Mixing Time of Vertex-Weighted Exponential Random Graphs

TL;DR

This paper analyzes the efficiency of Glauber dynamics for sampling vertex-weighted exponential random graphs, identifying regions with fast mixing times and a critical phase with intermediate mixing behavior.

Contribution

It characterizes the mixing time behavior of Glauber dynamics in different parameter regions for vertex-weighted exponential random graphs, including high, low, and critical temperature phases.

Findings

High temperature phase: mixing time is Θ(n log n)

Low temperature phase: mixing time is exponentially slow, e^{Ω(n)}

Critical curve: mixing time is O(n^{2/3})

Abstract

Exponential random graph models have become increasingly important in the study of modern networks ranging from social networks, economic networks, to biological networks. They seek to capture a wide variety of common network tendencies such as connectivity and reciprocity through local graph properties. Sampling from these exponential distributions is crucial for parameter estimation, hypothesis testing, as well as understanding the features of the network in question. We inspect the efficiency of a popular sampling technique, the Glauber dynamics, for vertex-weighted exponential random graphs. Letting $n$ be the number of vertices in the graph, we identify a region in the parameter space where the mixing time for the Glauber dynamics is $\Theta(n \log n)$ (the high temperature phase) and a complement region where the mixing time is exponentially slow on the order of $e^{\Omega(n)}$ (the low temperature phase). Lastly, we give evidence that along a critical curve in the parameter space the mixing time is $O(n^{2/3})$.