Kemeny's constant and the Lemoine point of a simplex

TL;DR

This paper presents a new geometric expression for Kemeny's constant of Markov chains, linking it to the distance between the circumcenter and Lemoine point in a related simplex, enhancing understanding of chain mixing properties.

Contribution

It introduces a novel geometric formulation of Kemeny's constant involving simplex points, connecting Markov chain invariants with geometric and resistance-based interpretations.

Findings

Derived a geometric expression involving simplex points

Connected Kemeny's constant with effective resistances and simplex geometry

Provided a new perspective on Markov chain mixing properties

Abstract

Kemeny's constant is an invariant of discrete-time Markov chains, equal to the expected number of steps between two states sampled from the stationary distribution. It appears in applications as a concise characterization of the mixing properties of a Markov chain and has many alternative definitions. In this short article, we derive a new geometric expression for Kemeny's constant, which involves the distance between two points in a simplex associated to the Markov chain: the circumcenter and the Lemoine point. Our proof uses an expression due to Wang, Dubbeldam and Van Mieghem of Kemeny's constant in terms of effective resistances and Fiedler's interpretation of effective resistances as edge lengths of a simplex.