lecture slides jackson networksonline week1 slides

7/28/2019 Lecture Slides Jackson NetworksOnline Week1 Slides

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SocialandEconomic

Networks:

ModelsandAnalysis

Matthew O. JacksonStanford University,Santa Fe Institute, CIFAR,

www.stanford.edu\~jacksonm

Copyright 2013 The Board of Trustees of The Leland Stanford J unior University. All Rights Reserved.Figures reproduced with permission from Princeton University Press.


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1.1:Introduction


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WhyStudyNetworks? Manyeconomic,political,andsocialinteractionsareshapedbythelocal

structureofrelationships: tradeofgoodsandservices,mostmarketsarenotcentralized!

sharingofinformation,favors,risk,...

transmissionofviruses,opinions...

accesstoinfoaboutjobs...

choicesofbehavior,education,...

politicalalliances,tradealliances

Socialnetworksinfluencebehavior

crime,employment,humancapital,voting,smoking,

networksexhibitheterogeneity,butalsohaveenoughunderlyingstructuretomodel

Pureinterestinsocialstructure

understandsocialnetworkstructure


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PrimaryQuestions:

Whatdoweknowaboutnetworkstructure?

Howdonetworksform?Dothe`rightnetworksform?

Howdonetworksinfluencebehavior?(andviceversa...)


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Synthesize

Manyliteraturesdealwithnetworks Sociology

Economics

ComputerScience

StatisticalPhysics

Math(randomgraph)...

Whathavewelearned?Whatareimportantareasforfutureresearch?


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ThreeAreasforResearch

Theory networkformation,dynamics,design...

hownetworksinfluencebehavior

coevolution? Empiricalandexperimentalwork

observenetworks,patterns,influence

testtheoryandidentifyregularities

Methodology

howtomeasureandanalyzenetworks


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CentralFocus

Modelsforanalyzingandunderstandingnetworks:

Randomgraphmethods

Strategic,gametheoretictechniques

hybrids,statisticalmodels


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Goals

Presumenopriorknowledge

Introduceyoutoavarietyofapproachestomodelingnetworks (morebreadththandepth)

Giveasenseofdifferentdisciplinestechniquesandperspectives


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Models Provideinsightintowhyweseecertainphenomena:

Whydosocialnetworkshaveshortaveragepathlengths?

Allowforcomparativestatics:

Howdoescomponentstructurechangewithdensity?Importantincontagion/diffusion/learning...

Predictoutofsample: Whatwillhappenwithanewpolicy(vaccine,R&Dsubsidy,

...)?

Allowforstatisticalestimation:

Istheresignificantclusteringonalocallevelordiditappearatrandom?


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Outline

PartI:BackgroundandFundamentals DefinitionsandCharacteristicsofNetworks(1,2) EmpiricalBackground(3)

PartII:NetworkFormation RandomNetworkModels(4,5) StrategicNetworkModels(6,11)

PartIII:NetworksandBehavior

DiffusionandLearning(7,8) GamesonNetworks(9)


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SocialandEconomic

Networks:

ModelsandAnalysis

Matthew O. Jackson

Stanford University,

Santa Fe Institute, CIFAR,www.stanford.edu\~jacksonmCopyright 2013 The Board of Trustees of The Leland Stanford J unior University. All Rights Reserved.


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1.2:ExamplesandChallenges


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Outline

PartI:BackgroundandFundamentals DefinitionsandCharacteristicsofNetworks(1,2) EmpiricalBackground(3)

PartII:NetworkFormation RandomNetworkModels(4,5) StrategicNetworkModels(6,11)

PartIII:NetworksandBehavior

DiffusionandLearning(7,8) GamesonNetworks(9)


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TwoExamples

Ideaofdata

Viewofapplications

Previewsomequestions


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ACCIAIUOL

ALBIZZI

BARBADORI

BISCHERI

CASTELLAN

GINORI

GUADAGNI

LAMBERTES

MEDICI

PAZZI

PERUZZI

PUCCI

RIDOLFI

SALVIATI

STROZZI

TORNABUON

Medici

GuadagniStrozzi

Padgett and Ansells(1993) Data (fromKent 1978)Florentine Marriages,1430s


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ACCIAIUOL

ALBIZZI

BARBADORI

BISCHERI

CASTELLAN

GINORI

GUADAGNI

LAMBERTES

MEDICI

PAZZI

PERUZZI

PUCCI

RIDOLFI

SALVIATI

STROZZI

TORNABUON

Medici .522

Guadagni .255Strozzi .104

Padgett and Ansells(1993) Data (fromKent 1978)Florentine Marriages,1430s


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.18

.12

.13

.11

.11

.05

Greece

France Germany

Portugal

ItalySpain

Elliott, Golub, J ackson (2012)


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WhatdoWeKnow? Networksplayroleinmanysettings

Jobcontacts,crime,risksharing,trade,politics,...

