Fitting Social Network Models Using Varying Truncation Stochastic Approximation MCMC Algorithm

Abstract

The exponential random graph model (ERGM) plays a major role in social network analysis. However, parameter estimation for the ERGM is a hard problem due to the intractability of its normalizing constant and the model degeneracy. The existing algorithms, such as Monte Carlo maximum likelihood estimation (MCMLE) and stochastic approximation, often fail for this problem in the presence of model degeneracy. In this article, we introduce the varying truncation stochastic approximation Markov chain Monte Carlo (SAMCMC) algorithm to tackle this problem. The varying truncation mechanism enables the algorithm to choose an appropriate starting point and an appropriate gain factor sequence, and thus to produce a reasonable parameter estimate for the ERGM even in the presence of model degeneracy. The numerical results indicate that the varying truncation SAMCMC algorithm can significantly outperform the MCMLE and stochastic approximation algorithms: for degenerate ERGMs, MCMLE and stochastic approximation often fail to produce any reasonable parameter estimates, while SAMCMC can do; for nondegenerate ERGMs, SAMCMC can work as well as or better than MCMLE and stochastic approximation. The data and source codes used for this article are available online as supplementary materials.

Key Words:

• Exponential random graph model
• Model degeneracy
• Monte Carlo MLE
• Trajectory averaging

ACKNOWLEDGMENTS

The authors thank the editor, the associate editor, and two referees for their detailed and constructive comments, which have led to significant improvement of this article. Liang’s research was partially supported by grants from the National Science Foundation (DMS-1007457 and CMMI-0926803) and the award (KUS-C1-016-04) conferred by King Abdullah University of Science and Technology (KAUST).