연결의 물리학: 일촌파도를 넘어 친구 추천의 시대로

연결의 물리학: 일촌파도를 넘어 친구 추천의 시대로

2014년 10월 28일 조선대학교 사범대학 부설 교과교육연구소 콜로키움

성균관대학교 에너지과학과 BK21+ 연구교수 이상훈

[email protected]://sites.google.com/site/lshlj82

• 들어가기: 친구 역설

• 네트워크란?

• 이웃 수는 왜 중요할까요?

• 이웃 수만 중요할까요?

• 물리학자가 왜 이런 걸?

• 중고등학생에게 가르쳐 볼까요?

• (시간이 되면) 현재 실제로 제가 진행중인 연구들이 궁금하신지?

발표 순서

친구

친구

Q. 제 친구들은 저보다 친구들이 많은 것 같아요. 기분 탓인가요?

친구

Q. 제 친구들은 저보다 친구들이 많은 것 같아요. 기분 탓인가요?

A. 아니오. 수학적으로 당연합니다.

친구

수학적으로

친구

수학적으로

Q. 여러분 친구 몇 명?

친구

수학적으로

Q. 여러분 친구 몇 명? 1명이요. ㅜㅜ

1명이요. ㅜㅜ 1명이요. ㅜㅜ

3명이요. ㅎㅎ

친구

수학적으로



3명이요. ㅎㅎ

Q. 여러분 친구는 보통 친구가 몇 명?

친구

수학적으로



3명이요. ㅎㅎ


3명이요. ㅜㅜ


다들 1명(저)밖에.. ㅋㅋ

친구

수학적으로



3명이요. ㅎㅎ


3명이요. ㅜㅜ



평균 친구의 수 = (1 + 1 + 1 + 3)/4 = 1.5명

친구

수학적으로



3명이요. ㅎㅎ


3명이요. ㅜㅜ



평균 친구의 수 = (1 + 1 + 1 + 3)/4 = 1.5명

친구들의 평균 친구 수 = (3 + 3 + 3 + 1)/4 = 2.5명

친구

수학적으로



3명이요. ㅎㅎ


3명이요. ㅜㅜ



평균 친구의 수 = (1 + 1 + 1 + 3)/4 = 1.5명

친구들의 평균 친구 수 = (3 + 3 + 3 + 1)/4 = 2.5명

친구들의 평균 친구 수 ≥ 평균 친구의 수

원문: “Perhaps the most important of them is that your friends just are not normal. No one’s friends are. By the very fact of being someone’s friend, friends select themselves. Friends are by definition friendly people, and your circle of friends will be a biased sample of the population because of it.” - Mark E. J. Newman, Social Networks 25, 83 (2003).

“어쩌면 가장 중요한 건 여러분의 친구는 평범하지 않다는 것이다. 누구의 친구도 그렇지 않다. 누군가의 친구라는 것 자체만으로도 친구들은 그들 스스로를 선택하는 것이다. 친구들은 정의상 친근한 사람들이며, 당신의 친구들이라는 집단은 그것 때문에 특별한 집단이다.” - 마크 뉴먼, Social Networks 25, 83 (2003).

친구 역설 (friendship paradox)

친구 수, 친구 수, 친구 수 . . .

친구 수, 친구 수, 친구 수 . . .

던바의 숫자(Dunbar’s number): 개인이 사회적 관계를 안정적으로 유지할 수 있는 사람의 숫자 (150 정도)

물리학자들이 흔히 하는 짓거리 단순화: 연결망 또는 네트워크(network)


점/대상/사람/컴퓨터/… (노드: node)


점/대상/사람/컴퓨터/… (노드: node)선/연결/인간관계/인터넷/… (링크: link)



이웃의 수(degree)= 2




이웃의 수 = 2

이웃의 수 = 4

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3




이웃의 수 = 2

이웃의 수 = 4

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3

이웃 수의 분포 (degree distribution): 2가 2개, 3이 4개, 4가 1개

역시 보통 이런 건 수학자들이 미리 해 놓음.. 그래프 이론 (graph theory)

쾨니스베르크(Königsberg)의 다리 문제: “한붓그리기” 문제



레온하르트 오일러 (Leonhard Euler)

“홀수의 이웃을 가진 점의 개수가 0개 또는 2개가 되어야 한붓그리기가 가능하다.”





이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 5





이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 5

홀수의 이웃을 가진 점의 개수 = 4

그렇군요. 이웃 수는 생각보다 더 의미가 있네요. 그럼 이웃 수를 일단 세 봅시다.


친구 0명인 사람 N(0)명, 1명인 사람 N(1)명, …, k명인 사람 N(k)명, …


친구 0명인 사람 N(0)명, 1명인 사람 N(1)명, …, k명인 사람 N(k)명, …p(k) = N(k)/N페이스북 사용자를 무작위로 뽑았을 때 친구가 k명일 “확률”

(N: 전체 페이스북 사용자 수)

Q. 확률 분포가 어떤 모양일까요?


A. 던바의 숫자 정도에 몰려 있지 않을까요?



(150 정도)

정규분포(normal distribution)정규분포(normal distribution)



사람의 키 분포처럼?

(150 정도)

정규분포(normal distribution)정규분포(normal distribution)

p(k) / k��

p(k) / e�(k�kaverage)2/�2

실제로는 멱급수(power-law) 분포인 네트워크가 많습니다!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 3 4 5 6 7 8 9

p(k)

k

normal distributionpower-law distribution

p(k) / k��

p(k) / e�(k�kaverage)2/�2

실제로는 멱급수(power-law) 분포인 네트워크가 많습니다!

이웃이 어마어마하게 많은 것들 [마당발(hub), 바람둥이, …]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2 3 4 5 6 7 8 9

p(k)

k

normal distributionpower-law distribution

이웃 수: 정규(normal)분포 vs 멱급수(power-law) 분포


ab = c ! b = loga c

평균 이웃 수 <k> = 대표값


평균 이웃 수 <k>로는 대표가 안 됨!




줄어드는 정도: 기울기 (가 이 분포의 특성을 더 잘 나타냄!)

p(k) / k��





p(k) / k��

(어떤 k에 대해서도!) 2k명의 이웃을 가진 사람의 수는 k명의 이웃을 가진 사람의 절반, …





p(k) / k��

(어떤 k에 대해서도!) 2k명의 이웃을 가진 사람의 수는 k명의 이웃을 가진 사람의 절반, …

이웃 수의 크기(scale)와 관계없이 성립: 척도 없는 네트워크 (scale-free network)

네트워크들이 진짜 그렇게 생겼나요?인터넷 (랜선 연결)

네트워크들이 진짜 그렇게 생겼나요?인터넷 (하이퍼링크를 통한 웹페이지 연결)

네트워크들이 진짜 그렇게 생겼나요?공동연구 네트워크 (네트워크를 연구하는 과학자들 간의 네트워크 구조)

