Fastest Mixing Markov Chain on Graphs with Symmetries

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (choose 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 number of optimization variables and size of matrices in solving the corresponding semidefinite programs. The problem can be considerably simplified and is often solvable by only exploiting symmetry. We obtain analytic or semianalytic 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. These approaches enable numerical solution of largescale instances, otherwise computationally infeasible to solve.