Kemeny's constant minimization for reversible Markov chains via structure-preserving perturbations

TL;DR

This paper explores how to optimize the connectivity of reversible Markov chains by applying structure-preserving perturbations to minimize Kemeny's constant, balancing improvements with feasibility constraints.

Contribution

It formulates the problem of minimizing Kemeny's constant under sparsity constraints as an optimization task and develops algorithms to find feasible solutions.

Findings

Estimation of the minimum achievable Kemeny's constant.

Identification of feasibility issues with certain perturbations.

Development of efficient algorithms for structured perturbations.

Abstract

Kemeny's constant measures the efficiency of a Markov chain in traversing its states. We investigate whether structure-preserving perturbations to the transition probabilities of a reversible Markov chain can improve its connectivity while maintaining a fixed stationary distribution. Although the minimum achievable value for Kemeny's constant can be estimated, the required perturbations may be infeasible. We reformulate the problem as an optimization task, focusing on solution existence and efficient algorithms, with an emphasis to the problem of minimizing Kemeny's constant under sparsity constraints.