ♣♦r ♥ ♣qñ♦ r♣♦ ♥♦♥♦rs ♦st♥♦s r♣♦ ♥t♥ó♥ P♦r ♦tr ♣rt ♣rs♥♠♦s r♠♥t♦s t♦rí ♦s ♣r ♠♦strr q ♦♣♥ó♥ ♥ó♥♣ ♣r♦♣rs s♦r r ② rs ♥ ♦♠♥♦s ♦ strs ♦♥r♥tr ♣♦r s♠♣ ♥ ♥tr ss ♠♠r♦s

♥ ♥str♦ ♥♦q s ♣♦tét♦ ♦r ♥ ①♣ó♥ ♣s ♣r ♦r♠ó♥ ♦♠♥♦s ♥♦♥♦rs ♥ st♦♥s t ♦rtr ♣r♦♠♦ ♥ ♦ s♠♥ó♥ ♦♣♥♦♥s ♣♦r ♦♥r ♥ ♣qñ♦ r♣♦ ♥t♥ó♥ ♠♥s♠♦ ♠tó♥ s♦ t③♦ s♦ ♦sr♦ trés ①♣r♥s ♦♥rts ♥s ♦♥ ♦♣ó♥ ♦♠♣♦rt♠♥t♦s ss ❬♥t♦ rsts ❪ s♠s♠♦rs♦s st♦s s♦s ♦♥②♥ q s ♦♣♥♦♥s rss rs♣t♦ ♥ó♥ ♥♥tr♥ ♥ ♠♥t ♣r♦♣♦ ♥ ♦♠♥s ♦♥ ♣r ♥♦♥r♥ tr r ♥ rts ♦r♥t♦♥s r♦ss ♥ ♥tó♥ ♦♥ t♥♥s ♥trsts ①tr♠s ♦ s♣ts♠♦ ①r♦ ♥ ♦♥tr s ♣♦íts ♣ús ❬② ♦s♦♥ ❪

♥ ♣ít♦ ♣rs♥t♠♦s ♥ ①t♥só♥ ♠♦♦ ♣r♦ q ♥♦♠♥♠♦s♠♦♦ ♥r③♦ ♦♥sr♠♦s ♥s tr♦é♥♦s r♥♦s♥ ♦s t♦rís ♣rs♦♥s ② ♥♦♣rs♦♥s ♥q ♠♦s stá♥ ♥♦r♦s♥ ♠♥s♠♦ s♠♥ó♥ ♦♣♥♦♥s s♦♦ ♦s ♣r♠r♦s s♦♥ ♣s

♣ít♦ ♦♥s♦♥s ♥s ② Prs♣ts

tr♥s♠tr ♥ ♥r♠ ♥♦s ♦♠♦ sr♠♣ó♥ ú♥ ♥tr♦ st sq♠ ♦♣t♠st ♦ q s♠♥② ♥t ♥s ♥♦r♦s♥ tr♥s♠só♥ ♥r♠ s ♦sr♥ tt♠♥t ♦s ♠s♠♦srst♦s q ♦♥ ♠♦♦ ♦r♥ ♦♥♠♥t ♥♦r♣♦r♠♦s ♥ ♠♥s♠♦ ♣tó♥ strtr ♥ r s♦ q ♦♠♣ñ ② ♦♠♣♠♥t ♠♥s♠♦ ♣tó♥ ♣♦r ♠tó♥ ♦t♦ s ♠♦r ♦s ♦s ♠♥s♠♦s ♥t♦s ♦♠♦ rs♣♦♥ss str③ó♥ ♥♦s ♥s ♠tó♥ s♦ ② ó♥ ♠sts ♣♦r ♦♠♦ ❬Prs♦♥ ❪ ♣tó♥ ♣♦r r♦♥①ó♥ ♥s ♣♦r ♦♠♦ r ♥ tr♥só♥s ♦♥s♥s♦ ♥♦♥♦r ♥♦r ♠♥♦s r♣t q ♥ s♦♥tr♦r ♦♥ ♥ ró♥ rít ♠ás ①t♥s P♦r ♦tr ♣rt s rtrístst♦♣♦ós ♦s strs ♠á①♠♦s ♦t♥♦s s r♥♥ ♦♥ rs♣t♦ ♦s é♥t♦ t♠ñ♦ ♦rrs♣♦♥♥ts r stát s r♥s t♦♣♦óss ♣s♠♥ ♥ s rtrísts ♣r♦♣ó♥ ♦♠♦ s ♥ ♥ r

tr tr♥t ♠♦♦ ♥r③♦ ♦rrs♣♦♥ ♥só♥ ♠♣ñs ♥♦r♠ó♥ ♥ ♦r ♥ó♥ s ♠♣ñs ♠ss s ♥tr♦♥♠♥t ♥ ♠♣♦ ①tr♥♦ φ q tú ♠♦♦ ss♦ ♦♥tr tr♥só♥ ♥♦r ♥♦♥♦r rtríst stst♦r s ♠♣ñs s q ♥♦ ♣r♦♠♥ ♥ó♥ ♦♠♣s ♥ sqr s♥ ♦♥♥r ♦s ♥♦♥♦rs ② ♣♦r ♦♥trr♦ s ♠♣ñs qí ♠♦s ♣r♦♣♦♥♥ ♥ ♥♦q ♦♥sr♦r q s♦♦ s tr q ♦s ♥♦rs ♠♥s ♦♣♥ó♥ ♦str♠♦s t♦r rt♠♥t ♠t♦r φ t♥t♦ ♥ rsó♥stát ♦♠♦ ♥ ♣tt ♥str♦ ♠♦♦ ♥r③♦ ♥ ♠r♦ t♦ φ rst ♠ás ♥t♥s♦ s♦r ♦s strs ♠♦♦ ♣tt♦ ♦ s ♠②♦r r strtr ♠♥t s ♥ ♠♣♦rt♥ st♦ s rtrísts t♦♣♦ós r♦ ♥ st s♦ ♥♦♥♦rs

Pr♦♠♥t ♦♥só♥ ♠ás ♠♣♦rt♥t ♠♦♦ ♦s ♣ít♦s ② s ♠♣t♦ str③ó♥ ♥♦♥♦s s♦r r ♥♣♦r ♠♥s♠♦ ♠tó♥ s♦ ♠♦ ♣♦r ♦♠♦ ♥ ♥♠♥ r♣♦ ♦s ♠♦♦s ♠♣♦ ♠♦ ♣r♥ q ♥ ♦rtr ♥ 95% srí s♥t ♣r ♣r♥r r♦ts sr♠♣ó♥ ♣r♦t♥♦ sí ♦s q ♥♦ ♥ s♦ ♥♦s ♣♦r r③♦♥s r③ ♠②♦r ♦s♦tr♦s ♠♦str♠♦s trés s♠♦♥s ♥♠érs q st ♠r ♣♦rí sr ♥s♥ts s ♦♥sr strtr ♦♥tt♦s s♦s ♥ ♦♥♥t♦ ♦♥ ♥ ♣r♦s♦ s♠♥ó♥ ♦♣♥♦♥s ♦ ♦st♥t ♥q ♦s t♦s str③ó♥ ♥♦♥♦s ♣r♦s ♣♦r ♥str♦ ♠♦♦ ♥ s♦ r♦s ♥ ♥♠r♦

ss ♦♣♦rt♥s ❬② té ❪ ú♥ rst♥ ①♣r♥s ♦♥tr♦sq ♦r③♥ ♥ ♥tí ts ♦♠♦ s r♥t♠♥t r③s ♣r s♦ r♣ ❬r② ❪ st ♦♥♦♠♥t♦ ♦♥rt♦ ♣r♠trí str♥s ♣♦íts ♥ó♥ ② ♠♣ñs ♣r♥ó♥ ♦③s ♥ rró♥ ♥t sr♠♣ó♥ ♦s rst♦s ♣rs♥t♦s ♥ ♦s ♣ít♦s ② ♥ s♦ ♥♦s ♣r s ♣ó♥ ❬s ❪

♦♠♦ tr♦ tr♦ ♠♦♦ s♠♥ó♥ ♦♣♥♦♥s ♣r♦♣st♦ ♥st tss ú♥ rqr ♥ ♥áss s♥s ♠ás ①st♦ rs♣t♦ ♦s r♥ts ♣rá♠tr♦s ♥♦r♦s P♦r ♦tr ♣rt sí ♦♠♦ s ♥ ♥♦r♣♦r♦ r♣♦s ♥t♥ó♥ t♠é♥ rstrí ♥trs♥t ①♣♦rr t♦ ♣r♦♠♦t♦rs ♥ó♥ q ♦♥ ♣rst♦ ♦♥♥③ ② t♦r r♥t s♦ s r ♦s ♠é♦s tr♦ s♣t♦ q ♣r♦②t♠♦s ♦rr ♥ tr♦ ♥♠t♦ s t♦ ♥ ♣♠ ♥♠♥♥t sr♠♣ó♥ ♦♠♦ ♥t♦r q ♥ ♥ó♥ ♣♦r t♠♦r ❬ ❪

