Metastable Mixing of Markov Chains: Efficiently Sampling Low Temperature Exponential Random Graphs
Abstract
In this paper we consider the problem of sampling from the lowtemperature 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 lowerdimensional critical set of parameters  when initialized at $G(n,p)$ for the right choice of $p$ has a metastable mixing time of $O(n^2\log n)$ to within total variation distance $\exp(\Omega(n))$.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.13153
 arXiv:
 arXiv:2208.13153
 Bibcode:
 2022arXiv220813153B
 Keywords:

 Mathematics  Probability;
 Mathematics  Statistics Theory
 EPrint:
 No figures. We don't do that around here