Fastest mixing Markov chain on graphs with symmetries

TL;DR

This paper presents methods to leverage graph symmetries for efficiently computing the fastest mixing Markov chains, reducing computational complexity and enabling solutions for large graphs.

Contribution

It introduces two approaches for symmetry exploitation in Markov chain optimization, including orbit theory and block-diagonalization, with applications to specific graph classes.

Findings

Symmetry exploitation reduces variables and matrix sizes in optimization.

Analytic solutions are obtained for certain classes of symmetric graphs.

The paper establishes a connection between orbit theory and block-diagonalization methods.

Abstract

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.