The role of Kemeny's constant in properties of Markov chains

Abstract

In a finite m-state irreducible Markov chain with stationary probabilities {πi} and mean first passage times mij (mean recurrence time when i = j) it was first shown, by Kemeny and Snell, that the sum, over j, of πj and mij is a constant, K, not depending on i. This constant has since become known as Kemeny’s constant. We consider a variety of techniques for finding expressions for K, derive some bounds for K, and explore various applications and interpretations of these results. Interpretations include the expected number of links that a surfer on the World Wide Web located on a random page needs to follow before reaching a desired location, as well as the expected time to mixing in a Markov chain. Various applications have been considered including some perturbation results, mixing on directed graphs and its relation to the Kirchhoff index of regular graphs.