Monte Carlo Maximum Likelihood for Exponential Random Graph Models: From Snowballs to Umbrella Densities

Abstract

Performing maximum-likelihood estimation for parameters in an exponential random graph model is challenging because of the unknown normalizing constant. Geyer and Thompson (1992) provide a Monte Carlo algorithm that uses samples from a distribution with known parameters to approximate the full likelihood, which is then maximized to estimate the MLE. We refine the approximation to use sample draws collected from differently parameterized distributions, increasing the effective sample size and improving the accuracy of the MLE estimate. Substantially lower estimation variance is demonstrated in simulated and actual network data. We also propose a new method for finding a starting point: scaling the MLE parameters of a small graph subsampled from the original graph. Through simulation with the triad model, this starting point produces convergence in many cases where the standard starting point (based on pseudolikelihood) fails to give convergence, though the reverse is also true.