# Exponential Random Graph Models for Social Networks Theory, Methods, and Applications

Intent of This BookSoftware and DataStructure of the BookSection I: RationaleSection II: MethodsSection III: ApplicationsSection IV: FutureHow To Read This BookAssumed Knowledge of Social Network AnalysisSection I RationaleWhat Are Exponential Random Graph Models?Exponential Random Graph Models: A Short DefinitionERGM TheoryBrief History of ERGMsNetwork Data Amenable to ERGMsFormation of Social Network StructureTie Formation: Emergence of StructureFormation of Social TiesNetwork Configurations: Consequential Network. Patterns and Related ProcessesLocal Network ProcessesDependency (and Theories of Network Dependence)Complex Combination of Multiple and Nested Social ProcessesFramework for Explanations of Tie FormationNetwork Self-OrganizationIndividual AttributesExogenous Contextual Factors: Dyadic CovariatesSimplified Account of an Exponential Random Graph Model as a Statistical ModelRandom GraphsDistributions of GraphsSome Basic Ideas about Statistical ModelingHomogeneityExample Exponential Random Graph Model AnalysisApplied ERGM Example: Communication in “The Corporation”ERGM Model and InterpretationMultiple Explanations for Network StructureSection II Methods Exponential Random Graph Model FundamentalsOutlineNetwork Tie-VariablesNotion of IndependenceERGMs from Generalized Linear Model PerspectivePossible Forms of DependenceBernoulli AssumptionDyad-Independent AssumptionMarkov Dependence AssumptionRealization-Dependent ModelsDifferent Classes of Model SpecificationsBernoulli ModelDyadic Independence ModelsMarkov ModelSocial Circuit ModelsOther Model SpecificationsConclusionDependence Graphs and Sufficient StatisticsOutlineDependence GraphHammersley-Clifford Theorem and Sufficient StatisticsSufficient Subgraphs for Nondirected GraphsDependence Graphs Involving AttributesConclusionSocial Selection, Dyadic Covariates, and Geospatial EffectsIndividual, Dyadic, and Other AttributesERGM Social Selection ModelsModels for Undirected NetworksModels for Directed NetworksConditional Odds RatiosDyadic CovariatesGeospatial EffectsConclusionAutologistic Actor Attribute ModelsSocial Influence ModelsPossible Forms of DependenceIndependent Attribute AssumptionNetwork-Dependent AssumptionsNetwork-Attribute-Dependent AssumptionsCovariate-Dependent AssumptionsDifferent Model Specifications and Their InterpretationIndependence ModelsNetwork Position Effects ModelsNetwork-Attribute Effects ModelsCovariate Effects ModelsConclusionExponential Random Graph Model Extensions: Models for Multiple Networks and Bipartite NetworksMultiple NetworksERGMs for Analyzing Two NetworksERGM Specifications for Two NetworksBipartite NetworksBipartite Network Representation and Special FeaturesERGM Specifications for Bipartite NetworksAdditional Issues for Bipartite NetworksLongitudinal ModelsNetwork DynamicsData StructureModelContinuous-Time Markov ChainTie-Oriented DynamicsDefinition of Dynamic ProcessStationary DistributionEstimation Based on ChangesConfigurations for NetworksRelations to Other ModelsReciprocity Model as PrecursorStochastic Actor-Oriented Models as AlternativesConclusionSimulation, Estimation, and Goodness of FitExploring and Relating Model to Data in PracticeSimulation: Obtaining Distribution of Graphs for a Given ERGMSampling Graphs Using Markov Chain Monte CarloMetropolis AlgorithmEstimationMaximum Likelihood PrincipleCurved ERGMsBayesian InferenceSolving the Likelihood EquationImportance Sampling: Geyer-Thompson ApproachStochastic Approximation: Robbins-Monro AlgorithmModifications for Longitudinal ModelTesting EffectsApproximate Wald TestAlternative TestsEvaluating Log-LikelihoodDegeneracy and Near-DegeneracyMissing or Partially Observed DataConditional Estimation from Snowball SamplesGoodness of FitApproximate Bayesian GOFIllustrations: Simulation, Estimation, and Goodness of FitSimulationTriangulationDegreesStars and Triangles TogetherEstimation and Model SpecificationSome Example Model SpecificationsGOFHow Do You Know Whether You Have a Good Model?What If Your Model Does Not Fit a Graph Feature?Should a Model Explain Everything?Section III Applications Personal Attitudes, Perceived Attitudes, and Social Structures: A Social Selection ModelPerceptions of Others and Social BehaviorData and MeasuresSocial Network QuestionsAttribute MeasuresAnalysesGoodness of FitModel SpecificationPurely Structural EffectsActor-Relation EffectsCovariate Network EffectsResultsExample 1: SchoolboysExample 2: Football TeamDiscussionHow To Close a Hole: Exploring Alternative Closure Mechanisms in Interorganizational NetworksMechanisms of Network ClosureData and MeasuresSetting and DataModel SpecificationResultsDiscussionInterdependencies between Working Relations: Multivariate ERGMs for Advice and SatisfactionMultirelational Networks in OrganizationsData, Measures, and AnalysesDescriptive ResultsMultivariate ERGM ResultsLow-AS BankHigh-AS BankDiscussionBrain, Brawn, or Optimism? Structure and Correlates of Emergent Military LeadershipEmergent Leadership in Military ContextAntecedents to Emergent LeadershipStructure of Emergent LeadershipSetting and ParticipantsModel SpecificationModeling IssuesPurely Structural EffectsActor-Relation EffectsResultsResults for Purely Structural EffectsResults for Actor-Relation EffectsDiscussionAutologistic Actor Attribute Model Analysis of Unemployment: Dual Importance of Who You Know and Where You LiveUnemployment: Location and ConnectionsData, Analysis, and EstimationDataAnalysisEstimationResultsDiscussionLongitudinal Changes in Face-to-Face and Text Message-Mediated Friendship NetworksEvolution of Friendship Networks, Communication Media, and Psychological DispositionsData and MeasuresSocial Network QuestionsActor-Relation MeasuresAnalysesModel SpecificationResultsResults for Face-to-Face Superficial NetworksResults for Face-to-Face Self-Disclosing NetworksResults for Text Message-Mediated Superficial NetworksResults for Text Message-Mediated Self-Disclosing NetworksDiscussionDifferential Impact of Directors’ Social and Financial Capital on Corporate Interlock FormationBipartite Society: The Individual and the GroupDirector Capital and Interlock FormationData and MeasuresSocial Network DataActor-Relation MeasuresAnalysesModel SpecificationIndependent Bivariate Attribute AnalysisPurely Structural EffectsModels with Attributes: Actor-Relation EffectsResultsResults for Independent Bivariate AnalysisResults for Purely Structural EffectsResults for Models Including Purely Structural and Actor-Relation EffectsDiscussionComparing Networks: Structural Correspondence between Behavioral and Recall NetworksRelationship between Behavior and RecallData and MeasuresDescription of NetworksData TransformationsModel SpecificationResultsVisualizationPreliminary Statistical AnalysisUnivariate ModelsModels of Recall Networks with Behavioral Networks as CovariatesMultivariate ModelsDiscussionSection IV Future Modeling Social Networks: Next StepsDistinctive Features of ERGMsModel SpecificationDependence HierarchyBuilding Model SpecificationsModels with Latent Variables: Hybrid FormsAssessing Homogeneity AssumptionsGeneral Issues for ERGMs