Albert

Albert

Nakarado

Barabasi

Jeong

Aleksiejuk

Holyst

Stauffer

Allaria

Arecchi

DigarboMeucci

Almaas

Kovacs

Vicsek

Oltvai

Krapivsky

Redner

Kulkarni

StroudAmaral

ScalaBarthelemy

Stanley

Meyers

Newman

Martin

Schrag

Antal

Arenas

Cabrales

Diaz−Guilera

Guimera

Vega−Redondo

DanonGleiser

Baiesi

Paczuski

BakSneppen

Banavar

Maritan

Rinaldo

Bianconi

Ravasz

Neda

Schubert

Barahona

Pecora

Barrat

Pastor−Satorras

Vespignani

Weigt

Gondran

Guichard

Battiston

Catanzaro

BenNaim

Frauenfelder

Toroczkai

Berlow

BernardesCosta

Araujo

Kertesz

Capocci

Boccaletti

Bragard

Mancini

Kurths

Valladares

Osipov

Zhou

Pelaez

Maza

Boguna

Bonanno

Lillo

Mantegna

Mendoza

Hentschel

Broder

Kumar

Maghoul

Raghavan

Rajagopalan

StataTomkins

Wiener

Bucolo

Fortuna

Larosa

Buhl

Gautrais

Sole

KuntzValverde

DeNeubourg

Theraulaz

CaldarelliDeLosRios

Munoz

Coccetti

CallawayHopcrof t

KleinbergStrogatz

Watts

Camacho

Servedio

Colaiori

Caruso

Latora

Rapisarda

Tadic

CastellanoVilone

ChatePikovsky

Rudzick

ChavezHwang

Amann

ClausetMoore

CohenBenAvraham

Havlin

Erez

Cosenza

Crucitt i

Frasca

Stagni

Usai

MarchioriPorta

DaFontouraCosta

DiAmbra

DeArcangelis

Herrmann

DeFraysseix

DeLucia

Bottaccio

Montuori

Pietronero

DeMenezes

Moukarzel

Penna

DeMoura

Motter

Grebogi

Dezso

Dobr in

Beg

Dodds

Muhamad

RothmanSabel

Donetti

Dorogovtsev

Goltsev

Mendes

Samukhin

Dunne

Williams Martinez

Echenique

GomezGardenes

Moreno

Vazquez

Ergun

Rodgers

Eriksen

SimonsenMaslov

Farkas

Derenyi

Ferrer−i−Cancho

JanssenKohler

Fink

Johnson

Carroll

Flake

Lawrence

Giles

Coetzee

Spata

Fortunato

Fronczak

Fronczak

Jedynak

Sienkiewicz

Garlaschelli

Castri

Loffredo

Gastner

Girvan

Goh

Ghim

Kahng

Kim

Lee

Oh

Floria

Gonzales

Sousa

Gorman

Gregoire

GrossKujala

Hamalainen

Timmermann

Schnitzler

Salmelin

Guardiola

Llas

Perez

Giralt

Mossa

Turtschi

Hari Ilmoniemi

Knuutila

Lounasmaa

Heagy

Herrmann

Provero

Hong

Roux

Holme

EdlingLiljeros

Ghoshal

Huss

Kim

YoonHan

Trusina

Minnhagen

Holter

Mitra

Cieplak

Fedroff

Hong

Choi

Park

LopezRuiz

Mason

Tombor

Jin

Jung

Kim

Park

Kalapala

Sanwalani

Chung

Kim

Kinney

Kumar

Leyvraz

SivakumarUpfal

Lahtinen

Kaski

Leone

Zecchina

Aberg

Liu

Lai

Hoppensteadt

Ye

Lusseau

Macdonald

Rigon

Giacometti

RodrigueziTurbe

Marodi

Dovidio

Marro

Dickman

Zaliznyak

Matthews

Mirollo

Vallone

Montoya

Moreira

AndradeGomez

Pacheco

Nekovee

VazquezPrada

Dasgupta

Nishikawa

Forrest

Balthrop

Leicht

Rho

Onnela

Chakraborti

Kanto

Jarisaramaki

RosenblumBassler

Corral

Park

Rubi

Smith

Pennock

Glover

Petermannn

Pluchino

PodaniSzathmary

Porter

Mucha

Warmbrand

RadicchiCecconi

Loreto

Parisi

Ramasco

Somera

Mongru

DarbyDowman

Rosvall

Rozenfeld

Schafer

Abel

Schwartz

Shefi

Golding

Segev

BenJacob

Ayali

Soffer

Kepler

Salazarciudad

Garciafernandez

Song

Makse

AharonyAdlerMeyerOrtmanns

Szabo Alava

Thurner

TassWeule

Volkmann

Freund

Tieri

Valensin

Castellani

Remondini

Franceschi

Kozma

Hengartner

Korniss

Torres

Garrido

Cancho

Vannucchi

Flammini

Vazquez

Czirok

Cohen

Shochet

Vragovic

Louis

Wuchty

Yeung

Yook

Tu

Yusong

Lingjiang

Muren

Zaks

Park

Collaborations Between Network Scientists

This figure shows a network of collaborationsbetween scientists working on networks. Itwas compiled from the bibliographies of tworeview articles, by M. Newman (SIAM Review2003) and by S. Boccaletti et al. (Physics Re-ports 2006). Vertices represent scientists whosenames appear as authors of papers in those bib-liographies and an edge joins any two whosenames appear on the samepaper. A small num-ber of other references were added by handto bring the network up to date. This figureshows the largest component of the resultingnetwork, which contains 379 individuals. Sizesof vertices are proportional to their so-called“community centrality.” Colors represent ver-tex degrees with redder vertices having higherdegree.

네트워크들이 진짜 그렇게 생겼나요?공동연구 네트워크 (네트워크를 연구하는 과학자들 간의 네트워크 구조)

Albert

Albert

Nakarado

Barabasi

Jeong

Aleksiejuk

Holyst

Stauffer

Allaria

Arecchi

DigarboMeucci

Almaas

Kovacs

Vicsek

Oltvai

Krapivsky

Redner

Kulkarni

StroudAmaral

ScalaBarthelemy

Stanley

Meyers

Newman

Martin

Schrag

Antal

Arenas

Cabrales

Diaz−Guilera

Guimera

Vega−Redondo

DanonGleiser

Baiesi

Paczuski

BakSneppen

Banavar

Maritan

Rinaldo

Bianconi

Ravasz

Neda

Schubert

Barahona

Pecora

Barrat

Pastor−Satorras

Vespignani

Weigt

Gondran

Guichard

Battiston

Catanzaro

BenNaim

Frauenfelder

Toroczkai

Berlow

BernardesCosta

Araujo

Kertesz

Capocci

Boccaletti

Bragard

Mancini

Kurths

Valladares

Osipov

Zhou

Pelaez

Maza

Boguna

Bonanno

Lillo

Mantegna

Mendoza

Hentschel

Broder

Kumar

Maghoul

Raghavan

Rajagopalan

StataTomkins

Wiener

Bucolo

Fortuna

Larosa

Buhl

Gautrais

Sole

KuntzValverde

DeNeubourg

Theraulaz

CaldarelliDeLosRios

Munoz

Coccetti

CallawayHopcrof t

KleinbergStrogatz

Watts

Camacho

Servedio

Colaiori

Caruso

Latora

Rapisarda

Tadic

CastellanoVilone

ChatePikovsky

Rudzick

ChavezHwang

Amann

ClausetMoore

CohenBenAvraham

Havlin

Erez

Cosenza

Crucitt i

Frasca

Stagni

Usai

MarchioriPorta

DaFontouraCosta

DiAmbra

DeArcangelis

Herrmann

DeFraysseix

DeLucia

Bottaccio

Montuori

Pietronero

DeMenezes

Moukarzel

Penna

DeMoura

Motter

Grebogi

Dezso

Dobr in

Beg

Dodds

Muhamad

RothmanSabel

Donetti

Dorogovtsev

Goltsev

Mendes

Samukhin

Dunne

Williams Martinez

Echenique

GomezGardenes

Moreno

Vazquez

Ergun

Rodgers

Eriksen

SimonsenMaslov

Farkas

Derenyi

Ferrer−i−Cancho

JanssenKohler

Fink

Johnson

Carroll

Flake

Lawrence

Giles

Coetzee

Spata

Fortunato

Fronczak

Fronczak

Jedynak

Sienkiewicz

Garlaschelli

Castri

Loffredo

Gastner

Girvan

Goh

Ghim

Kahng

Kim

Lee

Oh

Floria

Gonzales

Sousa

Gorman

Gregoire

GrossKujala

Hamalainen

Timmermann

Schnitzler

Salmelin

Guardiola

Llas

Perez

Giralt

Mossa

Turtschi

Hari Ilmoniemi

Knuutila

Lounasmaa

Heagy

Herrmann

Provero

Hong

Roux

Holme

EdlingLiljeros

Ghoshal

Huss

Kim

YoonHan

Trusina

Minnhagen

Holter

Mitra

Cieplak

Fedroff

Hong

Choi

Park

LopezRuiz

Mason

Tombor

Jin

Jung

Kim

Park

Kalapala

Sanwalani

Chung

Kim

Kinney

Kumar

Leyvraz

SivakumarUpfal

Lahtinen

Kaski

Leone

Zecchina

Aberg

Liu

Lai

Hoppensteadt

Ye

Lusseau

Macdonald

Rigon

Giacometti

RodrigueziTurbe

Marodi

Dovidio

Marro

Dickman

Zaliznyak

Matthews

Mirollo

Vallone

Montoya

Moreira

AndradeGomez

Pacheco

Nekovee

VazquezPrada

Dasgupta

Nishikawa

Forrest

Balthrop

Leicht

Rho

Onnela

Chakraborti

Kanto

Jarisaramaki

RosenblumBassler

Corral

Park

Rubi

Smith

Pennock

Glover

Petermannn

Pluchino

PodaniSzathmary

Porter

Mucha

Warmbrand

RadicchiCecconi

Loreto

Parisi

Ramasco

Somera

Mongru

DarbyDowman

Rosvall

Rozenfeld

Schafer

Abel

Schwartz

Shefi

Golding

Segev

BenJacob

Ayali

Soffer

Kepler

Salazarciudad

Garciafernandez

Song

Makse

AharonyAdlerMeyerOrtmanns

Szabo Alava

Thurner

TassWeule

Volkmann

Freund

Tieri

Valensin

Castellani

Remondini

Franceschi

Kozma

Hengartner

Korniss

Torres

Garrido

Cancho

Vannucchi

Flammini

Vazquez

Czirok

Cohen

Shochet

Vragovic

Louis

Wuchty

Yeung

Yook

Tu

Yusong

Lingjiang

Muren

Zaks

Park

Collaborations Between Network Scientists

This figure shows a network of collaborationsbetween scientists working on networks. Itwas compiled from the bibliographies of tworeview articles, by M. Newman (SIAM Review2003) and by S. Boccaletti et al. (Physics Re-ports 2006). Vertices represent scientists whosenames appear as authors of papers in those bib-liographies and an edge joins any two whosenames appear on the samepaper. A small num-ber of other references were added by handto bring the network up to date. This figureshows the largest component of the resultingnetwork, which contains 379 individuals. Sizesof vertices are proportional to their so-called“community centrality.” Colors represent ver-tex degrees with redder vertices having higherdegree.

