Half-graphs, other non-stable degree sequences, and the switch Markov chain

TL;DR

This paper identifies a new family of degree sequences for which the switch Markov chain mixes rapidly, expanding understanding beyond previously known stable classes, with implications for random graph generation.

Contribution

It introduces a non-trivial family of degree sequences not covered by prior stability conditions where the switch Markov chain still exhibits rapid mixing.

Findings

Rapid mixing for new degree sequence family

Connection to Tyshkevich-decompositions and strong stability

Extends known classes of sequences with efficient sampling

Abstract

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called $P$-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not $P$-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.