Markov Chain Monte Carlo Estimation of Exponential Random Graph Models

Abstract

his paper is about estimating the parameters of the exponential random graph model, also known as the p∗ model, using frequen- tist Markov chain Monte Carlo (MCMC) methods. The exponen- tial random graph model is simulated using Gibbs or Metropolis- Hastings sampling. The estimation procedures considered are based on the Robbins-Monro algorithm for approximating a solu- tion to the likelihood equation. A major problem with exponential random graph models re- sides in the fact that such models can have, for certain parameter values, bimodal (or multimodal) distributions for the sufficient statistics such as the number of ties. The bimodality of the ex- ponential graph distribution for certain parameter values seems a severe limitation to its practical usefulness. The possibility of bi- or multimodality is reflected in the possibility that the outcome space is divided into two (or more) regions such that the more usual type of MCMC algorithms, updating only single relations, dyads, or triplets, have extremely long sojourn times within such regions, and a negligible proba- bility to move from one region to another. In such situations, convergence to the target distribution is extremely slow. To be useful, MCMC algorithms must be able to make transitions from a given graph to a very different graph. It is proposed to include transitions to the graph complement as updating steps to improve the speed of convergence to the target distribution. Estimation procedures implementing these ideas work satisfac- torily for some data sets and model specifications, but not for all.