Utilizing Network Structure to Bound the Convergence Rate in Markov Chain Monte Carlo Algorithms

TL;DR

This paper introduces a method to improve convergence bounds for Markov Chain Monte Carlo algorithms in large networks by exploiting quasi-lumpable structures, enabling faster subset measure estimation.

Contribution

It provides a new theoretical bound for convergence rates in quasi-lumpable Markov chains, enhancing MCMC efficiency in clustered network structures.

Findings

Bounded convergence rate in quasi-lumpable chains

Numerical evidence of accelerated subset measure estimation

Applicable to networks with clustered node behavior

Abstract

We consider the problem of estimating the measure of subsets in very large networks. A prime tool for this purpose is the Markov Chain Monte Carlo (MCMC) algorithm. This algorithm, while extremely useful in many cases, still often suffers from the drawback of very slow convergence. We show that in a special, but important case, it is possible to obtain significantly better bounds on the convergence rate. This special case is when the huge state space can be aggregated into a smaller number of clusters, in which the states behave {\em approximately} the same way (but their behavior still may not be identical). A Markov chain with this structure is called {\em quasi-lumpable}. This property allows the {\em aggregation} of states (nodes) into clusters. Our main contribution is a rigorously proved bound on the rate at which the aggregated state distribution approaches its limit in quasi-lumpable Markov chains. We also demonstrate numerically that in certain cases this can indeed lead to a significantly accelerated way of estimating the measure of subsets. The result can be a useful tool in the analysis of complex networks, whenever they have a clustering that aggregates nodes with similar (but not necessarily identical) behavior.