♠♦s st♦ q ♦s s♣t♦s t♦♣♦ó♦s s rs s♦s rst♥ r♥ts ♥ ♣r♦♠s s♠♥ó♥ ♦♣♥♦♥s ② ♥r♠s ♥♦ss s♣♦r ♦ q rs♦s tr♦s s ♥ ♦♦ st♦ s♣í♦ t♥t♦ ss♣r♦♣s státs ♦♠♦ ♥á♠s s ♣r♦♣s t♦♣♦ós státs ♠ás♦srs ♥ s rs ♦r♥ s♦ s♦♥ t♦ ♦♥t str③ó♥♣r♦♠♦ C ♦♠♣r ♦♥ ♥s rs rrs ② st♥ ♠♥tr ♥♦♦s l ♦r♥ ♦rrs♣♦♥♥t rs t♦rs l ∼ lnN ♥♠r♦ r♥ts tr♦s ♦s ♥á♠ ♠r♦só♣ ♦s ♥ss♦s ♥ ♣r♦♣st♦ ♠♦♦s rs rs ♥ t♠♣♦ ♥♦ r rs ♠s q ♥♦ ♦♥t♠♣♥ sts ♣r♦♣s státs ♥ts ♠♥♦♥s ❬Prr tr♥♥ tr♥♥ ❪ ♥ st♦s ♣tr♥ ♦rrt♠♥t s ♣r♦♣s ♥á♠s ♦s ♥s r♥ ró♥ t ♥♦sts♥ s ♣r♦♣s státs s rs ♠s ♥ t♠♣♦ q ♥s♦ ♦srs ② ♠♦s t♥t♦ ♥ t ♦♠♦ ♥ ♣s♦ r♥t

♥ ♣ít♦ ♣r♦♣s♠♦s ♥ ♠♦♦ r♠♥t♦ rs q ♥♦r♣♦r ♥á♠ ♦r♠ó♥ ♥s trés ♦♥♣t♦ t s♦❬Prr ❪ t ♠♦♦ q s rs ♠s ♥ t♠♣♦ ♥rs sts♥ s ♣r♦♣s t♦♣♦ós ss Pr ♦ ♥♦r♣♦r♠♦s ♥ s♣t♦st ♠♦♠♥t♦ s♦s②♦ ♥ ♦s ♠♦♦s rs rs ♥ t♠♣♦ rátr ♥♦♠r♦♥♦ ♦s ♥ts s♦s st♦ s ♦♥sr♠♦s t♦s ♠♠♦r r♦ ♣③♦ ♥ ♦s ♥ts ♥ ♦♥tr♣♦só♥ ♦s ♠♦♦s qst♥ ♥á♠ ♦s ♥s s♦s r♥t♠♥t s♠♥ ♥ts ♠r♦♥♦s q s ♦♠♣♦rt♥ ♦♠♦ ♠♥♥ts t♦r♦s st♥♦ í♥♦s

♣ít♦ ♦♥s♦♥s ♥s ② Prs♣ts

♦♥ ♦tr♦s ♦s ③r ❬tr♥♥ ❪

♦str♠♦s q ♠♥s♠♦ rr trá♦ ♠♦♦ sr♣t♦♥ ♣ít♦ ♦r ♥tr♦r s ♦rr♦♥s t♦ ♦r♥ s♠♥t ♦srs ♥ s rs ♦r♥ s♦ ♠s ♥ t♠♣♦ Pr♦♣s♠♦s ♦sr♥ts ♠♦♦ ♣r ♣♦ó♥ ♥♦♦s ♦♥st♥t ② r♥t sr♠♦s ♥ít♠♥t s str♦♥s r♦ ♥ ♠♦s s♦s ② st♠♦s ♦♠♣♦rt♠♥t♦ s♥tót♦ s♦♦ s ③ ♠♦str♠♦s q s rtrístst♦♣♦ós s rs ♥rs ♠♥t ♥str♦ ♠♦♦ s ♦rrs♣♦♥♥ ♦♥s ♦srs ♥ ♦s s♦s st♦ ♥ sr♦ r s♦ ♦♦❬❱s♥t ❪ ② ♥ r ♦♥tt♦s rr ♦t♥ ♣r♦②t♦ ♦♦Pttr♥s ❬ttt♦ s ❪ ♥q ♠♦♦ ♥♦ stá ♥tr♦ ♥♥♥♥ ♥á♠ ♣rtr ♦r♠ó♥ ② ♦rr♦ ♥s r ♥♠r♦ tór♦ ♥tr♦ á ést ♣♦rí ♥tr♦rs

♦♥♠♥t ♠♦str♠♦s q ♠♥s♠♦ r♠♥t♦ rs qí♥tr♦♦ ♦♥t♥ ♠♦♦ rs ♦ts ❬♦♥ ❪ ♦♠♦ s♦♣rtr ♦s rst♦s ♣ít♦ r♦♥ ♣♦s ♥ ❬s ❪

♥♠♥t ♥ ♣ít♦ t③♠♦s s rs ♥rs ♥ ♣ít♦ ♣r♥t ♣r str ♥♥ tr♥st ♥ ♣♦r ♠♥s♠♦ rr trá♦ ♥ ♣rsst♥ sr♠♣ó♥ s♦r r srr♦♠♦s ♥♦rt♠♦ st♦ást♦ t♠♣♦ srt♦ s♦ ♥ ♥♦ ② t♣♦ ♣r str ♣r♦♣ó♥ sr♠♣ó♥ ♦♠♦ ♣rt st ♦rt♠♦ ♦r♥ t③♠♦s ♣r♦①♠ó♥ ♠t♥♦♠ ♥ ♣♦♦♥s ♦♥ ♥ rr t♠♣♦ ó♠♣t♦ ♥♦♥tr♠♦s q ♣rsst♥ s♠♥② ♥r♠♥tr ♣r♦ rr trá♦ q st ♦ s r ♣r ♦s♦s rtr♦s ♣rsst♥ ♥tr♦♦s

♦s rst♦s ♣ít♦ ♠str♥ q ♦♥♦♠♥t♦ s♣í♦ t♦♣♦♦í ♥ r s♦ ♣ t♥r ♥ ♠♣t♦ ♥♦t ♥ st♦ ♣rsst♥ sr♠♣ó♥ sí ♦♠♦ t♠é♥ ♦trs ♥r♠s ♥♦sstr♥s♠ts ♣♦r ♦♥tt♦ st♦s rst♦s srá♥ ♥♦s ♣r s ♣ó♥ r ♦♠♦ tr♦ tr♦ rst ♦♥srr ♥ ♠♦♦ ♠ás t♦q ♥♦r♣♦r ♥ ♥á♠ sr♠♥t♦ ② ♦rr♦ t♥t♦ ♥s ♦♠♦ ♥♦♦s P♦r ♦tr ♣rt ♠♦♦ t♠é♥ ♣♦rí ♥♦r♣♦rr ♦♦rts ♥③s ♣rr♥♠♥t ♣♦r ♦♠♦ st♦ s ♣r♦ ♥ ♥tr ♦s♥♦s ♠s♠ ♦♦rt ♣♦rí rstr s♣r♦r ♦rrs♣♦♥♥t ♣r ♦tr♦s ♦s ♣rt♥♥ts ♦♦rts st♥ts s♠s♠♦ ♣♦♠♦s ♥tr♦rstr♦♥s t♠♣♦s s♣r ♠ás rst t♣♦ ♠♠ ② str s♠♣♦rt♥ rt ♦♥ rs♣t♦ ♠♣t♦ ♦s t♦s t♦♣♦♦í

r ♠♦s ♦♠♥③♦ trr s♦r ♥s sts í♥s

♣é♥

♦♦ ♣♠♦ó♦

sr♠♣ó♥ s ♥ ♥r♠ r ♥♦s s ♣♦r ♥ ♣r♠①♦rs s♥♦ sr ♠♥♦ s ú♥♦ és♣ ♦♥♦♦ trt ♥ ♥r♠t♠♥t ♦♥t♦s ♥♠♥t♠♥t tr♥s♠t ♣rs♦♥ ♣rs♦♥ ♣♦rí ér trés s ♠r♦♦ts ü ♠ts r ♦ st♦r♥rt ♣r♥♣♠♥t ♥ñ♦s ♣qñ♦s ♥ ♣r♠r s♦r ② s ♥ s ♣r♥♣s ss ♠♦rt ♥♥t ♥ ❱ ♠♥str ♦♠♥t s ♥s é ♣r♦ ♥♠♥ ♦♥tr sr♠♣ó♥ s ♣♣rs ② r♦

