Estimation of exponential random graph models for large social networks via graph limits

Analyzing and modeling network data have become increasingly important in a wide range of scientific fields. Among popular models, exponential random graph models (ERGM) have been developed to study these complex networks. For large networks, however, maximum likelihood estimation (MLE) of parameters in these models can be very difficult, due to the unknown normalizing constant. Alternative strategies based on Markov chain Monte Carlo draw samples to approximate the likelihood, which is then maximized to obtain the MLE. These strategies have poor convergence due to model degeneracy issues. Chatterjee and Diaconis [1] propose a new theoretical framework for estimating the parameters of ERGM by approximating the normalizing constant using the emerging tools in graph theory-graph limits. In this paper, we construct a complete computational procedure built upon their results with practical innovations. More specifically, we evaluate the likelihood via simple function approximation of the corresponding ERGM's graph limit and iteratively maximize the likelihood to obtain the MLE. We also propose a new matching method to find a starting point for our iterative algorithm. Through simulation study and real data analysis of two large social networks, we show that our new method outperforms the MCMC-based method, especially when the network size is large (more than 100 nodes).