A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs

TL;DR

This paper proves that the switch chain Markov process rapidly mixes for sampling regular directed graphs, providing a polynomial bound on the mixing time by analyzing eigenvalues and extending existing methods.

Contribution

It establishes ergodicity and rapid mixing of the switch chain for regular directed graphs, introducing a new eigenvalue bound and extending multicommodity flow techniques.

Findings

The switch chain is ergodic for regular degree sequences.

The mixing time is polynomially bounded for regular directed graphs.

A new eigenvalue bound improves analysis of the chain's convergence.

Abstract

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree d, where d is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock, and by using it we avoid working with a lazy chain. A multicommodity flow argument is used to bound the second-largest eigenvalue of the chain. This argument is based on the analysis of a related Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill, but with significant extension required.