♦♦ ♦♠♣rt♠♥t st♦ást♦

♦ r♦ ♣rs♥t tss t③♠♦s ♥ ♠♦♦ ♣♠♦ó♦ st♦ást♦ ♦♠♣rt♠♥t t♣♦ s♣t①♣st♦♥t♦rtr♦♣r sr♠♣ó♥ ♦♦ q♦s ♥♦s ♥t♦s s♦♥ ♣s tr♥s♠tr ♥r♠ ♠♥trs q ♦s ♥♦s r♣r♦s qr♥ ♥♠♥♣r♠♥♥t st♦ rrtr♦ ♦♥sr♠♦s ♥ strtr ♦♥tt♦s ♥tr ♥ts sr♣t ♣♦r ♠♦ ♥ r ♦♠♣ G(N,L) ♦♠♣st N ♥♦♦s ② L ♥s ♥tr ♦s ♠s♠♦s ♦s t♦s ♠♦rá♦s s ♦♥sr♥s♣rs

s ♣r♦s tr♥só♥ ♥tr st♦s ♣♦r ♥♦ ② ♣r ♥ ♥tr♦ t♠♣♦r δt s ♥♥ ♥ r s♣ t♥ó♥ tr♥só♥ st♦ ss♣t rrtr♦ P (S → E) ♠♠♦s z(i) ♥t ♥♦s ♥t♦s ♣r ♥t i r ② 〈kp〉 r♦ ♠♦ ♥s♣rs♦♥s ♣♦r ♥t ♥t♦♥s ♣r♦ ♦♥t♦ ♥t i ♥ ♦♥tt♦ ♥ ♥tr♦ t♠♣♦r δt ♦♥ ♥♦ ss ♥♦s ♥t♦s ♥ ♣♦r δt c/〈kp〉 ♦♥ c/〈kp〉 ♦rrs♣♦♥ ts ♦♥t♦ ♣♦r ♥♦ ♣r♦ q ♥t i ♦♥tr ♥r♠ ♥ δt ♣r♦t♦

tt♣♦♥t♠♥trtstss♥

♣é♥ ♦♦ ♣♠♦ó♦

ss ♥♦s ♥♦s♦s s

P i(S → E) = 1− (1− δt c/〈kp〉)z(i) .

r♦ Pr♦s tr♥só♥ ♣♦r ♥♦ ♣r ♠♦♦ t③♦

r♥só♥ Pr♦S → E 1− (1− δt c/〈kp〉)z(i)E → I σδtI → R γδt

♥ st tr♦ ♦♥sr♠♦s c = 2.8 ♦r ♦♠♣t ♦♥ ♥ú♠r♦ r♣r♦t♦ ás♦ sr♠♣ó♥ 12 ≤ R0 ≤ 18 ♣r ♦s ♠♦♦s tr♠♥sts❬♥rs♦♥ ❪ P♦r ♦tr ♣rt ♦♥sr♠♦s q t♠♣♦ rtríst♦ tr♥só♥ ①♣st♦ ♥t♦ s TE = 8 ís ② ♥t♦ rrtr♦TI = 5 ís ❬♦② ♦♥♥ ❪ st ♠♦♦ s ♦rrs♣♦♥♥ts tss tr♥só♥ s ♥♥ ♦♠♦ σ = 1/TE = 0.125 ② γ = 1/TI = 0.2

♥♠♦s ♦♠♦ St Et It ② Rt s ♣♦♦♥s ♥ts ss♣ts ①♣st♦s ♥t♦s ② rrtr♦s t♠♣♦ t rs♣t♠♥t ♠♦♦ st♦ást♦ ♦♥ r ♦♠♣ ♣r♦♣st♦ r♣rs♥t ♥ ♣r♦s♦ r♦ ♣s♦ srt♦ δt ♦ ♦rt♠♦ t③ s♥ró♥♠♥t s ♣♦♦♥s st♦ ♣r ♥ ♣s♦ t♠♣♦r δt = 1/2 í s♥t ♠♦♦

♥t ss♣t i ∈ St s ①♣st♦ ♦♥ ♣r♦ ♥ s♦♦ ♦♥sr♥♦ q♦s ♥ts ♥t♦s j ♣rt♥♥ts s ♥t♦r♥♦ j ∈ It ♣♦ó♥ ss♣ts s t③ ♣r s♥t ♣s♦ t♠♣♦r St+δt

♥t ①♣st♦ i ∈ Et s ♥t♦ ♦♥ ♣r♦ σδt ♣♦ó♥ ♥ts ①♣st♦s s t③ ♣r s♥t ♣s♦ t♠♣♦rEt+δt

♥t ♥t♦ i ∈ It s r♣r ♦♥ ♣r♦ γδt ♣♦ó♥ ♥ts ♥t♦s s t③ ♣r s♥t ♣s♦ t♠♣♦r It+δt

t③ t♠♣♦ t = t+δt rt♦r♥♥♦ ♣s♦ st q Et+It = 0

♦♦ ♦♠♣rt♠♥t st♦ást♦

♦t♠♦s q ♦rt♠♦ ♥tr♦r ♥♦ t③ ♣♦ó♥ rrtr♦s ♦q ♠♥t♥rs ♣♦ó♥ t♦t N ♦♥st♥t st ♣ rrs ♣rtr s rst♥ts

♣é♥

r ♥tr♦ó♥ ♠♦♦

rs ♦ts

♦♦ rs ♦ts

s ♠♦♦s ♦♥ rs ♦ts ♦♥sr♥ ♥ ♣r♦♣ ♥trí♥s ♦s ♥♦♦s q tr♠♥ ♦♠♣t♠♥t s ♣r♦♣s r❬♦♥ ❪ ♠♦♦ ♦♠♥③ ♦♥ N ♥♦♦s s♦♥t♦s ♥♦ s♦♦ ♦♥ ♥ r ♦t h ♦t♥ ♠♥t ♥ stró♥ ♣r♦ ρ(h) P♦str♦r♠♥t ♦s ♥s s st♥ ♣rtr ♥ ♣r♦ ♦♥①ó♥ s♠étr r(hi, hj) st ♠♦♦ t♥t♦ ρ(h) ♦♠♦ r(hi, hj) tr♠♥♥ ♦♠♣t♠♥t s ♣r♦♣s t♦♣♦ós ♥ r t♦r r♦♥s ♣r♦♣s t♦♣♦ós s♦♥ ♦t♥s ♥ ♥ó♥ r ♦t h

② ♦ ♥ ♣r♦♣♦r g(k|h) ♣r♦ tr♥s♦r♠ó♥ s h r♦s k ♣r♦♣♦r g(k|h) r♣rs♥t ♣r♦ ♦♥♦♥ q ♥♥♦♦ ♦♥ r ♦t h rst ♥♠♥t ♦♥ r♦ k st ♦r♠ stró♥ r♦ ♣ sr srt ♦♠♦

P (k) =∑

g(k|h)ρ(h).

♠♦♦ s t♠é♥ ♣♦s ♦t♥r r♦ ♠♦ ♦s ♥♦s ♥♠t♦s ♥ ♥♦♦ ♦♥ r♦ k ♦♠♦ ❬♦♥ ❪

knn(k) = 1 +1

g(k|h)ρ(h)knn(h),

♦♥ knn(h) s ♥ ♦♠♦

knn(h) =N

ρ(h′)k(h′)r(h, h′),

♣é♥ r ♥tr♦ó♥ ♠♦♦ rs ♦ts

♦♥ k(h′) r♦ ♠♦ ♥ h′♥♦♦ ♦ ♣♦r

k(h) = N∑

ρ(h′)r(h, h′).