Critical behavior of the Ising model in annealed scale-free networks

Sang Hoon Lee,1 Meesoon Ha,1,2 Hawoong Jeong,1,3 Jae Dong Noh,4 and Hyunggyu Park2

1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea2School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

3Institute for the BioCentury, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea4Department of Physics, University of Seoul, Seoul 130-743, Korea

!Received 8 September 2009; published 25 November 2009"

We study the critical behavior of the Ising model in annealed scale-free !SF" networks of finite system sizewith forced upper cutoff in degree. By mapping the model onto the weighted fully connected Ising model, wederive analytic results for the finite-size scaling !FSS" near the phase transition, characterized by the cutoff-dependent two-parameter scaling with four distinct scaling regimes, in highly heterogeneous networks. Theseresults are essentially the same as those found for the nonequilibrium contact process in annealed SF networks,except for an additional complication due to the trivial critical point shift in finite systems. The discrepancy ofthe FSS theories between annealed and quenched SF networks still remains in the equilibrium Ising model, likesome other nonequilibrium models. All of our analytic results are confirmed reasonably well by numericalsimulations.

DOI: 10.1103/PhysRevE.80.051127 PACS number!s": 64.60.Cn, 89.75.Fb, 89.75.Hc

I. INTRODUCTION

Many aspects of our real world have been understood inthe context of complex networks #1,2$ and simple physicalmodels of critical phenomena on networks. Contrary to regu-lar lattices in the Euclidean space, complex networks arecharacterized by a highly heterogeneous structure as mani-fested in broad degree distributions. Recent studies on equi-librium or nonequilibrium systems have revealed that theheterogeneity is one of essential ingredients determining theuniversal feature of phase transitions and critical phenomena#3$.

The concept of the phase transition is well defined only inthe thermodynamic limit where the system size is taken toinfinity. So it is important to understand how finite-size ef-fects come into play near the transition. Such a task forphysical models on regular lattices has been successfully ac-complished by the standard finite-size scaling !FSS" theory#4$, based on the ansatz that a single characteristic lengthscale !correlation length" ! competes with the system’s linearsize L. Then, any physical observable depends only on adimensionless variable !=L /! in the scaling limit. Near asecond-order continuous transition, the correlation length di-verges as !%&"&−# with the reduced coupling constant " andthe finite-size effects become prominent.

The FSS theory for complex networks can be formulatedin a similar way: Since the Euclidean distance is undefined incomplex networks, one may take the volume scaling variableas !v=N /!v with the system size N !the total number ofnodes" and the correlated volume !v. The correlated volumediverges !v%&"&−# near the transition !#=#d in d dimensionallattices". For example, the magnetization of the Ising modelscales as

m!",N" = N−$/#%!"N1/#" , !1"

where the scaling function %!x"%O!1" for small x and x$ forlarge x with the order parameter exponent $.

The FSS theory with a single characteristic size has beentested numerically in many systems !see Ref. #3$ and refer-ences therein". In particular, the exact values for the FSSexponent # are conjectured #5$ by estimating the correlatedvolume !droplet" size for the nonequilibrium contact process!CP" and the equilibrium Ising model in random uncorrelatednetworks with static links, which are denoted as quenchednetworks.

However, considering a highly heterogeneous scale-free!SF" network, one should take into account not only a broaddegree distribution of P!k"%k−& but also the upper cutoff kcin degree, which scales as kc%N1/'. Without any constraint,kc is bounded naturally with 'nat=&−1. In general, one mayimpose a forced cutoff with '('nat. In the thermodynamiclimit, both N and kc diverge simultaneously and ' sets aroute to the limit. Therefore, one can suspect that the FSStheory may depend on the routes or equivalently on the valueof ', especially for networks with a broader distribution forsmall &.

For the quenched SF networks, it has been suggested thatthe FSS does not vary with ' for a weak forced cutoff !')&", which was confirmed numerically in various types ofSF networks #5,6$. However, in the annealed networks wherelinks are not fixed but fluctuate randomly in time, it wasrigorously shown that the CP model exhibits an anomalousFSS for any forced cutoff with 2)&)3 where a heteroge-neity !&"-dependent critical scaling appears #7–10$. More-over, the anomalous FSS is characterized by a cutoff!'"-dependent and two-parameter scaling with four distinctscaling regimes #10$, in contrast to the cutoff-independentand single-parameter scaling with three scaling regimes inthe standard FSS theory.

The anomalous FSS of the CP in the annealed SF net-works gives rise to a natural question: What is the mainingredient causing the anomaly? Some possible guesses maybe a nonequilibrium feature of the CP, absorbing nature !van-ishing activity" at criticality, or heterogeneity of networks#8,9$. In this paper, we answer to this question by studying

PHYSICAL REVIEW E 80, 051127 !2009"

1539-3755/2009/80!5"/051127!10" ©2009 The American Physical Society051127-1

네트워크들이 진짜 그렇게 생겼나요?생물체의 물질 대사 (metabolic) 네트워크

네트워크들이 진짜 그렇게 생겼나요?두뇌(brain)의 신경(neural) 연결 네트워크

네트워크들이 진짜 그렇게 생겼나요?두뇌(brain)의 신경(neural) 연결 네트워크

“우주에서 가장 복잡하다고 자기 자신에게 알려진 시스템 (the most complicated system in the universe known to itself)”

척도 없는 네트워크 (scale-free network): 왜 이렇게 여기저기 많을까요?

척도 없는 네트워크 (scale-free network): 왜 이렇게 여기저기 많을까요?

node failure

fc

0 1 Fraction of removed nodes, f

1 Relative size of largest cluster

S

제거된 노드의 비율

가장 큰 연결된 덩어리 (giant component) 의

상대적 크기

노드 제거 (고장)

무작위 네트워크 vs 척도 없는 네트워크

무작위 네트워크 vs 척도 없는 네트워크

오류 또는 공격에 대한 안정성(stability)

letters to nature

380 NATURE | VOL 406 | 27 JULY 2000 | www.nature.com

are believed to have a diameter of around six21. To compare the twonetwork models properly, we generated networks that have the samenumber of nodes and links, such that P(k) follows a Poissondistribution for the exponential network, and a power law for thescale-free network.

To address the error tolerance of the networks, we study thechanges in diameter when a small fraction f of the nodes is removed.The malfunctioning (absence) of any node in general increases thedistance between the remaining nodes, as it can eliminate somepaths that contribute to the system’s interconnectedness. Indeed, forthe exponential network the diameter increases monotonically withf (Fig. 2a); thus, despite its redundant wiring (Fig. 1), it is increas-ingly difficult for the remaining nodes to communicate with eachother. This behaviour is rooted in the homogeneity of the network:since all nodes have approximately the same number of links, theyall contribute equally to the network’s diameter, thus the removal ofeach node causes the same amount of damage. In contrast, weobserve a drastically different and surprising behaviour for thescale-free network (Fig. 2a): the diameter remains unchanged underan increasing level of errors. Thus even when as many as 5% of

the nodes fail, the communication between the remaining nodesin the network is unaffected. This robustness of scale-free net-works is rooted in their extremely inhomogeneous connectivitydistribution: because the power-law distribution implies that themajority of nodes have only a few links, nodes with smallconnectivity will be selected with much higher probability. Theremoval of these ‘small’ nodes does not alter the path structure ofthe remaining nodes, and thus has no impact on the overall networktopology.

