[book]Social and Ecnomic Networks(Princeton University Press 2010)
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SocialandEconomicNetworks1
MatthewO.Jackson
ThisDraft:March2008
Forthcoming:PrincetonUniversityPress
ccopyright:MatthewO.Jackson
Donotdistribute,post,orreproduceinanywaywithout
explicitpermissionoftheauthor
1Contents
Preface11
1Introduction17
1.1WhyModelNetworks?...........................17
1.2ASetofExamples:.............................18
1.2.1FlorentineMarriages........................18
1.2.2FriendshipsAmongHighSchoolStudents.............21
1.2.3RandomGraphsandNetworks..................24
1.2.4TheSymmetricConnectionsModel................32
1.3Exercises...................................36
2RepresentingandMeasuringNetworks39
2.1RepresentingNetworks...........................39
2.1.1NodesandPlayers.........................39
2.1.2GraphsandNetworks........................40
2.1.3PathsandCycles..........................43
2.1.4DirectedPaths,Walks,andCycles................45
2.1.5ComponentsandConnectedsubgraphs..............47
2.1.6Trees,Stars,Circles,andCompleteNetworks..........49
2.1.7Neighborhood............................51
2.1.8DegreeandNetworkDensity....................52
2.2SomeSummaryStatisticsandCharacteristicsofNetworks.......53
2.2.1DegreeDistributions........................53
2.2.2DiameterandAveragePathLength................55
2.2.3Cliquishness,Cohesiveness,andClustering............57
2.2.4Centrality..............................62
2.3Appendix:SomeBasicGraphTheory...................70
34CONTENTS
2.3.1Hall’stheoremandBipartiteGraphs...............70
2.3.2SetCoveringsandIndependentSets................71
2.3.3Colorings..............................73
2.3.4EulerianToursandHamiltonCycles...............74
2.4Appendix:EigenvectorsandEigenvalues.................77
2.4.1DiagonalDecompositions......................79
2.5Exercises...................................80
3EmpiricalBackgroundonSocialandEconomicNetworks83
3.1ThePrevalenceofSocialNetworks....................84
3.2ObservationsabouttheStructureofNetworks..............85
3.2.1DiameterandSmallWorlds....................85
3.2.2Clustering..............................88
3.2.3DegreeDistributions........................90
3.2.4CorrelationsandAssortativity...................97
3.2.5PatternsofClustering.......................99
3.2.6Homophily..............................100
3.2.7TheStrengthofWeakTies.....................102
3.2.8StructuralHoles...........................103
3.2.9SocialCapital............................104
3.2.10Di¤usion...............................104
4RandomGraph-BasedModelsofNetworks109
4.1StaticRandom-GraphModelsofRandomNetworks...........111
4.1.1PoissonandRelatedRandomNetworkModels..........111
4.1.2“SmallWorld”Networks......................112
4.1.3MarkovGraphsandpnetworks..................115
4.1.4TheCon…gurationModel......................117
4.1.5AnExpectedDegreeModel....................122
4.1.6SomeThoughtsaboutStaticRandomNetworkModels.....123
4.2PropertiesofRandomNetworks......................123
4.2.1TheDistributionoftheDegreeofaNeighboringNode.....124
4.2.2ThresholdsandPhaseTransitions.................127
4.2.3Connectedness............................133
4.2.4GiantComponents.........................139
4.2.5SizeoftheGiantComponentinPoissonRandomNetworks...139CONTENTS5
4.2.6GiantComponentsintheCon…gurationModel..........141
4.2.7DiameterEstimation........................145
4.3AnApplication:ContagionandDi¤usion.................148
4.3.1DistributionofComponentSizes.................151
4.4Exercises...................................154
4.5Appendix:UsefulFacts,Tools,andTheorems..............157
4.5.1SumsofSeries............................157
4.5.2eandStirling’sFormula......................158
4.5.3Chebyshev’sInequalityandtheLawofLargeNumbers.....158
4.5.4TheBinomialDistribution.....................159
4.5.5StochasticDominanceandMean-PreservingSpreads......160
4.5.6Domination.............................162
4.5.7Association.............................162
4.5.8MarkovChains...........................163
4.5.9GeneratingFunctions........................165
5GrowingRandomNetworks169
5.1UniformRandomness:anExponentialDegreeDistribution.......171
5.1.1Mean-FieldApproximations....................173
5.1.2ContinuousTimeApproximationsofDegreeDistributions...175
5.2PreferentialAttachment..........................176
5.3HybridModels...............................181
5.3.1Mean-FieldAnalysesofGrowingNetworkProcesses.......183
5.3.2MixingRandomandPreferentialAttachment..........183
5.3.3SimulationsasaCheckontheDegreeDistribution.......184
5.3.4FittingHybridDegreeDistributionstoData...........186
5.4SmallWorlds,Clustering,andAssortativity...............189
5.4.1Diameter..............................189
5.4.2PositiveAssortativityandDegreeCorrelation..........191
5.4.3ClusteringinGrowingRandomNetworks.............193
5.4.4AMeetings-BasedNetworkFormationModel..........194
5.4.5Clustering..............................196
5.5Exercises...................................200