Networkpositionandstructurematters

richsociologyliterature Medicis notthewealthiestnorthestrongestpolitically,

butthemostcentral

``SocialNetworkshavespecialcharacteristics

smallworlds,degreedistributions...


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Embeddednessof

EconomicActivity

Fewmarketsarecentralized, anonymous

Specificrelationshipsmatter...


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NetworksinLaborMarkets

MyersandShultz(1951) textileworkers: 62%firstjobfromcontact 23%bydirectapplication 15%byagency,ads,etc.

ReesandShultz(1970) Chicagomarket: Typist37.3% Accountant23.5% Materialhandler73.8% Janitor65.5%,Electrician57.4%

Granovetter(1974),Ioannides andLoury (2004)...


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OtherSettings Networksandsocialinteractionsincrime:

Reiss(1980,1988) 2/3ofcriminalscommitcrimeswithothers

Glaeser,Sacerdote andScheinkman (1996) socialinteractionimportantinpetty

crime,amongyouths,andinareaswithlessintacthouseholds NetworksandMarkets

Uzzi (1996) relationspecificknowledgecriticalingarmentindustry

Weisbuch,Kirman,Herreiner (2000) repeatedinteractionsinMarseillefishmarkets

SocialInsurance

Fafchamps andLund(2000) risksharinginruralPhilippines

DeWeerdt (2002) Tanzania,...

Diffusion

HybridcornadoptionRyanandGross(1943),Griliches (1957) DrugadoptionColeman,Katz,Menzel (1966)

Sociologyliterature interlockingdirectorates,aidstransmission,language,successofimmigrantgroups...


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TheChallenge: Howmanynetworksonjust30nodes?

Person1couldhave29possiblelinks,person2couldhave28notcounting1, .... total=435

So435possiblelinks, eachcouldeitherbepresentornot,so2x2x2...435times=2435networks

Atoms in the universe: between 2158 and 2246


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SocialandEconomic

Networks:

ModelsandAnalysis

Matthew O. Jackson




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1.3:DefinitionsandNotation


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SimplifyingtheComplexity

Globalpatternsofnetworks

degreedistributions,pathlengths...

SegregationPatterns

nodetypesandhomophily LocalPatterns

Clustering,Transitivity,Support

Positionsinnetworks

Neighborhoods,Centrality, Influence...


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RepresentingNetworks

N={1,,n} nodes,vertices,players

gin {0,1}nn adjacencymatrix (unweighted,possiblydirected)

gij =1 indicatesalink,tie,oredgebetweeniandj

Alternativenotation: ij in g alinkbetweeniandj Network(N,g)


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BasicDefinitions

Walkfrom i1toiK: asequenceofnodes(i1,i2,...,iK)andsequenceoflinks (i1i2,i2i3,...,iK1iK)suchthat

ik1ik in g foreachk

Convenienttorepresentitasthecorresponding

sequenceofnodes(i1,i2,...,iK) suchthatik1ik in g

foreachk


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BasicDefinitions

Path: awalk(i1,i2,...iK) witheachnodeik distinct

Cycle: awalkwhere i1=iK

Geodesic: ashortestpathbetweentwonodes


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Paths,Walks,Cycles...

4

1

326

7

5

Path (and a walk) from 1 to 7:1, 2, 3, 4, 5, 6, 7

4

1

326

7

5

Walk from 1 to 7 that is not a path:1, 2, 3, 4, 5, 3, 7

4

1

326

7

5

Simple Cycle (and a walk)

from 1 to 1: 1, 2, 3, 1

4

1

326

7

5

Cycle (and a walk) from 1 to 1:

1, 2, 3, 4, 5, 3, 1


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CountingWalks:

g=

0 1 0 11 0 0 1

0 0 0 1

1 1 1 0

g2 =

2

1 3

4

2 1 1 1

1 2 1 1

1 1 1 01 1 0 3

number of walks of length 2 from i to j


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CountingWalks:

g=

0 1 0 1

1 0 0 1

0 0 0 1

1 1 1 0

g3

=

2

1 3

4

2 3 1 4

3 2 1 4

1 1 0 34 4 3 2

number of walks of length 3 from i to j


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Components

(N,g)isconnectedifthereisapathbetweeneverytwonodes

Component: maximalconnectedsubgraph

(N,g)isasubsetof(N,g)

(N,g) isconnected i inNand ij ingimpliesj inN andij ing


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Anetworkwithfour

components:

5

2

4

3

1

10

7

69

8


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SocialandEconomic

Networks:

ModelsandAnalysis

Matthew O. Jackson




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1.4:Diameter


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Diameter,AveragePathLength

Howclosearenodestoeachother:

Howlongdoesittaketo

reachaveragenode?

