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

Abstract

We show that the swap Markov chain is rapidly mixing on $P$-stable degree sequences of simple, bipartite, and directed graphs. Consequently, we have rapid mixing on power-law distribution-bounded degree sequences with parameter $\gamma>2$ and on Erdős-Rényi graphs with arbitrary edge probability (aas).

Publication
arXiv preprint

Since 1997 a considerable effort has been spent to study the mixing time of swap (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 swap Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the swap 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 $\gamma>2$. The other one shows that the swap Markov chain on the degree sequence of an Erdős-Rényi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing.