Metastable mixing of Markov chains: Efficiently sampling low temperature exponential random graphs

Abstract

In this paper, we consider the problem of sampling from the low-temperature exponential random graph model (ERGM). The usual approach is via Markov chain Monte Carlo, but Bhamidi et al. showed that any local Markov chain suffers from an exponentially large mixing time due to metastable states. We instead consider metastable mixing, a notion of approximate mixing relative to the stationary distribution, for which it turns out to suffice to mix only within a collection of metastable states. We show that the Glauber dynamics for the ERGM at any temperature—except at a lower-dimensional critical set of parameters—when initialized at $\mathit{G}(\mathit{n},\mathit{p})$ for the right choice of p has a metastable mixing time of $\mathit{O}({\mathit{n}}^{2}log\mathit{n})$ to within total variation distance $exp(-\mathrm{\Omega }(\mathit{n}))$.