Exponential Random Graph Models with Big Networks: Maximum Pseudolikelihood Estimation and the Parametric Bootstrap

TL;DR

This paper compares maximum pseudolikelihood estimation and Monte Carlo maximum likelihood for large network ERGMs, demonstrating that bootstrapped MPLE offers a faster and accurate alternative for big networks.

Contribution

It introduces a parametric bootstrap method to improve MPLE accuracy and compares its performance to MCMLE in large network ERGMs.

Findings

Bootstrapped MPLE achieves similar accuracy to MCMLE.

Bootstrapped MPLE runs in one-fifth the time of MCMLE.

The methods are demonstrated on a U.S. Senate bills network.

Abstract

With the growth of interest in network data across fields, the Exponential Random Graph Model (ERGM) has emerged as the leading approach to the statistical analysis of network data. ERGM parameter estimation requires the approximation of an intractable normalizing constant. Simulation methods represent the state-of-the-art approach to approximating the normalizing constant, leading to estimation by Monte Carlo maximum likelihood (MCMLE). MCMLE is accurate when a large sample of networks is used to approximate the normalizing constant. However, MCMLE is computationally expensive, and may be prohibitively so if the size of the network is on the order of 1,000 nodes (i.e., one million potential ties) or greater. When the network is large, one option is maximum pseudolikelihood estimation (MPLE). The standard MPLE is simple and fast, but generally underestimates standard errors. We show that a resampling method---the parametric bootstrap---results in accurate coverage probabilities for confidence intervals. We find that bootstrapped MPLE can be run in 1/5th the time of MCMLE. We study the relative performance of MCMLE and MPLE with simulation studies, and illustrate the two different approaches by applying them to a network of bills introduced in the United State Senate.