The switch Markov chain for sampling irregular graphs

TL;DR

This paper proves that the switch Markov chain efficiently samples irregular graphs with certain degree constraints, extending rapid mixing results from regular to more general degree sequences.

Contribution

It establishes rapid mixing of the switch chain for irregular degree sequences with minimum degree at least 1 and bounded maximum degree, broadening applicability.

Findings

Proves rapid mixing for degree sequences with $d_{max} \,\leq\, \frac{1}{4}\sqrt{M}$

Mixing time is only slightly larger than in the regular case

Extends sampling efficiency to a wider class of irregular graphs

Abstract

The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{4}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only an order of $n$ larger than that established in the regular case, where $n$ is the number of vertices.