A decomposition based proof for fast mixing of a Markov chain over balanced realizations of a joint degree matrix

TL;DR

This paper proves that a swap Markov Chain Monte Carlo algorithm for sampling balanced graph realizations of a joint degree matrix mixes rapidly, with a polynomial upper bound on the relaxation time, using a novel factorization approach.

Contribution

It introduces a new factorization theorem for Markov chains and applies it to show rapid mixing of MCMC over balanced JDM realizations.

Findings

The Markov chain has polynomial mixing time.

The factorization theorem aids in analyzing complex Markov chains.

The approach uses conductance and Cheeger inequality methods.

Abstract

A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two degree groups are placed as uniformly as possible. We prove that a swap Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes $n$. To prove fast mixing, we first prove a general factorization theorem similar to the Martin-Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).