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

TL;DR

This paper explores Kemeny's constant in finite irreducible Markov chains, discussing its properties, bounds, and applications in mixing times, graph theory, and network analysis.

Contribution

It provides new expressions, bounds, and interpretations of Kemeny's constant, highlighting its significance in Markov chain mixing and related applications.

Findings

Kemeny's constant is independent of the starting state.

Derived bounds and expressions for Kemeny's constant.

Applications include perturbation analysis and graph mixing properties.

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.