Mixing time of exponential random graphs

TL;DR

This paper analyzes the mixing times of Markov chain Monte Carlo methods for exponential random graph models, revealing rapid mixing in high temperature regimes and slow mixing in low temperature regimes, impacting their practical use.

Contribution

It characterizes the mixing time regimes of exponential random graph models, providing rigorous insights into their sampling complexity and limitations.

Findings

High temperature regime: mixing time is Θ(n^2 log n)

Low temperature regime: mixing is exponentially slow

In high temperature, edges are asymptotically independent

Abstract

Exponential random graphs are used extensively in the sociology literature. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing, and more generally for understanding basic features of the network model itself. In practice sampling is typically carried out using Markov chain Monte Carlo, in particular either the Glauber dynamics or the Metropolis-Hasting procedure. In this paper we characterize the high and low temperature regimes of the exponential random graph model. We establish that in the high temperature regime the mixing time of the Glauber dynamics is $\Theta(n^2 \log n)$, where $n$ is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the Erd\H{o}s-R\'enyi random graph.