An informed agent that attempts to deliberately damage a net-work will not eliminate the nodes randomly, but will preferentiallytarget the most connected nodes. To simulate an attack we firstremove the most connected node, and continue selecting andremoving nodes in decreasing order of their connectivity k. Measur-ing the diameter of an exponential network under attack, we findthat, owing to the homogeneity of the network, there is nosubstantial difference whether the nodes are selected randomly orin decreasing order of connectivity (Fig. 2a). On the other hand, adrastically different behaviour is observed for scale-free networks.When the most connected nodes are eliminated, the diameter of thescale-free network increases rapidly, doubling its original value if5% of the nodes are removed. This vulnerability to attacks is rootedin the inhomogeneity of the connectivity distribution: the connec-tivity is maintained by a few highly connected nodes (Fig. 1b),whose removal drastically alters the network’s topology, and

0.00 0.04 0.08 0.120

1

2

3

0.0 0.10

1

<s>

and

S

0.0 0.2 0.40

1

2

0.0 0.2 0.40

1

2

10–1

100

101

102

0.0 0.4 0.80

1

f

ba

fcfc

FailureAttack

S <s>

InternetWWW

SFE

dc

Figure 3 Network fragmentation under random failures and attacks. The relative size ofthe largest cluster S (open symbols) and the average size of the isolated clusters ⟨s⟩ (filledsymbols) as a function of the fraction of removed nodes f for the same systems as inFig. 2. The size S is defined as the fraction of nodes contained in the largest cluster (that is,S ¼ 1 for f ¼ 0). a, Fragmentation of the exponential network under random failures(squares) and attacks (circles). b, Fragmentation of the scale-free network under randomfailures (blue squares) and attacks (red circles). The inset shows the error tolerance curvesfor the whole range of f, indicating that the main cluster falls apart only after it has beencompletely deflated. We note that the behaviour of the scale-free network under errors isconsistent with an extremely delayed percolation transition: at unrealistically high errorrates ( f max ! 0:75) we do observe a very small peak in ⟨s⟩ (⟨smax⟩ ! 1:06) even in thecase of random failures, indicating the existence of a critical point. For a and b werepeated the analysis for systems of sizes N ¼ 1;000, 5,000 and 20,000, finding that theobtained S and ⟨s⟩ curves overlap with the one shown here, indicating that the overallclustering scenario and the value of the critical point is independent of the size of thesystem. c, d, Fragmentation of the Internet (c) and WWW (d), using the topological datadescribed in Fig. 2. The symbols are the same as in b. ⟨s⟩ in d in the case of attack isshown on a different scale, drawn in the right side of the frame. Whereas for small f wehave ⟨s⟩ ! 1:5, at f w

c ¼ 0:067 the average fragment size abruptly increases, peaking at⟨smax⟩ ! 60, then decays rapidly. For the attack curve in d we ordered the nodes as afunction of the number of outgoing links, kout. We note that while the three studiednetworks, the scale-free model, the Internet and the WWW have different g, ⟨k⟩ andclustering coefficient11, their response to attacks and errors is identical. Indeed, we findthat the difference between these quantities changes only fc and the magnitude of d, Sand ⟨s⟩, but not the nature of the response of these networks to perturbations.

Exponentialnetwork

Scale-freenetwork(WWW,Internet)

Attack

Failure

Failure

Attack

f ≈ 0.05 f ≈ 0.18 f ≈ 0.45

100 101 102 103 0 2 4

100

10–2

100

10–2

100

10–1

10–4

10010–6

10–4

102 104

100 101 102 103

10–4

100

10–2

10–4

100

10–2

100 102 104 100 102 104

100

10–2

10–4

10–6

a b c

d e f

fc

Figure 4 Summary of the response of a network to failures or attacks. a–f, The clustersize distribution for various values of f when a scale-free network of parameters given inFig. 3b is subject to random failures (a–c) or attacks (d–f). Upper panels, exponentialnetworks under random failures and attacks and scale-free networks under attacksbehave similarly. For small f, clusters of different sizes break down, although there is still alarge cluster. This is supported by the cluster size distribution: although we see a fewfragments of sizes between 1 and 16, there is a large cluster of size 9,000 (the size of theoriginal system being 10,000). At a critical fc (see Fig. 3) the network breaks into smallfragments between sizes 1 and 100 (b) and the large cluster disappears. At even higher f(c) the clusters are further fragmented into single nodes or clusters of size two. Lowerpanels, scale-free networks follow a different scenario under random failures: the size ofthe largest cluster decreases slowly as first single nodes, then small clusters break off.Indeed, at f ¼ 0:05 only single and double nodes break off (d). At f ¼ 0:18, the networkis fragmented (b) under attack, but under failures the large cluster of size 8,000 coexistswith isolated clusters of sizes 1 to 5 (e). Even for an unrealistically high error rate off ¼ 0:45 the large cluster persists, the size of the broken-off fragments not exceeding11 (f).

© 2000 Macmillan Magazines Ltd

공격

고장

고장무작위 네트워크

척도 없는 네트워크

공격

척도 없는 네트워크 (scale-free network): 마당발(hub)의 연결 중요성

http://www.youtube.com/watch?v=r8RZZyrV9Lo

Scale-Free Networks Are Ultrasmall

Reuven Cohen* and Shlomo HavlinMinerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, Israel

(Received 1 July 2002; published 4 February 2003)

We study the diameter, or the mean distance between sites, in a scale-free network, having N sitesand degree distribution p!k" / k#!, i.e., the probability of having k links outgoing from a site. Incontrast to the diameter of regular random networks or small-world networks, which is known to bed$ lnN, we show, using analytical arguments, that scale-free networks with 2< !< 3 have a muchsmaller diameter, behaving as d$ lnlnN. For ! % 3, our analysis yields d$ lnN= lnlnN, as obtained byBollobas and Riordan, while for ! > 3, d$ lnN. We also show that, for any ! > 2, one can construct adeterministic scale-free network with d$ lnlnN, which is the lowest possible diameter.

DOI: 10.1103/PhysRevLett.90.058701 PACS numbers: 89.75.Hc, 02.50.–r, 89.75.Da

It is well known [1–4] that random networks, such asErdos-Renyi networks [5,6] as well as partially randomnetworks such as small-world networks [7], have a verysmall average distance (or diameter) between sites, whichscales as d$ lnN, where N is the number of sites. Sincethe diameter is small even for large N, it is common torefer to such networks as ‘‘small-world’’ networks. Manynatural and manmade networks have been shown topossess a scale-free degree distribution, includingthe Internet [8], World Wide Web [3,9], metabolic [10]and cellular networks [11], and trust cooperation net-works [12].

The question of the diameter of such networks isfundamental in the study of networks. It is relevant inmany fields regarding communication and computer net-works, such as routing [13], searching [14], and transportof information [13]. All those processes become moreefficient when the diameter is smaller. It also might berelevant to subjects such as the efficiency of chemical andbiochemical processes and spreading of viruses, rumors,etc., in cellular, social, and computer networks. Inphysics, the scaling of the diameter with the networksize is related to the physical concept of the dimension-ality of the system and is highly relevant to phenomenasuch as diffusion, conduction, and transport. The anom-alous scaling of the diameter in those networks is ex-pected to lead to anomalies in diffusion and transportphenomena on those networks. In this Letter we study thediameter of scale-free random networks and show thatit is significantly smaller than the diameter of regularrandom networks. We find that scale-free networks with2< !< 3 have diameter d$ lnlnN and thus can be con-sidered as ‘‘ultra small-world’’ networks.

We define the diameter of a graph as the average dis-tance between any two sites on the graph (unlike the usualmathematical definition of the largest distance betweentwo sites). Since no embedding space is defined for thosenetworks, the distance denotes the shortest path betweentwo sites (i.e., the smallest number of followed linksneeded to reach one from the other). If the network is

fragmented we are interested only in the diameter of thelargest cluster (assuming there is one).

To estimate the diameter we study the radius of suchgraphs. We define the radius of a graph as the averagedistance of all sites on the graph from the site with thehighest degree in the network (if there is more than one,we arbitrarily choose one of them). The diameter of thegraph, d, is restricted to

r & d & 2r; (1)

where r is the radius of the graph, defined as the averagedistance hli between the highest degree site (the origin)and all other sites.

A scale-free graph is a graph having degree distribu-tion, i.e., the probability that a site has k connections:

p!k" % ck#!; k % m;m' 1; . . . ; K; (2)

where c ( !!# 1"m!#1 is a normalization factor, and mand K are the lower and upper cutoffs of the distribution,respectively. The ensemble of such graphs has been de-fined in [15]. However, we refer here to the ensembleof scale-free graphs with the ‘‘natural’’ cutoff K %mN1=!!#1" [16–18].

We begin by showing that the lower bound on thediameter of any scale-free graph with ! > 2 is of theorder of lnlnN; then we show that for random scale-freegraphs with 2< !< 3 the diameter actually scales aslnlnN. It is easy to see that the lowest diameter for agraph with a given degree distribution is achieved by thefollowing construction: Start with the highest degree site,and then in each layer attach the next highest degree sitesuntil the layer is full. By construction, loops occur only inthe last layer. This structure is somewhat similar to agraph with assortative mixing [19]—since high degreesites tend to connect to other high degree sites.

In this kind of graph, the number of links outgoingfrom the lth layer (sites at distance l from the origin), "l,equals the total number of sites with degree between Kl,

P H Y S I C A L R E V I E W L E T T E R S week ending7 FEBRUARY 2003VOLUME 90, NUMBER 5

058701-1 0031-9007=03=90(5)=058701(4)$20.00 © 2003 The American Physical Society 058701-1

http://www.youtube.com/watch?v=r8RZZyrV9Lo

이웃(친구)의 수: 국소적(local) 특성

일촌의 수 (degree)

이제 공부를 좀 했으니 이 예제를 다시 봅시다.

