Are we there yet? When to stop a Markov chain while generating random graphs

TL;DR

This paper introduces two methods to determine the appropriate number of iterations needed to generate independent graph samples using Markov chains, ensuring proper statistical properties in network generation.

Contribution

It proposes two novel approaches for calculating the mixing time of Markov chain graph samplers based on edge occurrence time-series analysis.

Findings

Both methods show N proportional to the number of edges |E|.

Validated on real sparse graphs with 1,000 to 10,000 vertices.

Methods effectively determine when to stop the Markov chain for independent sampling.

Abstract

Markov chains are a convenient means of generating realizations of networks, since they require little more than a procedure for rewiring edges. If a rewiring procedure exists for generating new graphs with specified statistical properties, then a Markov chain sampler can generate an ensemble of graphs with prescribed characteristics. However, successive graphs in a Markov chain cannot be used when one desires independent draws from the distribution of graphs; the realizations are correlated. Consequently, one runs a Markov chain for N iterations before accepting the realization as an independent sample. In this work, we devise two methods for calculating N. They are both based on the binary "time-series" denoting the occurrence/non-occurrence of edge (u, v) between vertices u and v in the Markov chain of graphs generated by the sampler. They differ in their underlying assumptions. We test them on the generation of graphs with a prescribed joint degree distribution. We find the N proportional |E|, where |E| is the number of edges in the graph. The two methods are compared by sampling on real, sparse graphs with 10^3 - 10^4 vertices.