A structured condition number for Kemeny's constant

Abstract

Kemeny's constant is an interesting and useful quantifier describing the global average behaviour of a Markov chain. In this article, we examine the sensitivity of Kemeny's constant to perturbations in the transition probabilities. That is, we consider the problem of generating a condition number for Kemeny's constant, to give an indication of the size of the change in its value relative to the size of the perturbation. We provide a structured condition number and determine some illuminating upper and lower bounds which connect the conditioning of Kemeny's constant to well-studied condition numbers for the stationary vector of the Markov chain. We also investigate the behaviour of this structured condition number for several infinite families of Markov chains.