이웃의 수 = 2

이웃의 수 = 2

이웃의 수 = 4

이웃의 수= 3

이웃의 수 = 3

이웃의 수 = 3

이웃의 수 = 3


멀리서 볼까요?

대~충 보면..?

대~충 보면..?

대~충 잘 연결된 3개짜리 그룹

대~충 보면..?



대~충 보면..?


대~충 잘 연결된 4개짜리 그룹통계적으로

통계적으로

통계적으로

대~충 보면..?


대~충 잘 연결된 4개짜리 그룹통계적으로

통계적으로

통계적으로

중간 크기 성질 (mesoscopic property)

삼각관계

삼총사

삼총사도원결의 확률 = (유비가 관우와 친구가 될 확률)x(관우와 장비가 친구가 될 확률)x(장비가 유비와 친구가 될 확률)?!

장비관우

유비의 페이스북 친구 추천

장비관우


도원결의 확률 = (유비가 관우와 친구가 될 확률)x(관우와 장비가 친구가 될 확률)x(장비가 유비와 친구가 될 확률 >>>>>>>> 관우 없이 장비와 유비가 친구가 될 확률)

장비관우


도원결의 확률 = (유비가 관우와 친구가 될 확률)x(관우와 장비가 친구가 될 확률)x(장비가 유비와 친구가 될 확률 >>>>>>>> 관우 없이 장비와 유비가 친구가 될 확률)

뭉쳐 다니는 성질 (clustering):대충 통계적으로 유의미하게 뭉쳐진 집단

책 추천

여기에도 네트워크가?


양자간 (bipartite) 네트워크


대충 통계적으로 비슷한 사람들!



“공동출연 배우 네트워크”에서의 링크


“공동출연 배우 네트워크”에서의 링크“공동출연 배우 네트워크”에서의 그룹 (clique/hyperedge)

양자간 네트워크로부터 투영된(projected) 공동출연 배우 네트워크


속한 그룹의 수(“hyper”degree) = 2


네트워크를 청소년들에게 소개: 영국 옥스퍼드대 수학과의 교육 봉사활동http://www.youtube.com/watch?v=9dcdjcyA-8E

http://www.youtube.com/watch?v=9dcdjcyA-8E

다 좋습니다. 그런데 이것들이 물리학과 무슨 관련이?!

통계물리학: 미시적(micro) 성질 → 상호작용(interactions) → 거시적(macro) 성질

완벽하게 규칙적이거나 완벽하게 무작위인 “네트워크” 관계

자석 기체

미시적 성질 (원자 영역에서 이미 N/S 극)

거시적 성질 (자석)

미시적 성질 (원자, 분자)

거시적 성질 (압력밥솥)

규칙과 무작위가 섞인 복잡한(complex) 네트워크

완벽하게 규칙적이거나 완벽하게 무작위인 “네트워크” 관계

“물리학자들은 다른 사람들의 학문을 침범하기에 더 없이 적합한 사람들이다. 물론 대단히 똑똑한 탓도

있지만 일반적으로 연구대상에 대해 그다지 까다롭게 굴지 않기 때문이다. 물리학자들은 스스로를 아카

데미라는 정글의 제왕쯤으로 생각하는 경향이 있고, 자신들의 방법이 일반의 수준보다 높다고 여기면서

자신들의 영토를 물샐틈없이 수호한다. 하지만 그들의 또 다른 자아는 하이에나에 비견될 만한 것이어서,

쓸모가 있을 것 같으면 생각이나 기법을 기꺼이 빌려오고 남들이 풀지 못했던 문제의 뿌리를 뽑으며 즐거

워한다. 이런 태도는 약삭빨라 보일 수도 있지만 이전까지 물리학이 제외되어 있던 영역에 그들이 등장하

면서 위대한 발견이나 자극으로 이어지는 경우가 많다. 수학자들이 가끔 비슷한 행동을 하기는 해도 새로

운 문제의 냄새를 맡고 흥분한 굶주린 물리학자들처럼 맹렬하게 덤벼들지는 않는다…”

- 던컨 와츠 (Duncan J. Watts), “Small World: 여섯 다리면 건너면 누구와도 연결된다 (Six Degrees)”

강연 들어주셔서 감사합니다! =)

더 알고 싶으신 분들을 위한 책 추천

(혹시나!)제가 실제로 요즘 어떤 연구들을 하는지 관심이 있으시다면 …?!

https://sites.google.com/site/lshlj82/publications

2

grid element as

maxB2nnhd(A)

|r f (A, B)||ri(A, B)| , (2)

where nnhd(A) is the set of nodes in the nearest neighborsof A (four for each grid point) and Fig. S2 in SupplementalMaterial [8] shows the relative dispersion map for the same

simulation as in Fig. 1.Another way to define the relative dispersion is to use the

deformation gradient tensor [1] to get the instantaneous dis-placement instead of the final position by

W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)

where F(A) is the deformation gradient tensor at A, i.e.,

|F(A)ri(A, B)| =q{Fxx(A)[xi(B) � xi(A)] + Fxy(A)[yi(B) � yi(A)]}2 + {Fyx(A)[xi(B) � xi(A)] + Fyy(A)[yi(B) � yi(A)]}2. (4)

For all the measures, the distance measures such as ri(A, B)and the coordinates such as ri(A) take the shortest distanceamong all the possible distances considering the PBC, i.e., xitself and x± 2⇡ for x, and y itself and y± 2⇡ for y (hence ninecombinations in total).

Community Detection.—For W (1)AB(= W (1)

BA) in Eq. (1), theLouvain method [9, 10] with the Girvan-Newman nullmodel [11] and the resolution parameter � [12] is used forFig. S2 in Supplemental Material [8], where the number ofcommunities and the values of quality measure QGN are spec-ified for four di↵erent � values. The communities here de-scribe the groups of nodes where the intra-group interactionsare significantly stronger than the inter-group interactions.For the resolution parameter �, roughly speaking, the largerthe value of � is, the smaller (in terms of typical number of

FIG. 2. (color online). The ten communities represented by dif-ferent colors are found from the same simulated data illustrated inFig. 1, with the nearest neighbor interactions. The relative dispersionin Eq. (1) and the modularity in Eq. (5) with the resolution parameter� = 0.005 are used.

nodes in a community, thus larger number of communities)communities are identified.

The modularity for the Girvan-Newman null model, whichis the objective function QGN where the purpose is to find theset of communities {gA} that maximizes QGN, is given by

QGN =1

2m

X

AB

W (1)

AB � �kAkB

2m

!� (gA, gB) , (5)

where A and B are node indices, kA =P

B W (1)AB =

PB W (1)

BAis the sum of weights corresponding to the interactions con-nected to A, 2m =

PA kA is the total sum of weights in all the

interactions, � is the resolution parameter, �(gA, gB) = 1 if Aand B are in the same community and 0 otherwise, and theoverall factor 1/(2m) is used for the normalization conditionQ 2 [�1, 1].

For W (2)AB(, W (2)

BA) in Eq. (3), the Louvain method [9, 10]with the Leicht-Newman null model [13] and the resolutionparameter � is used for Fig. S3 in Supplemental Material [8],where the number of communities and the values of qualitymeasure QLN are specified for four di↵erent � values. Thecommunities here describe similar structures based on the rel-ative strength di↵erence between intra-group and inter-groupinteractions, but since the Leicht-Newman null model [13]considers the direction of the interactions, the method tendsto split the “source” and “sink” groups (but see Ref. [14]).

The modularity for the Leicht-Newman null model, whichis the objective function QLN where the purpose is to find theset of communities {gA} that maximizes QLN, is given by

QLN =1m

X

AB

0BBBB@W (2)

AB � �kin

A koutB

m

1CCCCA � (gA, gB) , (6)

where A and B are node indices, kinA =

PB W (2)

BA (koutA =

PB W (2)

AB) is the sum of incoming (outgoing) weights cor-responding to the interactions connected to A, respectively,m =

PA kin

A =P

A koutA is the total sum of weights in all the

interactions (same for incoming and outgoing weights), � isthe resolution parameter, �(gA, gB) = 1 if A and B are in thesame community and 0 otherwise, and the overall factor 1/mis used for the normalization condition Q 2 [�1, 1].

6

!

"

#

$

%

&

FIG. 1: (A) One of the fungal networks formed by Phanerochaete velutina after 30 days of growth across a compressed black-sand substratefrom a pentagonal arrangement of wood-block inocula. (B) Path of radiolabeled nutrient (14C-amino-isobutyrate) added at 30 days and imagedusing photon-counting scintillation imaging for 12 hours. (C) Merged overlay of panels (A) and (B) to highlight the path that is followedby the radiolabel. (D) We colour the edges of the manually-digitised network according to the logarithm of the conductance values. Edgethickness represents cord thickness. (E) We colour the edges according to the path score (PS) values of the fungal network. (F) MRF curvesfor conductance-based and PS-based weights. We show MRF curves for e↵ective energy (He↵), e↵ective entropy (S e↵), and e↵ective numberof communities (⌘e↵). See [15] for details, and note that the energy is proportional to the negative of optimised modularity. (In panels (D) and(E), the edges include nodes with degree 2 (as they are needed to trace the curvature of the cords). In the MRF analysis, we remove the nodeswith degree k = 2, and we adjust the weight of the edge that connects the remaining nodes to include the values of the intermediate links foreach k = 2 segment.

