Tree formulas, mean first passage times and Kemeny's constant of a Markov chain

TL;DR

This paper explores probabilistic and combinatorial tree formulas for Markov chain metrics, linking Green functions, hitting probabilities, and Kemeny's constant to spanning trees and forests, offering new derivations beyond classical theorems.

Contribution

It provides novel probabilistic and combinatorial derivations of tree formulas for Markov chain metrics, bypassing the traditional matrix tree theorem.

Findings

Tree formulas relate mean first passage times to spanning trees.

Kemeny's constant expressed as ratio of forest sums.

New derivations without matrix tree theorem.

Abstract

In this paper, we aim to provide probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilson's algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem. Let $m_{ij}$ be the mean first passage time from $i$ to $j$ for an irreducible chain with finite state space $S$ and transition matrix $(p_{ij}; i, j \in S)$. It is well-known that $m_{jj} = 1/\pi_j = \Sigma^{(1)}/\Sigma_j$, where $\pi$ is the stationary distribution for the chain, $\Sigma_j$ is the tree sum, over $n^{n-2}$ trees $\textbf{t}$ spanning $S$ with root $j$ and edges $i \rightarrow k$ directed to $j$, of the tree product $\prod_{i \rightarrow k \in \textbf{t} }p_{ik}$, and $\Sigma^{(1)}:= \sum_{j \in S} \Sigma_j$. Chebotarev and Agaev derived further results from {\em Kirchhoff's matrix tree theorem}. We deduce that for $i \ne j$, $m_{ij} = \Sigma_{ij}/\Sigma_j$, where $\Sigma_{ij}$ is the sum over the same set of $n^{n-2}$ spanning trees of the same tree product as for $\Sigma_j$, except that in each product the factor $p_{kj}$ is omitted where $k = k(i,j,\textbf{t})$ is the last state before $j$ in the path from $i$ to $j$ in $\textbf{t}$. It follows that Kemeny's constant $\sum_{j \in S} m_{ij}/m_{jj}$ equals to $ \Sigma^{(2)}/\Sigma^{(1)}$, where $\Sigma^{(r)}$ is the sum, over all forests $\textbf{f}$ labeled by $S$ with $r$ trees, of the product of $p_{ij}$ over edges $i \rightarrow j$ of $\textbf{t}$. We show that these results can be derived without appeal to the matrix tree theorem. A list of relevant literature is also reviewed.