♦♥♠♥t ♦♥t str③ó♥ ♠♦ ♦♠♦ ♥ó♥ r♦k ♣ sr srt♦ ♦♠♦

C(k) =1

ρ(h)g(k|h)C(h)

♦♥ C(h) s ♥

C(h) =∑

h′,h′′

P (h′|h)r(h′, h′′)P (h′′|h),

♦♥ P (h′|h) ♣r♦ ♦♥♦♥ ♦♥①ó♥ ♥tr ♥ h♥♦♦ ② ♥ h′♥♦♦ ♣♦r

P (h′|h) = Nρ(h′)r(h, h′)

♥ ♠♦♦ ♥tr♦♦ ♥ ♣ít♦ t a r♦ r ♦t ♦♥ ♣ t F (a) t♦♠♥♦ r stró♥ρ(h) ♥ ♥ ♦r♠ó♥ ♦♥t♥ ♥ ♠r♦ s rs rs ♥ t♠♣♦♦t♥s trés ♠♦♦ rst♥ ♥♦r♦♥s ♦ s ♦rr♦♥s ♦s ♥s ♣♦r ♠♥s♠♦ rr trá♦ ①♣t♦ ♣r s♦ ♦♥ q = 0 s♠s♠♦ ♥ ♣ít♦ ♠♦s ♦t♥♦ Nk|a ♦♠♦ ♥q♥t ♣r♦♣♦r ♦rrs♣♦♥♥t ♠♦♦ rs ♦ts ♣rqr q ≥ 0 ♥ ♠r♦ ♥♦ ♥♦s ♣♦s rr s ♦rr♦♥s r♦ t♦ ♦r♥ s♦♦ ♣rtr ♦♥♦ ♣r♦ ♦♥①ó♥ r(a, a′)

♦♠♦ ♦rr ♣r ♠♦♦ rs ♦ts st♦ s sí ♦ q ♠♥s♠♦ ♦♥①♦♥♦ ♥tr♦ ♦rr♦♥s r♦ ♠ás á s♠♣sts ♣♦r ♠♥s♠♦ ♥♦ ♣♦r t s♦ ♦s ♥ts ♦♦st♥t ♣♠♦s srr t♦s s ♣r♦♣s t♦♣♦ós s rs ♥③ ♦♥♦s F (a) ② r(a, a′) ♣r s♦ ♣rtr ♦♥ q = 0

P♦r út♠♦ r♠♦s ♦s ♥♠♥t♦s ♥rs ♣r ♥ ró♥ ♥ó♥ s ♥ st s♦ ♣rtr t♥♠♦s q = 0 ② ♣♦rst r③ó♥ ♣♦♠♦s ♦t♥r ♥ ①♣rsó♥ ♣r♦①♠ ♣r ♣r♦rℓ(i, j) q ♦s ♥♦♦s i ② j ♦♥ ts ai ② aj rs♣t♠♥t rst♥♦♥t♦s ♦ ℓ ♥s r♦s sℓ(i, j) = 1−rℓ(i, j) ♣r♦

♦♦ rs ♦ts

q ♥♦ s st③ ♥ ♦♥①ó♥ ♥tr q♦s ♦s ♥♦♦s ♥♦ ℓ ♥sr♦♥ r♦s sí ♣♦♠♦s srr sℓ(i, j) ♦♠♦ ❬tr♥♥ ❪

sℓ(i, j) =∑

Pℓ(zi)Pℓ(zj)

1− 1

N − n

1− 1

N −m

♦♥ Pℓ(zi) Pℓ(zj) ♣r♦ q zi zj ♥s ♠r♥ i j ♦ ℓ ♥s r♦s ♣♦r

Pℓ(zi) =

aiN〈a〉

1− aiN〈a〉

)ℓ−zi

s♥♦ ai/(N〈a〉) ♣r♦ r ♥♦♦ i ♦♥ t ai ♥trt♦ ♣♦ó♥ ♦♥sr♥♦ zi, zj ≪ N ♥ t♥♠♦s ♥t♦♥s

zi,j∏

1− 1

N − n

1− 1

♦ sstt②♥♦ ♥ ♥t♦ ♦♥ ♣r♦①♠ó♥ ♦t♥♠♦s rℓ(ai, aj) ♦♥

rℓ(ai, aj) = 1− sℓ(ai, aj) ≃ 1−(

1− aiN2〈a〉

1− ajN2〈a〉

q ♣ sr ♣r♦①♠ ♣♦r

rℓ(ai, aj) ≃ 1− exp

− ℓ

N2〈a〉(ai + aj)

trés ♦rrs♣♦♥♥ ρ(h) → F (a) r(h, h′) → rℓ(a, a′) g(k|h) → Nk|a

② r♠♣③♥♦ ①♣rsó♥ ♥ s s ② rr♠♦s ♥ó♥ s ó♥ s♥♦ ♣r♦♠♥t♦ sr♣t♦ ♥❬tr♥♥ ❪

♦rí

❬rt ❪ rt ♦♥ ♥ rs ♠tr ♦ t ❲♦r

❲ ❲ tr ♦ ♣s r ♣á

❬rt ❪ rt ♥ rás ttst ♠♥s ♦ ♦♠♣①

♥t♦rs ♦ P②s ♦ ♥♦ ♣s ♥r② r ♣á

❬♥rs♦♥ ❪ ♥rs♦♥ ♥ ② ♥t♦s sss ♦ ♠♥s②♥♠s ♥ ♦♥tr♦ ①♦r ❯♥rst② Prss ❯ r ♣á ②

❬♥r ❪ ♥r ♦♦② ♦ ♠♥s tt ♦♥ ❲ ♦ Pt♦ ♥t♦s♠ ♥ ♠tt❱♥t♦♥ rt② rs ss st② t ♥ ♥qt② ♦r

❲♦r t r♥ ♦ ♥♦ ♣s rr② r ♣á

❬①r♦ ❪ ①r♦ ss♠♥t♦♥ ♦ tr ♦ t ♦

♦♥r♥ ♥ ♦ P♦r③t♦♥ ♦♥t s♦t ♦ ♥♦ ♣s r ♣á ②

❬② ❪ ♦r♠♥ ② sttst ♠t♦ ♦ st♠t♥ t ♣r♦s

♦ ♥t♦♥ ♥ ♥t♦♥ ♦ ♥ ♥t♦s ss r ♣á

❬② ❪ ♦r♠♥ ② ♥ st♠t♥ t t♥t ♥ ♥t♦s ♣r♦s

♦ ♠ss ♠s t t♦ ss♣ts ♦♥② ♦♠tr ♣s r ♣á

❬② ❪ ♦r♠♥ ② ♦♠ st♦st ♠♦s ♦r s♠ ♣♠s ♥

r ♣♦♣t♦♥s ♣♣ ttsts ♣s r ♣á

❬② ❪ ♦r♠♥ ②t ♠t♠t t♦r② ♦ ♥t♦ssss ♥ ts ♣♣t♦♥s rs r♥ ♦♠♣♥② t r♥♦♥trt ❲②♦♠ s P r ♣á ②

❬rs ❪ rs ♥ rt ♠r♥ ♦ s♥ ♥ r♥♦♠

♥t♦rs ♥ ♦ ♣s r ♣á

♦rí

❬rs ❪ rs ♥ rt ♠r♥ ♦ s♥ ♥ r♥♦♠

♥t♦rs ♥ ♦ ♥♦ ♣ r ♣á ②

❬rás ❪ rt rás ♣②ss ♦ t ❲ P②ss❲♦♥♥ ♦r♥ ② r ♣á

❬r② ❪ ❱t♦r r② ♠♦ ♠s③ ♥♣♥ ♦♦♥ ♦♥tt ♥② ♦♥♥ ♦ ♠r ❯③♥♥ ♥ r téP♦st ♥t♦r ss♦rttt② ♦ ♥♥③ ♥t♦♥ t s♦♦

♠♣t♦♥s ♦r ♦tr rs ♥ r ♠♠♥t② P♦ ♦ ♥♦ ♣ r ♣á

❬rrt ❪ rrt rté♠② Pst♦rt♦rrs ♥ ❱s♣♥♥ rttr ♦ ♦♠♣① ❲t t♦rs Pr♦♥s ♦ t t♦♥ ♠② ♦ ♥s ♦ ♥♦ ♣s r♣á

❬rttt ❪ rttt tr♠♥st ♥ st♦st ♠♦s ♦r rrr♥t

♣♠s ♥ Pr♦♥s ♦ t tr r② s②♠♣♦s♠ ♦♥ ♠t♠t sttsts ♥ ♣r♦t② ♦♠ ♣ ❯♥rst② ♦ ♦r♥ Prss r② r ♣á

❬rttt ❪ r rttt ss ♣r♦t② ♥ ♦♠♠♥t② s③♦r♥ ♦ t ♦② ttst ♦t② rs ♥r ♣s r ♣á ②

❬ ❪ rs s♦♥ P ♥ ♥ r♥ r♦♣

♥trst rss s♥trst ♥ s♠♣♦① ♥t♦♥ ♣♦② Pr♦♥s♦ t t♦♥ ♠② ♦ ♥s ♦ ♥♦ ♣s r ♣á

❬ ❪ rs ♥ r♥ ❱♥t♦♥ ♥ t t♦r② ♦

♠s Pr♦♥s ♦ t t♦♥ ♠② ♦ ♥s ♦ t ❯♥ttts ♦ ♠r ♦ ♥♦ ♣s r ♣á

❬ ❪ rs ♠tt♦♥ ②♥♠s ♣rt ♥t♥ ♦r

Pr♦ ♦ ♦ ♥♦ ♣s st r♣á

♦rí

❬ ❪ rs ♥ ♠t ttr②② ♦t♦♥r② ♠

t♦r② ♥ s♦ r♥♥ ♥ tr♠♥ ♦ ♥ srs ♥♦ P♦♦♠♣tt♦♥ ♦♦② ♦ ♥♦ ♣ r ♣á

❬r♠♥ ❪ Ptr r♠♥ ♠s ♦♦② ♥ tr♥ t♦ ♥s♦ t♦♥ trtr ♦ ♦s♥t ♦♠♥t ♥ ① t♦rs♠r♥ ♦r♥ ♦ ♦♦♦② ♦ ♥♦ ♣s r♣á