4

FIG. 1. Examples of (a) ergodic and (b) nonergodic clans. We color the regions of South Korea based on the fraction of the total populationcomposed of members of the clan in the year 2000. We use arrows to indicate the origins of the two clans: Gimhae on the left and Ulsan(“Hakseong” is the old name of the city) on the right. In this map, we use the 2010 administrative boundaries [38]. See the appendices fordiscussions of data sets and data cleaning.

marries from clan i into any other clan, N is the total popu-lation, and si j is the exclusive population within a circle ofradius ri j centered on the centroid of clan i. Note that mem-bers of clans i and j are not included in computing si j [43]. Asbefore, mi is the population of clan i, members of clan i marryinto clan j, and clan j keeps the marriage records. In contrastto the gravity model, the radiation model does not include anyexternal parameters. Importantly, this renders it unable to de-scribe the geographically-independent situation that we needto consider in our study (and which we can obtain by setting� = 0 in the gravity model).

For both the gravity and radiation models, we use censusdata from the year 2000 [37] as a proxy for past populations.This allows us to compute the quantities ri j, mi, and si j. Ourapproximation is supported by previously reported estimatesof stability in Korean society. Historically, most clans havegrown in parallel with the total population, so we assumethat the relative sizes of clans have remained roughly constant[23]. In both Eqs. (1) and (2), only the relative sizes mi/Nand si j/N matter for calculating the flux (up to a constant ofproportionality).

B. Human di↵usion and ergodicity analysis

One way to quantify the notion of clan ergodicity is toexamine what we call the “clan-density anomaly”, whichdescribes the local deviation in density of members of a

given clan. The clan-density anomaly is �i(r, t) = ci(r, t) �[mi(t)/N(t)]⇢(r, t) at position r = (x, y) and time t, whereci(r, t) is the (spatially and temporally varying) local clan con-centration (i.e., the clan population density), mi(t) is the totalclan population, ⇢(r, t) is the local population density (i.e.,the total population of all clans at point r and time t, di-vided by the di↵erential area), and N(t) is the total popula-tion of all of the clans at time t. If a clan were to occupy aconstant fraction of the population everywhere in the coun-try, then �i = 0 everywhere because its local concentrationwould be ci = (mi/N)⇢. (This situation corresponds to perfectergodicity.) The range of typical values for the clan-densityanomaly depends on a clan’s aggregate concentration in thecountry. Examining the anomaly relative to clan concentra-tion, the year-2000 numbers for �i/(mi⇢/N) range from �1700to 7400 for Kim from Gimhae and from �19000 to 87000 forLee from Hakseong. Clearly, the distribution of the latter ismuch more heterogeneous (see Fig. 17 in Appendix I).

Combining the notion of clan-density anomaly with tra-ditional arguments—flow ideas based on Ohm’s law and“molecular weights for population” are mentioned explicitlyin [6, 10]—about migration from population gradients [2–10] suggests a simple Fickian law [51] for human transporton long time scales. We propose that the flux of clan mem-bers is Ji / r�i, so individuals move preferentially awayfrom high concentrations of their clans. This implies that@ci/@t = r · Ji / r2�i (where we have assumed that the con-stant of proportionality is independent of space), which yields

4

!!!!!!!

!

!!!!!!!!!!!!!!!

Entrance*hole*2*

Initial,area,–,2,rabbits,

Not!sure!–!failed!bolt!hole?!

Collapsed!+!widened!out.!Could!be!passing!place!

Secondary!entrance!

Exploratory!tunnel!–!hit!rock?!

Secondary,area,

Tertiary,area,or,extension,of,secondary,

PRIMARY*HUB*

SECONDARY*HUB*

A,B,

C,

Resting,area,

Entrance*hole*1*

Entrance*hole*3*

FIG. 2: (color online) The photo (courtesy of Hannah Sneyd) of rabbit warren excavation site, where some known characteristic regions aremarked.

!

Entrance!hole!1!

Entrance!hole!2!

Breeding!chamber!

No!idea!

Bolt!hole!

PRIMARY*HUB*

SECONDARY*HUB*

Tertiary!Entrance!

Secondary!entrance!!

Possible!boltEhole!turned!entrance!!Secondary!entrance!!

C,

B,

A,

D,

Tertiary!!entrance!

Tertiary!Entrance!

Tertiary!Entrance!

!

Resting!area!

PRIMARY*HUB*

SECONDARY*HUB*

Collapsed*+*widened*out.*Could*be*passing*place*

E,

Entrance!

FIG. 3: (color online) The rabbit warren’s 3D structures (courtesy of Hannah Sneyd) from di↵erent angles, where some important locationsfor transportation and breeding chambers A–E are indicated.

between banks (Wb0b from bank b to b0) should refer to theinterbank exposures from the lending bank to the borrowingbank in principle, but the data is only available in the unit of acountry c to a bank b, Ecb =

Pb02C(c) Wb0b where the set C(c)

is composed of the list of banks belonging to the country c.Therefore, for each Ecb, we equally distribute it to each bankb0 in that country c (except for the lending bank b itself) asWb0b = Ecb/|C(c) \ {b}|.

Figure 7 shows the CS (the resolution �↵ = �� = 0.01 for↵ 2 [0, 1] and � 2 [0, 1] is used) and PS (with optimal path-ways maximizing the sum of weights) of the interbank net-

work, where a few very large PS values dominate the system:GB089, BE004, FR013, DE017, and ES060 (sorted by the PSvalues). Similar to another non-transportation-based dolphinsocial network in Sec. III A, the CS and PS are correlated toeach other (Pearson: 0.430, Spearman: 0.499) more than orcomparable to CS vs BC (Pearson: 0.122, Spearman: 0.538)and PS vs BC (Pearson: 0.212, Spearman: 0.573).

https://sites.google.com/site/lshlj82/publications

2014년 노벨 생리의학상

iStockphoto/porcorex

Exploring Maps with Greedy Navigators

Sang Hoon Lee and Petter Holme

Phys. Rev. Lett. 108, 128701 (2012)Published March 22, 2012

Synopsis: Greed is Good

Many a tourist has, perhaps happily, gotten lost in the twists and turns along the way to Venice’s Piazza San

Marco. How navigable a city is—or could be with an extra footbridge or better-placed signs—is something

network models try to quantify. Now, writing in Physical Review Letters, two scientists show how one such

model could better account for the way humans actually go about reaching a destination.

Sang Hoon Lee and Petter Holme at Umeå University in Sweden focus on a type of “greedy” navigation model,

where at each point on a map, a navigator heads in the direction most in line with her destination (say a tall

building in the distance) and only backtracks if she can’t move to a point that hasn’t already been visited. The

model thus assumes a navigator has more information than one making random decisions, but doesn’t have at

hand any “smart” technology telling her the overall shortest route.

Using maps of New York, Boston, and the Swiss Rail System, as well as the maze at Leeds Castle in England,

the authors compare the distance traveled by a greedy navigator with that taken by a random navigator and the

actual shortest path. Not surprisingly, greedy navigators get to where they are going in a much shorter distance

than random travelers, though this advantage almost vanishes in the disorienting twists and turns in a maze.

Such models could be used to figure out the impact of blocking off certain bridges, tunnels, or roads on drivers

or pedestrians trying to navigate a city. What do Lee and Holme advise to keep a greedy navigator’s trip as

short as possible in Boston? Keep the Harvard Bridge open. – Jessica Thomas

ISSN 1943-2879. Use of the American Physical Society websites and journals implies that the user has read

and agrees to our Terms and Conditions and any applicable Subscription Agreement.

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Vilse i Venedig – Icelabforskare har modellen[2012-03-23] Vi vet alla hur lätt det kan vara att gå vilse i vissa storabyggnader, säg ett sjukhus. Även om vi gått igenom den några gånger förrhjälper det inte mycket. Hur kan man mäta hur lättnavigerat ett områdeär? Det är något Umeåfysiker undersöker i en artikel i den prestigefylldatidskriften Physical Review Letters.

Postdoktor Sang Hoon Lee ochuniversitetslektor Petter Holme, vidinstitutionen för fysik, har byggt ennätverksmodell för hur människortar sig fram för att nå en specifikdestination. Båda forskarna tillhörden kreativa forskningsmiljön Icelabvid Umeå universitet.