Howfastwillinformationspread?...

Howdoesitdependonnetworkdensity?


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Diameter

Diameter largestgeodesic(largestshortestpath) ifunconnected,oflargestcomponent...

Averagepathlength

(lesspronetooutliers)


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Diameter:

diameter is eithern/2 or (n-1)/2

diameter is on order of2 log2(n+1)

K levels has n = 2K+1-1 nodesso, K = log2(n+1) -1diameter is 2K


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Smallaveragepathlengthanddiameter

Milgram (1967)letterexperiments

median5forthe25%thatmadeit CoAuthorshipstudies

Grossman(2002)Mathmean7.6,max27,

Newman(2001)Physicsmean5.9,max20 Goyal etal(2004)Economicsmean9.5,max29

WWW

Adamic,Pitkow (1999) mean3.1(85.4%possibleof50Mpages)

Facebook

Backstrom etal(2012) mean4.74(721millionusers)


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Neighborhood

andDegree

Neighborhood:Ni(g)={j|ij ing}

(usualconventioniinoting)

Degree: di =#Ni(g)


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ErdosRenyi (1959,1960)

RandomGraphs startwithnnodes

eachlinkisformedindependentlywithsomeprobabilityp

Servesasabenchmark``G(n,p)


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SequencesofNetworks

Linksaredenseenoughsothatnetworkisconnectedalmostsurely:

d(n) (1+)log(n)some >0

d(n)/n 0:networkisnottoocomplete


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TheoremonNetwork

Structure

Ifd(n) (1+)log(n)some >0 andd(n)/n 0

Thenforlargen,averagepathlengthanddiameter

areapproximatelyproportionalto log(n)/log(d)

(Provenforincreasinglygeneralmodels:

ErdosRenyi59 MoonandMoser1966,Bollobas1981;ChungandLu01;Jackson08;...)


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TheoremonNetwork

StructureIfd(n) (1+)log(n)some >0 andd(n)/n 0

AvgDist(n) P 1

log(n)/log(d(n))

samefordiameter


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SocialandEconomicNetworks:

ModelsandAnalysis

Matthew O. Jackson




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1.5:DiameterandTrees


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Diameter

Boundscanbedifficult theoremsarenarrow,butintuitioniseasy

Letsstartwithaneasycalculation

Cayley Tree: eachnodebesidesleaveshasdegreed


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Intuition:

1 step: Reach d nodes,


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Ideas:


then d(d-1),


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Ideas:


then d(d-1),

then d(d-1)2,


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Ideas:


then d(d-1),

then d(d-1)2, d(d-1)3, ...

After lsteps, totals roughly dl


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Whatifnotatree,butERrandomgraph?

allhavesamedegree reallyarerandom showthatfractionofnodesthathave

nearlyaveragedegreeisgoingto1

somelinksmaydoubleback

mostnodesuntilthelaststeparestillnotreached!


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ModelsandAnalysis

Matthew O. JacksonStanford University,



55/101

1.6:DiametersofRandom

Graphs


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TheoremonNetwork

StructureIfd(n) (1+)log(n)some >0 andd(n)/n 0

AvgDist(n) P 1

log(n)/log(d(n))

samefordiameter


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Movingoutl linksfromrootineachdirectionreaches d+d(d1)+.... d(d1)l 1 nodes

Thisis d((d1)l 1)/(d2) nodes orroughly(d1)l

Toreachn1,needroughly(d1)l =n or

l ontheorderof log(n)/log(d)


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Whatifnotatree,butErdosRenyirandomgraph?

allhavesamedegree reallyarerandom

showthatfractionofnodesthathavenearlyaveragedegreeisgoingto1

E[d]>(1+ )log(n)


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Chernoff Bounds:

Xisbinomialvariablethen

Pr(E[X]/3 X 3E[X]) 1 eE[X]


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Chernoff Bounds:

Xisbinomialvariablethen

Pr(E[X]/3 X 3E[X]) 1 eE[X]

http://en.wikipedia.org/wiki/Chernoff_bound


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Chernoff Bounds: Linksbinomialimplies

Probabilitythatnodehasdegreeclosetoaverage:

Pr(d/3 i 3d) 1 ed

Pr (d/3 3d) (1 ed

)n

(missingsteps:degreesnotquiteind.)


