Improvements in Simulation, Convergence Monitoring, and Modeling of Exponential Random Graph Models for Social Network Analysis Open Access
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Exponential Random Graph Models (ERGMs) are one class of models usedto describe observations from a social network. ERGMs have been usedin many applications and are popular due to their ability to modeltransitivity of associations among other social networkfactors. This dissertation makes improvements in three areas relatedto the use of ERGMs. The first contribution is a stratified sampling proposal for use incomputation of ERGM parameter estimates. Parameters of ERGMs are computationally intensive to estimate due to the need to use simulation to deal with the intractablenormalizing constants that appear in probability densityfunctions. One approach to dealing with the normalizing constants isto sample new networks. Metropolis Hastings (MH) algorithms are usedto sample networks by randomly sampling pairs of nodes in a network and flipping the association betweenthe pairs on or off. This is very inefficient due to the sparsity oftypical networks. Current sampling methods improve on naive schemes bysampling pairs with different rates based on the current existence ofedges between members of the pair. The novel proposal in this dissertation isto sample pairs with probabilities conditional on higher-orderdescriptions of the social network. This leads to a new proposaldistribution for MH sampling for ERGMs. The innovation aims to solve abad mixing problem when a transitivity effect is modeled inERGMs. The second contribution is in the area of monitoring convergence ofiterative simulations to produce draws from a social network givenparameter values. As was mentioned, social networks are randomlygenerated in order to deal with a normalizing constant duringestimation. Current methods generate a single simulated sequence ofnetworks for a proscribed number of iterations. The quality of the MCMC sampling is not adequately monitored during the process. Instead methods of monitoring multiple, over-dispersedsequences of MCMC simulations are applied in the ERGM context. Theproposal is a new use of existing methods from the MCMCliterature. Methods are applied using both the traditional and newstratified sampling proposals. It is shown that monitoring multipleparallel sequences for convergence gives a clearer assessment ofconvergence and some advantage in estimation. The third contribution is in the area of modeling of social networkdata. Global measures of transitivity, such as GWESP, are included inmost social network modeling efforts. Here local measures are proposed that give more flexibility tomodeling. A global measure might not be appropriate for all parts of asocial network. A local version of transitivity is appropriate inmany network contexts in which subsets of the network are identifiableand have few if any connections between them. Global measures also arecomputationally expensive. Instead of examining all possible subsets of the networkwhen computing a statistics, a local version of the statisticeliminates cross-subset computation. Therefore the local measuresyield computational advantages as well.