Lee och Holme har använtdatorsimulerade navigatörer, med demest grundläggande egenskaperna

hos människor, vilka försöker ta sig från en punkt till en annan i en miljö de interiktigt känner. Dessa navigatörer har en känsla av riktning och ett minne av var devarit men ingen mental karta över området.

– Genom att jämföra hur snabbt dessa hitta fram till sina mål med den kortastevägen man skulle fått från en GPS kan man förstå hur lättnavigerad en stadsdeleller en byggnad är, säger Petter Holme.

Forskarna har testat sin metod på olika verkliga kartor – från Manhattan ochBoston, till Schweiz järnvägsnät och en labyrint utanför Leeds Castle i England.Naturligtvis lyckas labyrinten lura navigatörerna så de inte navigerar mycket bättreän en person som vandrar slumpmässigt.

Vilken nytta kan denna modell få i praktiken?– Modellen skulle kunna användas för att underlätta för fotgängare i urbanamiljöer. Det kan hända att ett område blir lättare att navigera om man stänger avvissa vägar och passager, säger Petter Holme, som själv har ett mediokertlokalsinne.

Om Icelab:Integrated Science Lab, IceLab, bildades år 2010 och är en enhet för forskning ochutbildning som bygger teorier och metoder för att förstå levande system. Mananvänder verktyg från fysik, matematik och datavetenskap. IceLab bedrivertvärvetenskaplig forskning och utbildning i skärningspunkten mellan fysik,matematik, samhällsvetenskap och livskunskap. Det finns cirka 20 heltidsanställdai forskargruppen, alla med olika bakgrund vilket bidrar till en unik forskningsmiljö.

http://www.org.umu.se/icelab/

Mer om studien:Artikeln är publicerad i Physical Review Letters 108 2012 och har titelnExploring Maps with Greedy Navigators. Författare är Sang Hoon Lee och PetterHolme.

http://prl.aps.org/abstract/PRL/v108/i12/e128701

För mer information, kontakta gärna:Petter HolmeTelefon: 070-229 83 96E-post: [email protected]

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Hur navigatörerna hittar in till mitten av en trädgårdslabyrint.

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2014년 노벨 생리의학상

족보와 인구조사 자료를 바탕으로 한 인구 이동 조사3

FIG. 1. Examples of (a) ergodic and (b) non-ergodic clans. We color the regions of South Korea based on the fraction of the populationcomposed of members of the clan in the year 2000. We use arrows to indicate the origins of the two clans: Gimhae on the left and Ulsan(“Hakseong” is the old name of the city) on the right. In this map, we use the 2010 administrative boundaries [30]. See the Appendix fordiscussions of data sets and data cleaning.

into clan j, and clan j keeps the marriage records.For both the gravity and radiation models, we use census

data from the year 2000 [29] as a proxy for past populations.This allows us compute the quantities ri j, mi, and si j. Ourapproximation is supported by previously reported estimatesof stability in Korean society: historically, most clans havegrown in parallel with the total population, so we assumethat the relative sizes of clans have remained roughly constant[23]. In both Eqs. (1) and (2), only the relative sizes mi/Nand si j/N matter for calculating the flux (up to a constant ofproportionality).

B. Human Di↵usion and the Ergodicity Analysis

One way to quantify the notion of clan ergodicity is toexamine what we call the “clan density anomaly,” whichdescribes the local deviation in density of members of agiven clan. The clan density anomaly is �i(r, t) = ci(r, t) �[mi(t)/N(t)]⇢(r, t) at position r = (x, y) and time t, whereci(r, t) is the (spatially and temporally varying) local clan con-centration (i.e., the clan population density), mi(t) is the to-tal clan population, ⇢(r, t) is the local population density (i.e.,the total population of all clans at point r and time t, dividedby the di↵erential area), and N(t) is the total population ofall the clans at time t. If a clan were to occupy a constantfraction of the population everywhere in the country, then�i = 0 everywhere because its local concentration would be

ci = (mi/N)⇢. (This situation corresponds to perfect ergodic-ity.) The range of typical values for the clan density anomalydepends a clan’s aggregate concentration in the country. Ex-amining the anomaly relative to clan concentration, the year-2000 numbers for �i/(mi⇢/N) range from �1700 to 7400 forKim from Gimhae and from �19000 to 87000 for Lee fromHakseong. Clearly, the distribution of the latter is much moreheterogeneous (see Fig. 13).

Combining the notion of clan density anomaly with tra-ditional arguments—flow ideas based on Ohm’s law and“molecular weights for population” are mentioned explicitlyin [6, 10]—about migration from population gradients [2–10] suggests a simple Fickian law [43] for human transporton long time scales: we propose that the flux of clan mem-bers is Ji / r�i, so individuals move preferentially awayfrom high concentrations of their clans. This implies that@ci/@t = r · Ji / r2�i (where we have assumed that the con-stant of proportionality is independent of space), which yieldsthe di↵usion equation

@�i

@t= Di r2�i . (3)

We thereby identify the constant of proportionality as an av-erage di↵usion constant Di with dimensions [length2/time].This prediction of di↵usion of clan members is consistent withpast theories that posited human di↵usion (e.g., cultural [44]and demic [45] di↵usion). An important distinction is thatwe are proposing a process of di↵usive mixing of clans rather

김해 김씨의 분포 학성 이씨의 분포

족보와 인구조사 자료를 바탕으로 한 인구 이동 조사3

FIG. 1. Examples of (a) ergodic and (b) non-ergodic clans. We color the regions of South Korea based on the fraction of the populationcomposed of members of the clan in the year 2000. We use arrows to indicate the origins of the two clans: Gimhae on the left and Ulsan(“Hakseong” is the old name of the city) on the right. In this map, we use the 2010 administrative boundaries [30]. See the Appendix fordiscussions of data sets and data cleaning.

into clan j, and clan j keeps the marriage records.For both the gravity and radiation models, we use census

data from the year 2000 [29] as a proxy for past populations.This allows us compute the quantities ri j, mi, and si j. Ourapproximation is supported by previously reported estimatesof stability in Korean society: historically, most clans havegrown in parallel with the total population, so we assumethat the relative sizes of clans have remained roughly constant[23]. In both Eqs. (1) and (2), only the relative sizes mi/Nand si j/N matter for calculating the flux (up to a constant ofproportionality).

B. Human Di↵usion and the Ergodicity Analysis

One way to quantify the notion of clan ergodicity is toexamine what we call the “clan density anomaly,” whichdescribes the local deviation in density of members of agiven clan. The clan density anomaly is �i(r, t) = ci(r, t) �[mi(t)/N(t)]⇢(r, t) at position r = (x, y) and time t, whereci(r, t) is the (spatially and temporally varying) local clan con-centration (i.e., the clan population density), mi(t) is the to-tal clan population, ⇢(r, t) is the local population density (i.e.,the total population of all clans at point r and time t, dividedby the di↵erential area), and N(t) is the total population ofall the clans at time t. If a clan were to occupy a constantfraction of the population everywhere in the country, then�i = 0 everywhere because its local concentration would be

ci = (mi/N)⇢. (This situation corresponds to perfect ergodic-ity.) The range of typical values for the clan density anomalydepends a clan’s aggregate concentration in the country. Ex-amining the anomaly relative to clan concentration, the year-2000 numbers for �i/(mi⇢/N) range from �1700 to 7400 forKim from Gimhae and from �19000 to 87000 for Lee fromHakseong. Clearly, the distribution of the latter is much moreheterogeneous (see Fig. 13).

Combining the notion of clan density anomaly with tra-ditional arguments—flow ideas based on Ohm’s law and“molecular weights for population” are mentioned explicitlyin [6, 10]—about migration from population gradients [2–10] suggests a simple Fickian law [43] for human transporton long time scales: we propose that the flux of clan mem-bers is Ji / r�i, so individuals move preferentially awayfrom high concentrations of their clans. This implies that@ci/@t = r · Ji / r2�i (where we have assumed that the con-stant of proportionality is independent of space), which yieldsthe di↵usion equation

@�i

@t= Di r2�i . (3)

We thereby identify the constant of proportionality as an av-erage di↵usion constant Di with dimensions [length2/time].This prediction of di↵usion of clan members is consistent withpast theories that posited human di↵usion (e.g., cultural [44]and demic [45] di↵usion). An important distinction is thatwe are proposing a process of di↵usive mixing of clans rather

김해 김씨의 분포 학성 이씨의 분포

6

0 2000

0.02

number of regions occupied

pro

babili

ty d

ist. (

clans) (a)

0 2500

0.02

radius of gyration (km)pro

babili

ty d

ist. (

clans) (c)

0 2000

0.02

number of regions occupied

pro

babili

ty d

ist. (

indiv

iduals

)

(b)

0 2500

0.02

radius of gyration (km)

pro

babili

ty d

ist. (

indiv

iduals

)

(d)

FIG. 3. Distribution of the number of di↵erent administrative regionsoccupied by clans. (a) Probability distribution of the number of dif-ferent administrative regions occupied by a Korean clan in the year2000. (b) Probability distribution of the number of di↵erent admin-istrative regions occupied by the clan of a Korean individual selecteduniformly at random in the year 2000. The di↵erence between thispanel and the previous one arises from the fact that clans with largerpopulations tend to occupy more administrative regions. Note thatthe rightmost bar has a height of 0.17 but has been truncated for vi-sual presentation. (c) Probability distribution of radii of gyration (inkm) for clans in 2000. (d) Probability distribution of radii of gyra-tion (in km) for clans of a Korean individual selected uniformly atrandom in 2000. The di↵erence between this panel and the previ-ous one arises from the fact that clans with larger populations tendto occupy more administrative regions. Solid curves are kernel den-sity estimates (from Matlab R2011a’s ksdensity function with aGaussian smoothing kernel of width 5).

have jokbo are fairly ergodic, so the variables associated withthe j indices (i.e., the grooms) in Eqs. (1) and (2) have alreadylost much of their geographical precision, which is consistentboth with the values ↵ = 0 and � = 0 (the population prod-uct model). Again see the scatter plots in Fig. 2, in which wecolor each clan according to the number of di↵erent admin-istrative regions that it occupies. Note that the three di↵erentergodicity diagnostics are only weakly correlated (see Fig. I2).

