# Smaller Universes for Uniform Sampling of 0,1-matrices with fixed row and column sums

## TL;DR

This paper investigates Markov chain algorithms for sampling 0,1 matrices with fixed row and column sums, aiming to understand why practical convergence is faster than theoretical bounds suggest.

## Contribution

It demonstrates that ignoring node labels in Markov chains leads to faster convergence, providing insights into the discrepancy between theory and practice.

## Key findings

- Projected Markov chains converge faster than original chains
- Ignoring node labels simplifies sampling without loss of generality
- Experimental results show much fewer steps are needed for good samples

## Abstract

An important problem arising in the study of complex networks, for instance in community detection and motif finding, is the sampling of graphs with fixed degree sequence. The equivalent problem of generating random 0,1 matrices with fixed row and column sums is frequently used as a quantitative tool in ecology. It has however proven very challenging to design sampling algorithms that are both fast and unbiased. This article focusses on Markov chain approaches for sampling, where a close-to-random graph is produced by applying a large number N of small changes to a given graph. Examples are the switch chain and Curveball chain, which are both commonly used by practitioners as they are easy to implement and known to sample unbiased when N is large enough. Within theoretical research, much effort has gone into proving bounds on N. However, existing theoretical bounds are impractically large for most applications while experiments suggest that much fewer steps are needed to obtain a good sample. The contribution of this article is twofold. Firstly it is a step towards better understanding of the discrepancy between experimental observations and theoretically proven bounds. In particular, we argue that while existing Markov chain algorithms run on the set of all labelled graphs with a given degree sequence, node labels are unimportant in practice and are usually ignored in determining experimental bounds. We prove that ignoring node labels corresponds to projecting a Markov chain onto equivalence classes of isomorphic graphs and that the resulting projected Markov chain converges to its stationary distribution at least as fast as the original Markov chain. Often convergence is much faster, as we show in examples, explaining part of the difference between theory and experiments...