# The switch Markov chain for sampling irregular graphs and digraphs

## TL;DR

This paper proves that the switch Markov chain efficiently samples irregular undirected and directed graphs with certain degree constraints, extending rapid mixing results beyond regular graphs.

## Contribution

It establishes new rapid mixing bounds for the switch chain on irregular graphs and digraphs with specific degree conditions, broadening applicability.

## Key findings

- Rapid mixing for undirected graphs with degree constraints
- Rapid mixing for directed graphs with positive degrees
- Mixing time bounds are close to regular case results

## Abstract

The problem of efficiently sampling from a set of (undirected, or directed) 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, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree $d_{\max}$ which satisfies $3\leq d_{\max}\leq \frac{1}{3}\, \sqrt{M}$, where $M$ is the sum of the degrees. The mixing time bound obtained is only a factor $n$ larger than that established in the regular case, where $n$ is the number of vertices. Our result covers a wide range of degree sequences, including power-law graphs with parameter $\gamma > 5/2$ and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by $\frac{1}{4}\, \sqrt{m}$, where $m$ is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of $m^2$ larger than that established in the regular case.