The Role of Kemeny's Constant in Properties of Markov Chains

Abstract

In a finite state irreducible Markov chain with stationary probabilities \pi_i and mean first passage times m_(ij) (mean recurrence time when i = j) it was first shown by Kemeny and Snell (1960) that \sum_j \pi_j m_(ij) is a constant K, not depending on i. This constant has since become known as Kemeny's constant. A variety of techniques for finding expressions and various bounds for K are derived. The main interpretation focuses on its role as the expected time to mixing in a Markov chain. Various applications are considered including perturbation results, mixing on directed graphs and its relation to the Kirchhoff index of regular graphs.