Fastest Mixing Reversible Markov Chain: Clique Lifted Graphs and Subgraphs

TL;DR

This paper investigates how to optimize the mixing time of reversible Markov chains on various graph structures, introducing clique liftings and subgraph analysis to achieve faster convergence to a stationary distribution.

Contribution

It demonstrates that the FMRMC problem on clique lifted graphs reduces to the base graph, and provides methods to determine optimal transition probabilities on subgraphs independently.

Findings

Optimal mixing times are identical for clique lifted graphs and base graphs.

The FMRMC problem can be solved via semidefinite programming for various topologies.

Transition probabilities on subgraphs can be optimized independently.

Abstract

Markov chains are one of the well-known tools for modeling and analyzing stochastic systems. At the same time, they are used for constructing random walks that can achieve a given stationary distribution. This paper is concerned with determining the transition probabilities that optimize the mixing time of the reversible Markov chains towards a given equilibrium distribution. This problem is referred to as the Fastest Mixing Reversible Markov Chain (FMRMC) problem. It is shown that for a given base graph and its clique lifted graph, the FMRMC problem over the clique lifted graph is reducible to the FMRMC problem over the base graph, while the optimal mixing times on both graphs are identical. Based on this result and the solution of the semidefinite programming formulation of the FMRMC problem, the problem has been addressed over a wide variety of topologies with the same base graph. Second, the general form of the FMRMC problem is addressed on stand-alone topologies as well as subgraphs of an arbitrary graph. For subgraphs, it is shown that the optimal transition probabilities over edges of the subgraph can be determined independent of rest of the topology.