❬ss ❪ ss ♥ ♠♠♥t② ♥ ♥♦t♦♥ ♠♥ r♣á

❬♦♥ ❪ ❱ ♦♥ ♠ ♠♦tt ♥ st ♥♦♥ ♦ ♦♠♠♥ts ♥ r ♥t♦rs tt ♣P r ♣á

❬♦♥ ❪ r♥ ♦♥ ♥ ♦♠♦ Pst♦rt♦rrs ss ♦ ♦rr

t r♥♦♠ ♥t♦rs t ♥ rs P②s ♦ ♣ r ♣á ②

❬♦ñá ❪ rá♥ ♦ñá ♦♠♦ Pst♦rt♦rrs rt í③r♥ ① r♥s ♦s ♦ s♦ ♥t♦rs s ♦♥ s♦ st♥ t

t♠♥t P②s ♦ ♥♦ Pt ♣ ♦♠r r ♣á

❬♦r ❪ ♦r ♥ r♥ ♦s ♥ ♦♦ ♦♠♣①t②

♥ ♠ss ②♥♠s Pr♦ ♦ ♦ ♥♦ ♣s ♥r② r ♣á ②

❬♦r ❪ ♥♠♥ ♦r ♥ r②♥ r♥ ♣ ♣rsst♥ ♥ ②

♥♠s ♦ ♠ss ♣♠s P♦s♦♣ r♥st♦♥s ♦ t ♦②♦t② ♦ ♦♥♦♥ rs ♦♦ ♥s ♦ ♥♦ ♣s r ♣á ②

❬♦♥ ❪ ♦rt ♦♥ rst♦♣r rss s♦♥ ♦♥s ♠ r♠r ♠r♦♥ r♦ ♠ tt ♥ ♠s ♦r

♠♦♥♣rs♦♥ ①♣r♠♥t ♥ s♦ ♥♥ ♥ ♣♦t ♠♦③t♦♥

tr ♦ ♥♦ ♣s ♣t♠r r ♣á ②

♦rí

❬r♦♥ ❪ tr♥ r♦♥ s♥♥ ♦♥ r② ♠s② s♦♥ ♦♥ r♥ rs ❱♥♥t ♠♦♥ r♦ r♠ rsr♥ s ❯ ♣r♥ts s♦♥♠♥ ♦t ♠ss♠♠♣s

r ♥ ②rs tr t ts♠ ♦♥tr♦rs②

qtt ♥②ss ❱♥ ♦ ♥♦ ♣s r♣á

❬rss ❪ rss rrt rss ♥ s

♥t♦♥ ♥ ts♠ ♦♥tr♦rs② ♥ ❯♥t ♥♦♠

♥t ♦♠♠♥t② ♦tr ♦r r ♦ rs ♦♠♠♥t♦♥ ❱♥ ♦ ♥♦ ♣s r ♣á

❬r♦ ❪ ss♦ r♦ t♥ ②sár③ r♥♥♦ r♥♦ ♥sús ó♠③rñs ♦t♦♥r② ❱♥t♦♥ ♠♠ ♥ ♦♠

♣① t♦rs ♦ ♦ s r ♣á ②

❬st♥♦ ❪ st♥♦ rs ♥ ❱s♣♥♥ ♦♥qr♠

Ps r♥st♦♥ ♥ ♦ ♦r ♦ ♥♥ P②s tt ♦ ♥♦ ♣s r ♣á

❬ttt♦ ❪ r♦ ttt♦ ❲♦tr ❱♥ ♥ r♦ ♥ rrt ❱tt♦r♦③③ ♥r♥ç♦s P♥t♦♥ ♥ ss♥r♦ ❱s♣♥♥ ②♥♠s

♦ ♣rs♦♥t♦♣rs♦♥ ♥trt♦♥s r♦♠ strt s♥s♦r ♥t♦rsP♦ ♦♥ ♦ ♥♦ ♣ r ♣á ②

❬♥t♦ ❪ ♠♦♥ ♥t♦ ♣r ♦ ♦r ♥ ♥ ♥♥ ♦

t♦r ①♣r♠♥t ♥ ♦ ♥♦ ♣s r ♣á

❬♥t♦ ❪ ♠♦♥ ♥t♦ ♥ ①♣r♠♥t st② ♦ ♦♠♦♣② ♥ t ♦♣

t♦♥ ♦ t ♦r ♥ ♦ ♥♦ ♣s ♠r r ♣á ②

❬♦ ❪ ♥♦♦♥ ♦ t ②rs ♥ r s♦ r♥s♣ ♥ ♠♦

t② sr ♠♦♠♥t ♥ ♦t♦♥s s♦ ♥t♦rs ♥ Pr♦♥s ♦t t ♥tr♥t♦♥ ♦♥r♥ ♦♥ ♥♦ s♦r②♥ t ♠♥♥ ♣s r ♣á

❬rsts ❪ ♦s rsts ♥ ♠s ♦r ♦ ♦♥t♦♥

t♦r② ①♠♥♥ ②♥♠ s♦ ♥t♦rs ♥ ♠♥ ♦r ttsts ♥ ♠♥ ♦ ♥♦ ♣s r ♣á

♦rí

❬♦♥ ❪ ♦♥ ♥ ♥ r t♦rs r ❯trs♠ P②s tt ♦ ♣ r ♣á

❬♦♦♠r ♠♦♥ ❪ P♦ ♦♦♠r ♠♦♥ ♥ rá♥ ♦ñá str♥♦ r♥♦♠ sr ♥t♦rs P②s ♦ ♥♦ Pt ♣ st r ♣á

❬♦♥♥ ❪ ♥r ♦♥♥ P♠♥ ♦♥ ♥ ♦② tt ♥ ♥ r②♥ r♥ s♦♥ t ♠♣t ♦ t♥ t♠ s

trt♦♥s ♦♥ t ♣rsst♥ ♦ ♠ss ♦r♥ ♦ ♦② ♦t②♥tr ♣ rs r ♣á ②

❬♦♥♦r ❪ ♦♥♦r ♦ t③ tt r♥s♦ r♥♦♦♥çs ♣♣♦ ♥③r ♥ ss♥r♦ ♠♠♥ P♦t ♣♦r

③t♦♥ ♦♥ tttr ♥ ❲ r ♣á ②

❬r♥ ❪ r♥ rs str♦♠ ♥ ♦s② rt r♥ tt♥♦r ♥ ♦♥ ♥r ♥rr♥ s♦ ts r♦♠ ♦

r♣ ♦♥♥s Pr♦♥s ♦ t t♦♥ ♠② ♦ ♥s♦ ♥♦ ♣s r ♣á

❬s♥ ❪ s♥ ♥ ♦r♥♦t ♠r♥ ♦ ♠

❲♦r r♦♠ ♦ ♥trt♦♥s ♦♥ q♥t♥ t♦rs P②s tt ♦ ♥♦ ♣ r ♣á ②

❬♦♠í♥③ ❪ ♥ ♦♠í♥③ r ♦r♥r r♥ rr r♥♦r rst♥ s ♦♥ ② s Ps♥ ♦ ♥ ♦s ♥ rtí♥③ ♦s♣ ♦st r ♦sqr ♥ r♠♥ ③s r tr ♦ ss ♥ ♦♠♠♥t② t ❱♥t♦♥

♦r ♠♣t♦♥s ♦r t ❱♥t♦♥ ♥ ♥t♦ssss ♦ ♥♦ ♣s ♦♠r r ♣á

❬♥♦r♦ ❪ rt♦ ♥♦r♦ Pr♦ ♥r ♥ Pr♦ P♦tt ♠

♣t ♦ ♥ s ts ♦♥ t ♥tr st♦r② ♦ ♠♠♥③t♦♥ ♣r♦

r♠♠s ♥ ♠tt♦♥♠ ♣♣r♦ ♦r ♦ ♦ ♥♦ ♣s r r ♣á

❬♥♦r♦ ❪ rt♦ ♥♦r♦ ♥r Pr♦ ♥ P♦tt Pr♦ ♥

tr♣② ♦ P ♥tr♥t♦♥ ♥ Prt ♦s ♥ tr♠♥♥ t

t♦♠ ♦ ❱♥t♦♥ Pr♦r♠♠s P♦ ♦ ♥♦ ♣ r ♣á

♦rí

❬♦r♦♦ts ❪ ♦r♦♦ts str♥ ♦ ♦rrt ♥t♦rs P②s tt ♦♥♥ ♦t ttr P②s ♦ ♥♦ Pt ♣ rr② r ♣á

❬♦rs♦ ❪ ♦ sr ♦rs♦ ♥ s ♦♠♠♥t② tt♦♥ ♥

t♦rs rt♦♥ ♥ ♦s ♦ ♥♦ ♣s r ♣á

❬ ❪ t♥ ♥ ① P♥t♥ t② ♠♥♥ s♥s♥ ♦♠♣①

s♦ s②st♠s Prs♦♥ ♥ ❯qt♦s ♦♠♣t♥ ♦ ♥♦ ♣s r ♣á ②

❬ ❪ t♥ ① ♥② P♥t♥ ♥ ③r ♥rr♥r♥s♣ ♥t♦r strtr ② s♥ ♠♦ ♣♦♥ t Pr♦♥s ♦t t♦♥ ♠② ♦ ♥s ♦ ♥♦ ♣s r ♣á ②