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Chernoff Bounds:

Pr (d/3 3d) (1 ed)n

If d>(1+ )log(n) then

Pr (d/3 3d) (1 1/n1+)n

exp(n) 1


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So:

If d>(1+ )log(n) then

Pr (d/3 3d)


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Thus:

If d>(1+ )log(n) thenwithprob 1:

log(n)/log(3d)


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Avg distanceanddiameter:

Larged: log(3d)&log(d/3)tendtolog(d)

log(n)/log(3d)


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somelinksmaydoubleback

mostnodesuntilthelaststeparestillnotreached,so

mostlinksstillreachingnewnodes!

Afterkstepsreached arounddk nodesandn dk still

unreached

ifk


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Ideas:

Most at maximum distance

(100, 10000, 1000000,

100000000...)

so average distance is

actually same order as

diameter


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ModelsandAnalysis

Matthew O. Jackson




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1.7:DiametersintheWorld


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TheoremonNetwork

Structure

Ifd(n) (1+)log(n)some >0 andd(n)/n 0

Thenforlargen,averagepathlengthanddiameter

areapproximatelyproportionalto log(n)/log(d)

(Provenforincreasinglygeneralmodels:

ErdosRenyi59 MoonandMoser1966,Bollobas

1981;ChungandLu01;Jackson08;...)


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SmallWorlds/SixDegreesof

Separation

n=6.7billion(worldpopulation)

d=50(friends,relatives...)

log(n)/log(d)isabout6!!


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AddHealthdataset

Schoolsvaryinaveragedegree

andhomophily

Doesdiametermatchlog(n)/log(d)?

Examinedataanddiameter


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0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

AverageShortestPathvs Log(n)/Log(d) 84HighSchools AdHealth

AverageShortestPathvsLog(n)/Log(d)

Linear(AverageShortestPathvsLog(n)/Log(d))

GolubandJackson(2012)


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ErdosNumbers NumberoflinksincoauthorshipnetworktoErdos

Had 509coauthors, morethan1400papers

2004auctionofcoauthorshipwithWilliamTozier(Erdos #=4)onEBay,winnerpaid>1000$

KevinBaconsite....


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Density: AverageDegree

HSFriendships(CJP09) 6.5

Romances (BMS03) 0.8

Borrowing(BCDJ12) 3.2

Coauthors(Newman01,GLM06)

Bio 15.5

Econ 1.7

Math 3.9

Physics 9.3

Facebook(Marlow09) 120


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ModelsandAnalysis

Matthew O. Jackson




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1.8:DegreeDistributions


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DegreeDistributions

Averagedegreetellsonlypartofthestory:


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DegreeDistribution,G(n,p):

probabilitythatnodehasdlinksisbinomial

[(n1)!/(d!(nd1)!)]pd (1p)nd1

Largen,smallp,thisisapproximatelyaPoisson distribution:[(n1)d /d!] pd e(n1)p

hencename``Poissonrandomgraphs


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Randomnetworkp=.02,50nodes


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Note

manyisolatednodes

severalcomponents

nocomponenthasmorethanasmallfraction

ofthenodes,juststartingtoseeonelargeoneemerge


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RandomNetworkp=.08,50nodes


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Distributionoflinkspernode:Fattails(Price1965)

Morehighandlowdegreenodesthanpredictedatrandom CitationNetworks toomanywith0citations,

toomanywithhighnumbersofcitationstohavecitationsdrawnatrandom

``Fattailscomparedtorandomnetwork

Relatedtoothersettings(wealth,citysize,

wordusage...):Pareto(1896),Yule(1925),Zipf(1949),Simon(1955),


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Degree NDwwwAlbert,Jeong,Barabasi(1999)

Log(Degree)

log(freq)


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ScaleFreeDistributions

P(d)=cda

log(P(d))=log(c) a log(d)

Bearman, Moody, and Stovels04 Hi h S h l R


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04 High School RomanceData


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RomanceNetwork

Bearman et al HS Networ k

-3

-2.5

-2

-1.5

-1

-0.5

0

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

Log Degre e

LogCCDF

Series1

Series2

fit: Uniform at Random .99

fit: Power .84

S i l d E i


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ModelsandAnalysis

Matthew O. Jackson




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1.9:Clustering

Cl i


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Clustering

Whatfractionofmyfriendsarefriendsofeachother?

