Bayesian Inference of Exponential-family Random Graph Models for Social Networks
University of Washington, United States -- Washington, Ph.D., (2011)Abstract
Exponential-family random graph model (ERGM) has been widely applied in the fields of social network analysis, genetics (like protein interaction networks), information theory and more broadly. Because of the intractability of the likelihood function, Markov Chain Monte-Carlo (MCMC) algorithms are typically applied to approximate the likelihood (Geyer and Thompson 1992). However, ERGMs still suffer from inferential degeneracy and computational deficiency. In this dissertation, we apply Bayesian inference to ERGM to resolve model degeneracy and bias-reduction problems. We implement efficient MCMC algorithms for parameter estimation. We particularly are interested in conjugate priors of exponential families and the conjugacy properties of ERGM. We carry out simulation studies to show the superiority of the estimators under Bayesian framework over those based on Monte-Carlo likelihood approximation and pseudo-likelihood.
The second part of this dissertation focuses on model selection of exponential-family random graph models. Hunter, Goodreau and Handcock (2006) developed procedures for the goodness-of-fit of ERG models based on graphical diagnostics. However, the problems of model selection for ERG models have yet not been well studied. We investigate model selection for ERG models using both likelihood ratio tests and Bayesian methods. We propose a novel systematic procedure to conduct likelihood ratio tests to compare ERG models. Given two sets of models, we evaluate the likelihood ratio statistic, explore its sampling distribution and calculate the Monte-Carlo p-values at the end. Meanwhile, we develop a numerical algorithm to estimate the Bayes factor for given models. Finally, likelihood ratio tests and Bayesian model selection are tested and compared using real social network data.
In the third part, we propose a variational approach to solve likelihood approximation and subsequent parameter estimation problems for ERGMs. We study the mean parameter space under a new model specification and employ the naive mean field method in order to approximate mean parameters and subsequent likelihood functions. Preliminary results from simple 2-star models show that when dependence structure is not strong, the application of variational representation of ERGM models is able to calculate mean value parameters through optimization instead of expensive MCMC simulation. In addition, the variational approach returns comparable estimates of natural parameters to the maximum likelihood method.
Finally, as an application to real time data, we study networks of friendship and collaborations in a cohort of graduate students. We collected network data using monthly surveys and wearable sensors. Then we apply Exponential Random Graph Models to nine months of data on four types of relationships. This model-based approach allows us to observe outcomes of homophily and transitivity within each month, as well as describe patterns in homophily and transitivity over time. Lastly, we compare these dynamic patterns across different substantive relations (work collaborations, telephone conversations, social visits outside of school, and face-to-face conversations).