February 2024 Metastable mixing of Markov chains: Efficiently sampling low temperature exponential random graphs
Guy Bresler, Dheeraj Nagaraj, Eshaan Nichani
Author Affiliations +
Ann. Appl. Probab. 34(1A): 517-554 (February 2024). DOI: 10.1214/23-AAP1971


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 G(n,p) for the right choice of p has a metastable mixing time of O(n2logn) to within total variation distance exp(Ω(n)).

Funding Statement

This work was supported in part by NSF CAREER award CCF-1940205 and NSF Award DMS-2022448.


G.B. and D.N. gratefully acknowledge the hospitality of the Simons Institute for Theoretical Computer Science during Fall 2020 and Fall 2021.


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Guy Bresler. Dheeraj Nagaraj. Eshaan Nichani. "Metastable mixing of Markov chains: Efficiently sampling low temperature exponential random graphs." Ann. Appl. Probab. 34 (1A) 517 - 554, February 2024. https://doi.org/10.1214/23-AAP1971


Received: 1 October 2022; Revised: 1 March 2023; Published: February 2024
First available in Project Euclid: 28 January 2024

MathSciNet: MR4696284
zbMATH: 07829149
Digital Object Identifier: 10.1214/23-AAP1971

Primary: 60J10
Secondary: 68Q87

Keywords: efficient sampling , exponential random graphs , Glauber dynamics , Markov chain

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.34 • No. 1A • February 2024
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