Cli(g) =#{kj ing |k,jinNi(g)} / #{kj |k,jinNi(g)}

Averageclustering:

Clavg(g)=i Cli(g)/nFreq

of thislink?

i j

k

Cl i


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Clustering

Whatfractionofmyfriendsarefriends?

Cli(g) =#{kj ing |k,jinNi(g)} / #{kj |k,jinNi(g)}

Averageclustering: Clavg(g)=i Cli(g)/n

Overallclustering:

Cl(g)=i #{kj ing |k,jinNi(g)} / i #{kj |k,jinNi(g)}

Diff i Cl t i


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DifferencesinClustering

Average tends to 1

Overall tends to 0

Clustering in a Poisson


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ClusteringinaPoissonRandomNetwork

AverageandOverallclusteringtendto0,ifmaxdegreeisboundedandnetworkbecomeslarge:

Cl(g)=i #{kj ing |k,jinNi(g)} / i #{kj |k,jinNi(g)}

issimplyp Ifdegreeisbounded,thenp(n1)isbounded

Sopgoesto0asngrows

High? Clustering Coefficients


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High?ClusteringCoefficients Prisonfriendships

.31 (MacRae 60)vs .0134

coauthorships .15 math(Grossman02)vs .00002,

.09 biology(Newman01)vs .00001,

.19 econ(Goyal etal06)vs .00002,

www .11forweblinks(Adamic 99)vs .0002

Freqof thislink?

1 2

3

Clustering: .46Random: .29


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Padgett and Ansells data1430s Florentine marriagesand business dealings

Random: .29

Week 1 Wrap


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Week1 Wrap Manyrelationshipsare``networkedandunderstanding

networkstructurecanhelpunderstandbehaviorand

outcomes

Networksarecomplex,butcanbepartlydescribedbysomecharacteristics

degreedistributions

clustering

diameter...

Treelikestructuresaregeneratedbyrandomlinksleadtoshortpaths

Manyobservedsocialnetworksaremoreclusteredthanwouldariseatrandom

Week1:References InOrderMentioned


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Jackson,M.O.(2008)SocialandEconomicNetworks,PrincetonUniversityPress

Padgett,J.F.,andC.K.Ansell(1993)RobustActionandtheRiseoftheMedici,14001434,AmericanJournalofSociology98:12591319. Kent,D.(1978)TheRiseoftheMedici:FactioninFlorence14261434,Oxford:OxfordUniversityPress. Elliott,M.L.,B.GolubandM.O.Jackson(2012) ``FinancialNetworksandContagion,SSRN workingpaper2175056

MyersC.A.,andG.P.Shultz(1951)TheDynamicsofaLaborMarket,NewYork:PrenticeHall. ReesAJ,ShultzGP.1970.WorkersinanUrbanLaborMarket.Chicago:Univ.ChicagoPress

GranovetterM.1973.Thestrengthofweakties.Am.J.Sociol.78:1360 80

Ioannides YM,DatcherLoury L.2004.Jobinformationnetworks,neighborhoodeffectsandinequality.J.Econ.Lit.424:1056 93

ReissAJ.1980.Understandingchangesincrimerates.InIndicatorsofCrimeandCriminalJustice:QuantitativeStudies.Washington,DC:Bur.JusticeStat.

ReissAJ.1988.Cooffendingandcriminalcareers.InCrimeandJustice:AReviewofResearch,Vol.10,ed.MTonry.Chicago:Univ.Chicago

Press Glaeser E,Sacerdote B,Scheinkman J.1996.Crimeandsocialinteractions.Q.JEcon.111:507 48

Uzzi B.1996.Thesourcesandconsequencesofembeddedness fortheeconomicperformanceoforganizations:thenetworkeffect.Am.Sociol.Rev.61:67498

Weisbuch,G.,A.Kirman,andD.Herreiner (2000)MarketOrganization,Economica 110:411436.