Our observations of clan bimodality for Korea contrastsharply with our observations for family names in theCzech republic, where most family names appear to be non-ergodic [25] (see Fig. I3). One possible explanation of theubiquity of ergodic Korean names is the historical fact thatmany families from the lower social classes adopted (or evenpurchased) names of noble clans from the upper classes nearthe end of the Joseon dynasty (19th–20th centuries) [20, 52].At the time, Korean society was very unstable, and this pro-cess might have, in essence, introduced a preferential growthof ergodic names.

In Fig. 4, we show the distribution of the di↵usion constants

−5 200

0.7

diffusion constant (km2/year)

pro

ba

bili

ty d

istr

ibu

tion

FIG. 4. Distribution of estimated di↵usion constants (in km2/year)computed using 1985 and 2000 census data and Eq. (3). Thesolid curve is a kernel density estimate (from Matlab R2011a’sksdensity function with default smoothing). See the Appendixfor details of the calculation of di↵usion constants.

that we computed by fitting to Eq. (3). Some of the values arenegative, which presumably arises from finite-size e↵ects inergodic clans as well as basic limitations in estimating di↵u-sion constants using only a pair of nearby years. In Fig. I4,we show the correlations between the di↵usion constants andother measures.

C. Convection in Addition to Di↵usion as Another Mechanismfor Migration

The assumption that human populations simply di↵use is agross oversimplification of reality. We will thus consider theintriguing (but still grossly oversimplified) possibility of si-multaneous di↵usive and convective (bulk) transport. In thepast century, a dramatic movement from rural to urban areashas caused Seoul’s population to increase by a factor of morethan 50, tremendously outpacing Korea’s population growthas a whole [53]. This suggests the presence of a strong at-tractor or “sink” for the bulk flow of population into Seoul, ashas been discussed in rural-urban labor migration studies [54].The density-equalizing population cartogram [55] in Fig. I5clearly demonstrates the rapid growth of Seoul and its sur-roundings between 1970 and 2010.

If convection (i.e., bulk flow) directed towards Seoul hasindeed occurred throughout Korea while clans were simulta-neously di↵using from their points of origin, then one oughtto be able to detect a signature of such a flow. In Fig. 5(a),we show what we believe is such a signature: we observethat the fraction of ergodic clans increases with the distancebetween Seoul and a clan’s place of origin. This would be un-expected for a purely di↵usive system or, indeed, in any othersimple model that excludes convective transport. By allow-ing for bulk flow, we expect to observe that a clan’s mem-bers preferentially occupy territory in the flow path that islocated geographically between the clan’s starting point andSeoul. For clans that start closer to Seoul, this path is short;for those that start farther away, the longer flow path ought

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Sang Hoon Lee/Sungkyunkwan University

Matchmaker, Matchmaker, Make Me a Match: Migration of Populations via Marriages in the Past

Sang Hoon Lee ( ), Robyn Ffrancon, Daniel M. Abrams, Beom Jun Kim ( ), and Mason A. PorterPhys. Rev. X 4, 041009 (2014)Published October 16, 2014

Synopsis: Wedding Registries Reveal MigrationPaths

Patterns in human movement have been studied using traffic data, mobile phone records, and even dollar billcirculation. A new investigation uses centuries-old Korean family books, called jokbos ( in Korean), as arecord of emigration by female brides. The analysis of this movement, as well as more recent census data,shows that both diffusion- and convection-like phenomena may play a role in mixing different populations.

Understanding human mobility is important for improving city planning and for responding to outbreaks ofdisease. Previous modeling work has suggested that the statistical patterns in human movement resemblecertain physical phenomena, such as the diffusion of molecules in liquids. However, these patterns are drawnprimarily from short-term (a day to at most a year) tracking data, so it’s unclear whether the same models applyto longer-term migrations that affect cultural and genetic dissemination.

In Korea, marriages have traditionally involved the relocation of the bride to her groom’s home. Sang Hoon Leeof Sungkyunkwan University, Korea, and his colleagues analyzed of these marriages catalogued inten jokbos that date as far back as the th century. The books don’t provide specific locations, so theresearchers inferred migration paths from the geographic regions associated with each bride and groom’s clannames. Because the bride migration rate between regions was primarily dependent on clan populationdensities, the team tried to explain migration with a model in which Koreans moved randomly away (diffused)from high concentrations of their own clan. However, the estimated diffusion rate was too slow to explain howwell certain clans have spread throughout the country. The team concluded that a directed convective-like flowtowards the capital city Seoul enhanced the mixing rate. According to the authors, the combination of diffusionand convection may explain other human flow patterns.

This research is published in Physical Review X.

–Michael Schirber

ISSN 1943-2879. Use of the American Physical Society websites and journals implies that the user has readand agrees to our Terms and Conditions and any applicable Subscription Agreement.

200 00013

Sang Hoon Lee/Sungkyunkwan University

Matchmaker, Matchmaker, Make Me a Match: Migration of Populations via Marriages in the Past

Sang Hoon Lee ( ), Robyn Ffrancon, Daniel M. Abrams, Beom Jun Kim ( ), and Mason A. PorterPhys. Rev. X 4, 041009 (2014)Published October 16, 2014

Synopsis: Wedding Registries Reveal MigrationPaths

Patterns in human movement have been studied using traffic data, mobile phone records, and even dollar billcirculation. A new investigation uses centuries-old Korean family books, called jokbos ( in Korean), as arecord of emigration by female brides. The analysis of this movement, as well as more recent census data,shows that both diffusion- and convection-like phenomena may play a role in mixing different populations.

Understanding human mobility is important for improving city planning and for responding to outbreaks ofdisease. Previous modeling work has suggested that the statistical patterns in human movement resemblecertain physical phenomena, such as the diffusion of molecules in liquids. However, these patterns are drawnprimarily from short-term (a day to at most a year) tracking data, so it’s unclear whether the same models applyto longer-term migrations that affect cultural and genetic dissemination.

In Korea, marriages have traditionally involved the relocation of the bride to her groom’s home. Sang Hoon Leeof Sungkyunkwan University, Korea, and his colleagues analyzed of these marriages catalogued inten jokbos that date as far back as the th century. The books don’t provide specific locations, so theresearchers inferred migration paths from the geographic regions associated with each bride and groom’s clannames. Because the bride migration rate between regions was primarily dependent on clan populationdensities, the team tried to explain migration with a model in which Koreans moved randomly away (diffused)from high concentrations of their own clan. However, the estimated diffusion rate was too slow to explain howwell certain clans have spread throughout the country. The team concluded that a directed convective-like flowtowards the capital city Seoul enhanced the mixing rate. According to the authors, the combination of diffusionand convection may explain other human flow patterns.

This research is published in Physical Review X.

–Michael Schirber

ISSN 1943-2879. Use of the American Physical Society websites and journals implies that the user has readand agrees to our Terms and Conditions and any applicable Subscription Agreement.

200 00013

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How wedding registries reveal migration paths

Oxford Mathematics Professor Mason Porter and formerpostdoctoral student Sang Hoon Lee, nowof Sungkyunkwan University in Korea, have found a newway of analysing population mix. In the past patterns inhuman movement have been studied using traffic data,mobile phone records, and even dollar bill circulation.Their new investigation uses centuries-old Korean familybooks, called jokbos ( in Korean), as a record ofemigration by female brides. The analysis of thismovement, as well as more recent census data, showsthat both diffusion- and convection-like phenomena mayplay a role in mixing different populations.

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연결의 물리학: 일촌파도를 넘어 친구 추천의 시대로

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Transcript of 연결의 물리학: 일촌파도를 넘어 친구 추천의 시대로