Generalized Markov chain tree theorem and Kemeny’s constant for a class of non-Markovian matrices

Abstract

Given an ergodic Markov chain with transition matrix P and stationary distribution π, the classical Markov chain tree theorem expresses π in terms of graph-theoretic parameters associated with the graph of P. For a class of non-stochastic matrices M2 associated with P, recently introduced by the first author in Choi (2020) and Choi and Huang (2020), we prove a generalized version of Markov chain tree theorem in terms of graph-theoretic quantities of M2. This motivates us to define generalized version of mean hitting time, fundamental matrix and Kemeny’s constant associated with M2, and we show that they enjoy similar properties as their counterparts of P even though M2 is non-stochastic. We hope to shed lights on how concepts and results originated from the Markov chain literature, such as the Markov chain tree theorem, Kemeny’s constant or the notion of hitting time, can possibly be extended and generalized to a broader class of non-stochastic matrices via introducing appropriate graph-theoretic parameters. In particular, when P is reversible, the results of this paper reduce to the results of P.