❬rös ❪ P rös ♥ ré é♥② ♥ ♥♦♠ r♣s Pt♦♥st♠t ♦ ♣ r ♣á ②

❬rös ❪ P rös ♥ ré é♥② ♥ t ♦t♦♥ ♦ ♥♦♠ r♣sPt♦♥ ♦ t t♠t ♥sttt ♦ t ♥r♥ ♠② ♦♥ ♣s r ♣á

❬r♦♥③ ❪ t r♦♥③ P♦tr r♦♥③ ♥ ♥s③ ♦②st r♣t ♥t ♥ r♥♦♠ ♥t♦rs P②s ♦ ♥♦ Pt ♣ ♦♠r r ♣á ②

❬ ❪ ♥ ♥ ♦s♥♦♦♠ ♦♥ ❲♥ ♥ rt♥ ♦♠tt♦♥ ②♥♠s ♦ ♥t♦♥ ♦r ♦♥ s♦ ♥t♦rs Pr♦♥s ♦♦ ♥s ♦ ♥♦ ♣s r♣á ②

❬stñ② ❪ P stñ② s♥ ♠② Prr ♦r♥♥♥r ♦t P ♦t ❲♠ ♥ ♥ r ♥r♦r② ❲ ss❯♥t tts ♥r② ② ❲ ♦rt② ♥ ♠♦rtt② ② r♣♦rt ♦ ♥♦ ♣ r ♣á

❬s♣ ❪ ♥ s♣ ♥r t♦ ♦r ♠r② ♠

t♥ t t♦st ♠ ♦t♦♥ ♦ ♦♣ ♠ t♦♥s ♦♠♣t P②s ♦ ♣ r ♣á

♦rí

❬♦♥çs ❪ r♥♦ ♦♥çs Prr ♦ ♥ ❱s♣♥♥ ss♥r♦♦♥ ❯srs tt② ♦♥ ttr t♦rs ❱t♦♥ ♦ ♥rs

♠r P♦ ♦ ♥♦ ♣ r ♣á

❬♦♥③③ ❪ ♦♥③③ ♦ ♥ rs ❯♥rst♥♥ ♥

♠♥ ♠♦t② ♣ttr♥s tr ♥♦ ♣s ♥ r ♣á ②

❬r ❪ r r ♥ t ♥ Ptr rt ①

♣♦r♥ ♥t♦r strtr ②♥♠s ♥ ♥t♦♥ s♥ t♦r❳ ♥Pr♦♥s ♦ t t P②t♦♥ ♥ ♥ ♦♥r♥ P② ♣s Ps♥ ❯ st r ♣á

❬♥ ❪ s♥ ♥ rrt t ❲r st ♦♠ ♥❱③♥ r ♦str ♦r♥ ♥ str r ♥ ♦rt ♥♥♥♥ ss ♦tr t tr♥s ♠r♥ ♥ts ♦ ♥♦ ♣s r r ♣á

❬♦♠ ❪ Pttr ♦♠ ♥ ♦♠ ♠ r♦♥ sr ♥t♦rs tt♥ str♥ P②s ♦ ♥♦ ♣ r ♣á ②

❬P ❪ r ♣á

❬s ❪ ♦r♥③♦ s tt té ♥ rrt r♦ ttt♦ ♥r♥ç♦s P♥t♦♥ ♥ ❲♦tr ❱♥ ♥ r♦ ❲ts ♥ r♦ ♥②

ss ♦ t♦ ♦r ♥t♦rs ♦r♥ ♦ ♦rt ♦♦②♦ ♥♦ ♣s r ♣á ②

❬♦s♦♥ ❪ ♦rt ♦s♦♥ P ❱ r♦♥s ♥ r♦r② P♦♥ t①♦♥♦♠② ♦ rs♦♥♥ s ♥ t ♥t♥ ♠♦♠♥t ❱♥♦ ♥♦ ♣s r ♣á

❬r♥t② ❪ r②s r♥t② ♥♥s r②t♥ s t tr ❲② r ♣á

❬♥ ❪ ♥ ♥ ♥ ♦rrs ♦rrt♦♥ ♦

s ♦r ♦♦ ♣♠s Pr♦♥s ♦♦ ♥s ♦ ♥♦ ♣s st r ♣á

♦rí

❬♥ ❪ ♥ ♥ r♥ ss ①t♥t♦♥ ♥ ♦♠♠♥t②

s③ ♠♦♥ t ♣rsst♥ ♦ ♠ss ♥ ♦ ♥♦ ♣s r ♣á ②

❬♥ ❪ ♥ ♥ r♥ t ♦ rt② ♥ ♥t♦♥

♣r♦ ♦♥ t ♣rsst♥ ♥ s♣t s♣r ♦ ♥t♦s sss t♠t ♦s♥s ♦ ♥♦ ♣s r ♣á ②

❬♠♠ ❪ ♦♥st♥t♥ ♠♠ ❱ít♦r í③ ú ♦r ♥ ① ♥ ♦♥qr♠ tr♥st♦♥s ♥ ♦♠♣① ♥t♦rs ♠♦ ♦ s♦

♥trt♦♥ P②s tt ♦♥♥ ♦t ttr P②s ♦ ♥♦ Pt ♣ rr② r ♣á ②

❬♠ ❪ Ptr ♠ ♥ t♥ r♥r r ♦sr ②♥♠s r

s s♥s ♥ s♦ ♠t♣① ♥t♦rs P②s ♦ ♣ ♥ r ♣á ②

❬♥t ❪ ♥t t♥♦r r♣s ♣t♦r♠ ♦r ♦♠♥t♦r ♦♠♣t♥ Prss ❨♦r ❨ ❯ r ♣á

❬♦ss♥ts ❪ ♦r ♦ss♥ts ♥ ♥♥ ❲tts ♠♣r ♥②ss

♦ ♥ ♦♥ ♦ t♦r ♥ ♦ ♥♦ ♣s ♥r② r ♣á ②

❬r♣s② ❪ P r♣s② ♥r ♥ ②r③ ♦♥♥tt② ♦

r♦♥ ♥♦♠ t♦rs P②s ttrs ♦ ♥♦ ♣s r ♣á ②

❬r♣s② ❪ P r♣s② ♦rs ♥ ♥r r strt♦♥s♦ r♦♥ ♥t♦rs P②s tt ♦ ♥♦ ♣s ♥ r ♣á

❬③rs ❪ P ③rs ♦rt rt♦♥t r♥s♣ s

s♦ ♣r♦ss sst♥t ♥ ♠t♦♦♦ ♥②ss r♦♠ ♥♦♥tr♦ ♥ ♠♦r♥ s♦t② ♦ ♥♦ ♣s r ♣á ②

❬③r ❪ ③r ① ♥② P♥t♥ ♠ ♥♥ r rt s③♦ rs ♦♥ rr ♦s rsts ♦sr ♦♥trt♦r ♠s ♦r ②r♦♥ t♠♥♥t ♥ t ♥t♦r t

