Towards random uniform sampling of bipartite graphs with given degree sequence

TL;DR

This paper analyzes a Markov chain for sampling bipartite graphs with fixed degree sequences, demonstrating polynomial mixing time for semi-regular sequences through a novel canonical path construction.

Contribution

It introduces a new approach to bounding mixing times using canonical paths, specifically for semi-regular bipartite degree sequences.

Findings

Mixing time is polynomial in the number of vertices for semi-regular sequences.

A new canonical path construction improves analysis of the Markov chain.

Provides theoretical bounds on the efficiency of uniform sampling.

Abstract

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of {\em semi-regular} degree sequence. The novelty of our approach lays in the construction of the canonical paths in Sinclair's method.