The mixing time of the switch Markov chains: a unified approach

Abstract

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any P -stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent γ > 1 + √ 3. The other one shows that the switch Markov chain on the degree sequence of an Erds-Rnyi random graph G(n, p) is asymptotically almost surely rapidly mixing if p is bounded away from 0 and 1 by at least 5 log n n−1 .

Asymptotic enumeration of hypergraphs by degree sequence

“…The present paper provides such formulae. In addition, enumeration results were used in [10] for uniformly sampling graphs with a given power law degree sequence using Monte Carlo Markov chains (MCMC). Uniform sampling of hypergraphs using MCMC has not advanced as far theoretically as the graph case, and our enumeration results may have implications there as well.…”

Sampling hypergraphs with given degrees

“…The chain has been used in many contexts, including contingency tables [14], and was first applied to bipartite graphs by Kannan, Tetali and Vempala [22]. The mixing time of the switch chain has been shown to be polynomial for various bipartite and general degree sequences, see for example [1,10,16,19,25,27]. If a Markov chain leads to an FPAUS then we say that the Markov chain is rapidly mixing.…”

Cutoff for Rewiring Dynamics on Perfect Matchings

“…Analysis of the switch chain has a long and rich history; far too much to discuss in totality here. Recent overviews can be found in the introductions of the recent papers [AK19, AK20], [Erd+19] or [TY20]. It was introduced by Kannan, Tetali and Vempala [KTV97,KTV99] in the late 90s, making it over 20 years old.…”