♦rí

♦♠♥ ♦ ♦♠♣tt♦♥ s♦ s♥ ♥ ❨♦r ❨♦ ♥♦ ♣ r ♣á ②

❬r♠♥ ❪ rst♥ r♠♥ ♥ ♠ ♦s ♥♦r♠t♦♥ ♦♥t♦♥ ♥

♠♣r t② ♦ t ♣r ♦ s ♦♥ ♥ ttr ♦ t

♦rs ❲ ♦ ♣s r ♣á ②

❬♥♦ ❪ ♥♦ ♥ ♦♥ ♥r ♥

♣rt♦♥ ♣r♦♠ ♦r s♦ ♥t♦rs ♦ ♥♦ ♣s r ♣á

❬♦② ❪ ♥ ♦② st strt♦♥s ♦ ♥t♦s ♣r♦s ♥ ♣

♠ ♠♦s ♥♥ ♣ttr♥s ♦ ♣rsst♥ ♥ ②♥♠s ♦rt♣♦♣t♦♥ ♦♦② ♦ ♥♦ ♣s r ♣á ②

❬♦♥♦♥ ❪ ❲②♥ P ♦♥♦♥ ♥ ♠s ❨♦r rr♥t ♦trs ♦

♠ss ♥♣♦① ♥ ♠♠♣s s♦♥ rt♦♥ ♥ ♦♥tt rts♠r♥ ♦r♥ ♦ ♣♠♦♦② ♦ ♥♦ ♣s r ♣á

❬s♦ ❪ s♦ ♥ ♥♣♣♥ ♣t② ♥ stt② ♥ t♦♣♦♦②

♦ ♣r♦t♥ ♥t♦rs ♥ ♦ ♥♦ ♣ r♣á

❬r ❪ ♦♥ r ♥ ② tt ♦ t ❲♦rs ❱♥s♥ ♠♠♥③t♦♥ ❲♦r t r♥③t♦♥ r ♣á

❬② ❪ ♦♠s ② ♥ ♦ss r♠♥ ❵str♥ ♦ ①♠♣t♦♥s s

♦t t♦♥ trt t♦ r ♠♠♥t② ❱♥ ♦ ♥♦ ♣s r ♣á ②

❬ ❪ rt ♦ ♥③♦ rs ❨♦♥s ♥ ♦ r♥ ♥ ②rs ♥ s♦♥ P ♥

♠♣t ♦ ♠tt♦♥ ♦♥ ♥t♦♥ ♦r ♥ s♦ ♦♥tt ♥t♦rs

P♦ ♦♠♣t ♦ ♦ ♥♦ ♣ r ♣á ②

❬Prs♦♥ ❪ r Prs♦♥ ②♥♥ ♠t♦♥ ♥ ♠s ♦♦rs ♦ tr ♦♠♦♣② ♥ ♦ t♦rs ♥♥ ♦♦♦ ♥♦ ♣s r ♣á ②

♦rí

❬s ❪ s ñ ♥ ♦rs♦ tt♦♥ ♦ ♦♠♠♥t②

strtrs ♥ ♥t♦rs ♦ ♦♣t♠③t♦♥ P②s ♦ ♥♦ ♣s ♠r r ♣á ②

❬s ❪ s ♥ ♦rs♦ tr♥t ♣♣r♦ t♦ ♦♠♠♥t②

tt♦♥ ♥ ♥t♦rs P②s ♦ ♥♦ Pt ♣ ♥ r ♣á

❬s ❪ s ♥ ♦rs♦ ♠♦r② ts ♥ strtr ♥

s♦ ♥t♦rs t tt②r♥ ♥ts ♦r♥ ♦ ttst ♥s ♦r② ♥ ①♣r♠♥t ♦ ♣ r ♣á

❬s ❪ ♥rés s ♥ ♦ ♦rs♦ ❱♥t♦♥ ♥ ♣

trst ♠♦ ♦r t ss♠♥t♦♥ ♦ ♥t♦♥ ♦r t ①tr♥

♥tr♥t♦♥ r❳ ♣r♣r♥t r❳ r ♣á ②

❬ ❪ ❱ ③ ♦r ♥ ♠♠ ♥r② ♥

♠trt st♦st ♠♦s ♦ ♦♥s♥ss ♦r♠t♦♥ ♦♠♣t♥ ♥♥ ♥♥r♥ ♦ ♥♦ ♣s ♦ r ♣á

❬r♠ ❪ t♥② r♠ ♠ ❲♦r Pr♦♠ Ps②♦♦② ♦②♦ ♥♦ ♣s r ♣á ②

❬♦♦② ❪ ♦♦② ♥ rt ♣♦♥t ♦r r♥♦♠ r♣s t

♥ r sq♥ r ♣á ②

❬♦♦② ❪ ♦♦② ♥ s③ ♦ t ♥t ♦♠♣♦♥♥t ♦

r♥♦♠ r♣ t ♥ r sq♥ ♦♠♥ Pr♦ ♦♠♣t♦ ♣ r ♣á

❬♥ ❪ ♥ ♥ P s Prr ♦ s ♦sé ♥r ♦♠♦ ♥ ♥ r♥á♥ s r♥s ♦ ♣♦r

r strt♦♥ ♥ t tr♦♥t② ♦ ♠♥ tt② ♥ s♦ ♥t♦rs

♣ ♦ ♣ r ♣á ②

❬♠♥ ❪ ♠♥ strtr ♦ s♥t ♦♦rt♦♥

♥t♦rs P ♦ ♥♦ ♣s r ♣á

❬♠♥ ❪ ♠♥ tr♦t③ ♥ ❲tts ♥♦♠

r♣s t rtrr② r strt♦♥s ♥ tr ♣♣t♦♥s P②s ♦ ♥♦ ♣ ② r ♣á

♦rí

❬♠♥ ❪ ♠♥ ss♦rtt ①♥ ♥ t♦rs P②s tt ♦ ♣ r ♣á ②

❬♠♥ ❪ ♠♥ ❲tts ♥ tr♦t③ ♥♦♠ r♣

♠♦s ♦ s♦ ♥t♦rs Pr♦ t ❯ ♦ ♣♣ ♣s rr② r ♣á ②

❬♠♥ ❪ ♠♥ ①♥ ♣ttr♥s ♥ ♥t♦rs P②s ♦ ♣ r ♣á ②

❬♠♥ ❪ ♠♥ ♥ ②♦♥ Pr ❲② s♦ ♥t♦rs r

r♥t r♦♠ ♦tr t②♣s ♦ ♥t♦rs P②s tt ♦♥♥ ♦tttr P②s ♦ ♥♦ Pt ♣ ♣t♠r r♣á

❬♠♥ ❪ r ♠♥ ♥♦♠ r♣s s ♠♦s ♦ ♥t♦rs♣s ❲②❱ ❱r ♠ ♦ r ♣á

❬♠♥ ❪ ♠♥ strtr ♥ ♥t♦♥ ♦ ♦♠♣① ♥t

♦rs ♦ ♥♦ ♣s r ♣á ②

❬♠♥ ❪ ♠♥ ♥ r♥ ♥♥ ♥ t♥ ♦♠

♠♥t② strtr ♥ ♥t♦rs P②s ♦ ♥♦ ♣ rr② r ♣á ②

❬P♣♦♣♦♦s ❪ rs♦s P♣♦♣♦♦s s♠ ts ♥s rr♥♦ rá♥ ♦ñá ♥ ♠tr r♦♦ P♦♣rt② rss s♠rt②

♥ r♦♥ ♥t♦rs tr ♦ ♥♦ ♣s ♣t♠r r ♣á

❬Prr ❪ ♠② Prr ❲②♥ ts st♦ ②♥ s♠ rtá♥③ P ♦t s ♦ Ptr ♦r♠♥ ♦rt r♥ rs ♥ rs ❲ r♦♥ ♠♣t♦♥s ♦ ♠ss

♦tr ♥ ♥♥ ♦r sst♥ ♠♥t♦♥ ♦ ♠ss ♥ t ❯♥t

tts ♥♥ ♦r♥ ♦ ♥ ♦ ♥♦ ♣s r ♣á

❬Pst♦rt♦rrs ❪ ♦♠♦ Pst♦rt♦rrs ① ❱á③q③ ♥ ss♥r♦ ❱s♣♥♥ ②♥♠ ♥ ♦rrt♦♥ ♣r♦♣rts ♦ t ♥tr♥tP②s tt ♦ ♥♦ ♣ r ♣á ②

♦rí

❬Pst♦rt♦rrs ❪ ♦♠♦ Pst♦rt♦rrs ♥ ss♥r♦ ❱s♣♥♥♣♠ ♣r♥ ♥ r t♦rs P②s tt ♦ ♣s ♣r r ♣á

❬Pr ❪ ♥♥ Pr tr♥ ♠♥ ♠ ♦ ♥ ♥ ♦r t♦rs ss♦t t ♣t ♦ ♠ss ♠♠♣s ♥ r ♥ ♥ s ♦ s♥ ♥t♥ ♥s ♥ ♦♥t♠♣♦rr②

❯ ♦♦rt ♣r♦s♣t ♦♦rt st② ♠ ♦ ♥♦ ♣s r ♣á

❬Prr ❪ Prr ♦♥çs Pst♦rt♦rrs ♥ ❱s♣♥♥ tt②r♥ ♠♦♥ ♦ t♠ r②♥ ♥t♦rs ♣ ♦ ♣ r ♣á ②

❬P ❪ P ❯

❲ r ♣á ②

❬P♦t♥ ❪ s♥ P♦t♥ ♥ t♥② P♦t♥ s♦rt st♦r② ♦

♥t♦♥ ❱♥s ♦ ♣s r ♣á

❬ ❪ st♥♦ ♦♥ ❱ ♦rt♦ ♥ Prs♥♥ ♥ ♥t②♥ ♦♠♠♥ts ♥ ♥t♦rs Pr♦♥s ♦ tt♦♥ ♠② ♦ ♥s ♦ ♥♦ ♣ r♣á

❬s③ ❪ s③ ♥ rs rr ♦r♥③t♦♥ ♥ ♦♠

♣① ♥t♦rs P②s ♦ ♣ r ♣á ②

❬♥t❱t♦r ❪ ♥ ♥t❱t♦r ♥ ♠r ❱♥ rs ♥t ♥♠ ♥ t st ♠ rst P♦s♦♣ r♥st♦♥s ♦ t♦② ♦t② ♦♦ ♥s ♦ ♥♦ ♣ r ♣á

