A stopping criterion for Markov chains when generating independent random graphs

TL;DR

This paper proposes a practical stopping criterion for Markov chain-based graph generation, ensuring independence of samples by relating chain length to the number of edges, supported by theoretical and empirical evidence.

Contribution

It introduces a mathematically justified method to determine Markov chain length proportional to the number of edges for independent graph sampling.

Findings

Chain length proportional to number of edges suffices for independence.

Empirical convergence of graph properties at the proposed chain length.

Provides a practical guideline for practitioners in network generation.

Abstract

Markov chains are convenient means of generating realizations of networks with a given (joint or otherwise) degree distribution, since they simply require a procedure for rewiring edges. The major challenge is to find the right number of steps to run such a chain, so that we generate truly independent samples. Theoretical bounds for mixing times of these Markov chains are too large to be practically useful. Practitioners have no useful guide for choosing the length, and tend to pick numbers fairly arbitrarily. We give a principled mathematical argument showing that it suffices for the length to be proportional to the number of desired number of edges. We also prescribe a method for choosing this proportionality constant. We run a series of experiments showing that the distributions of common graph properties converge in this time, providing empirical evidence for our claims.