Fafchamps M,LundS.2003.RisksharingnetworksinruralPhilippines.J.Dev.Econ.71:261 87

DeWeerdt J.2004.Risksharingandendogenousnetworkformation.InInsuranceAgainstPoverty,ed.SDercon.Oxford:OxfordUniv.Press

Ryan,B.,andN.C.Gross(1943)TheDiffusionofHybridSeedCorninTwoIowaCommunities,RuralSociology8:1524. Griliches,Z.(1957)HybridCorn:AnExplorationintheEconomicsofTechnologicalChange,Econometrica25(4):501522. Coleman,J.S.,E.Katz,andH.Menzel(1966)MedicalInnovation:ADiffusionStudy,Indianapolis,Ind.:BobbsMerrill.

Week1:ReferencesContd


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Milgram,S.(1967)TheSmallWorldProblem,PsychologyToday2:6067. Grossman,J.W.(2002)TheEvolutionoftheMathematicalResearchCollaborationGraph,inProceedingsofthe33rdSoutheastern

ConferenceonCombinatorics (Congressus Numerantium,Vol.158). Newman,M.E.J.(2001)Scientificcollaborationnetworks. I.NetworkconstructionandfundamentalresultsPhys.Rev.E64,016131

Goyal,S.,M.vanderLeij,andJ.L.MoragaGonzalez (2006)Economics:AnEmergingSmallWorld,JournalofPoliticalEconomy114(2):403412.

Adamic,L.A.(1999)TheSmallWorldWeb,inProceedingsoftheECDL.LectureNotesinComputerScience1696,Berlin:SpringerVerlag.

Backstrom,L.,P.Boldiy,M.Rosay,.J.Ugander S.Vignay (2012)``FourDegreesofSeparation arXiv 1111.4570v3

Erdos,P.,andA.Renyi (1959)OnRandomGraphs,Publicationes Mathematicae Debrecen6:290297. (1960)OntheEvolutionofRandomGraphs,PublicationoftheMathematicalInstituteoftheHungarianAcademyofSciences5:1761. (1961)OntheStrengthofConnectednessofaRandomGraph,ActaMathamatica AcademyofSciencesofHungarica 12:261267. JW Moon,L Moser ( 1966)``Almostall(0,1)matricesareprimitive,Studia Sci.Math.Hungar,

BBollobs (1981) ``DiameterofRandomGraphs, Trans.Am.Math.Soc.,1981 ams.org

Chung,F.,andL.Lu(2002)TheAverageDistancesinRandomGraphswithGivenExpectedDegrees,ProceedingsoftheNationalAcademyofSciences99:1587915882.

Jackson,M.O.(2008b)AverageDistance,Diameter,andClusteringinSocialNetworkswithHomophily,arXiv:0810.2603v1,intheProceedingsoftheWorkshopinInternetandNetworkEconomics(WINE2008),LectureNotesinComputerScience,EditedbyC.

PapadimitriouandS.Zhang,SpringerVerlag,BerlinHeidelberg.

Golub,B.andM.O.Jackson (2012)``NetworkStructureandtheSpeedofLearning:MeasuringHomophilyBasedonits Consequences,AnnalsofEconomicsandStatistics 107/108JulyDec.,2012.

Week1:ReferencesContd


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Currarini,S.,M.O.Jackson,andP.Pin(2010)Identifyingtherolesofracebasedchoiceandchanceinhighschoolfriendshipnetworkformation,intheProceedingsoftheNationalAcademyofSciences,107(11):4857 4861

Bearman,P.,J.Moody,andK.Stovel (2004)ChainsofAffection:TheStructureofAdolescentRomanticandSexualNetworks,Chicago,UniversityofChicago,manuscript.

Banerjee,A.,A.G.Chandrasekhar, E.Duflo, M.O.Jackson(2012)``DiffusionofMicrofinance, NBERWorkingpaperw17743

Marlow,C.(2009).MaintainedRelationshipsonFacebook. mimeo

PriceDJS.1965.Networksofscientificpapers.Science149:51015

Pareto,V.(1896)CoursdEconomiePolitique,Geneva:Droz Yule,G.(1925)AMathematicalTheoryofEvolutionBasedontheConclusionsofDr.J.C.Willis,F.R.S.PhilosophicalTransactionsofthe

RoyalSocietyofLondonB213:2187. Zipf,G.(1949)HumanBehaviorandthePrincipleofLeastEffort,Cambridge,Mass.:AddisonWesley. Simon,H.(1955)OnaClassofSkewDistributionFunctions,Biometrika 42(34):425440. Albert,R.,H.Jeong,andA.L.Barabasi (1999)DiameteroftheWorldWideWeb,Nature401:130131. MacRae,J.(1960)DirectFactorAnalysisofSociometric Data,Sociometry23:360371.

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