❬té ❪ r té ♥ st♥ ♦♥♦r t ♦ ♦♣♥♦♥

str♥ ♦♥ ss ♦trs ♦ ♥tr ♦ ♥♦ ♣s ♠r r ♣á ②

❬té ❪ r té ♥ s♥ ♥ ssss♥ ♥t♦♥s♥t♠♥ts t ♦♥♥ s♦ ♠ ♠♣t♦♥s ♦r ♥t♦s ss

②♥♠s ♥ ♦♥tr♦ P♦ ♦♠♣t ♦ ♦ ♥♦ ♣ t♦r r ♣á

♦rí

❬♠♦♥ ❪ ♥ ♠♦♥ t♣♥ P rt ♥ ♥t②r sr② rrt rss ♥ s② ♦♠♣s♦r②

♥t♦♥ ♥ ♦♥s♥t♦s ♦r ♣♦s♦♣ ①♠♣t♦♥s ♣st ♣rs♥t ♥

tr ♥t ♦ ♥♦ ♣s r ♣á

❬♥③ ❪ tr ♥③ ♥ strtr ♠♦ ♦ ♣r♥ ♣♦st

♥t♦♥ ♠ss tr♥s♠ss♦♥ t♠t ♥ ♥ ♦♦②♦ ♥♦ ♣s r ♣á

❬rr♥♦ ❪ ♥s rr♥♦ ♥ r♥ ♦♥ ♥♥ str♥ ♥

r♥♦♠ ♥t♦rs t rtrr② r strt♦♥s P②s ♦ ♣ ② r ♣á

❬♦♥ ❪ ♦♥② ♦♥ tr♥s s ♠ss ♣♠ rts ♦r♥ ♦ ♥♦ ♣ r ♣á

❬♠♦♥ ❪ rrt ♠♦♥ ♥ ss ♦ s strt♦♥ ♥t♦♥s ♦♠tr ♣s r ♣á ②

❬♦♥ ❪ ♦♠♥ ♦♥ ❩ ♦s ♠♠ ♥ rtás③ó rás ♠ts ♦ Prtt② ♥ ♠♥ ♦t② ♥ ♦ ♥♦ ♣s r ♣á ②

❬tr♥♥ ❪ tr♥♥ ♥r r♦♥ ♥ ♦♠♦ Pst♦rt♦rrs ♦♥ ♠♥ ②♥♠s ♦ t♦ ♥trt♦♥ ♥t♦rs

P②s tt ♦ ♥♦ ♣ ♣r r ♣á ②

❬tr♥♥ ❪ tr♥♥ ♥ ♦♠♦ Pst♦rt♦rrs ♦♣♦♦

♣r♦♣rts ♦ t♠♥trt tt②r♥ ♥t♦r P②s ♦ ♣ ♥ r ♣á ②

❬r♠♥ ❪ r♠♥ rt rs② r②♥♥ s♠ rt♥③ ♦♥ ♠r② st♦♥ P ♦tr♥ ❲trs♦♥t♦ ♥ rs ❲ r♦♥ ss ♦tr ♥

② ♥t ♣♦♣t♦♥ ♥ ♦ r♦ ♦ t ♥t♥t♦♥②

♥r♥t Ptrs ♦ ♥♦ ♣s ♣r r ♣á

❬③ ❪ ③ ♥ ♠♦tt ♥ t♥ r♥r trt♦

♥ ♦r♥③t♦♥ ♦ rs s♦ ♥t♦rs ♥ ♥ ♦♥♥ ♦r Pr♦

♦rí

t ❯ ♦ ♥♦ ♣s st r ♣á ②

❬③ ❪ ③ ♥ t♥ r♥r sr♥ s♦ ②♥♠s ♥

♠ss ♠t♣②r ♦♥♥ ♠ ♦ t♦rs ♦ ♥♦ ♣s r ♣á

❬②♦r ❪ r♥t ②♦r ③t r P② rr♥t♦♥ rrst♥ Ptr♦♣♦♦s s ♦t② ♥ ♥ P♥ ❲t ts♠ ♥ ♠ss ♠♠♣s ♥ r ♥ ♥♦ ♣♠♦♦

♥ ♦r s ss♦t♦♥ ♥t ♦ ♥♦ ♣s r ♣á

❬♦♦♥♥ ❪ tt ♦♦♥♥ P ♥♥ r r♠ä ör②ö♥♥ ♥ ♠♠♦ s ♠♦ ♦r s♦ ♥t♦rs P②s ttst ♥s ♥ ts ♣♣t♦♥s ♦ ♥♦ ♣s r ♣á

❬❯♥r ❪ ♦♥ ❯♥r r♥ rrr rs str♦♠ ♥ ♠r♦♥r♦ ♥t♦♠② ♦ t ♦♦ ♦ r♣ t r①♦♠♠♥t ♣s rs t r ♣á ②

❬❱á③q③ ❪ ❱á③q③ Pst♦rt♦rrs ♥ ❱s♣♥♥ rst♦♣♦♦ ♥ ②♥♠ ♣r♦♣rts ♦ t ♥tr♥t P②s ♦ ♥♦ ♣ r ♣á

❬❱á③q③ ❪ ① ❱á③q③ r♦♥ ♥t♦r t ♦ rs Prr♥t

tt♠♥t str♥ rr② ♥ r ♦rrt♦♥s P②s ♦ ♥♦ ♣ r ♣á ②

❬❱rs ❪ ❱rs ♦ ♠ ♥s r♥r♥♦ P♦ ♥ ♦♠s rr♥t ♣♠s ♥ s♠ ♦r ♥t

♦rs ♦r♥ ♦ ♦rt ♦♦② ♦ ♥♦ ♣s r ♣á ②

❬❱s♥t ❪ ♠ ❱s♥t ♥ s♦ ②♦♥ ♥ P rs♥ ♠♠ ♥ t ♦t♦♥ ♦ sr ♥trt♦♥ ♥ ♦♦ ♥ ♦♥r♦r♦t ♥ ♥r rs♥♠rt② trs ❲ ♣s r ♣á ②

♦rí

❬❲ ❪ ♥r ❲ ♠♦♥ r ♥r ♥t♦♥② ♦♥♥♥ ss♦♥ ♦s♥ r r♦t③ ♠r P ♦♥ ♦♠s♦♥ Ptr r②t ②♠♣♦

♥♦r ②♣r♣s ♥♦♥s♣ ♦ts ♥ ♣rs ♦♣♠♥t

s♦rr ♥ r♥ ♥t ♦ ♥♦ ♣s r ♣á ②

❬❲ ❪ tr ♦ ss ♥ ❲s ♦ ② ♣♦rt ♦ t

♥s rs♣♦♥ t♦ t ♦tr ♣♣♦rt t♥q t♦r r ♣á

❬❲tts ❪ ❲tts ♥ tr♦t③ ♦t ②♥♠s ♦ s♠♦r

♥t♦rs tr ♥♦ ♣s r ♣á ②

❬❲ ❪ ❲ ❯ P ❲♦r t r♥③t♦♥ r ♣á ②

❬❳ ❪ ♥ ❳ ♥ ♠♥ ♦♠♣tt♦♥ ♣♣r♦ t♦ rt

r③♥ t ♠♣t ♦ s♦ ♥♥ ♦♥ ♥s ♥t♦♥ s♦♥

♠♥ P♦ ♦ ♥♦ ♣ r ♣á ②

❬❨♥ ❪ ♦♥ ❨♥ ♥ r s♦ ♦♥ ♥♦r♠t♦♥ s♦♥ ♥

♠♣t ♥t♦rs ♥ t ♥♥ t ♥tr♥t♦♥♦♥r♥ ♦♥ ♣s r ♣á

❬❨ ❪ ❯♥② ❨ ♠t♠t t♦r② ♦ ♦t♦♥ s ♦♥ t

♦♥s♦♥s ♦ r ❲s P♦s♦♣ r♥st♦♥s ♦ t♦② ♦t② ♦ ♦♥♦♥ rs ♦♥t♥♥ P♣rs ♦ ♦♦rtr ♣s r ♣á ②

❬❩r② ❪ ❲ ❲ ❩r② ♥ ♥♦r♠t♦♥ ♦ ♠♦ ♦r ♦♥t ♥

ss♦♥ ♥ s♠ r♦♣s ♦r♥ ♦ ♥tr♦♣♦♦ sr ♦ ♣s r ♣á

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