Hitting Time Quasi-metric and Its Forest Representation

Abstract

Let $\hat m_{ij}$ be the hitting (mean first passage) time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $\Gamma$ be the weighted digraph whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. It holds that $$ \hat m_{ij}= q_j^{-1}\cdot \begin{cases} f_{ij},&\text{if }\;\; i\ne j,\\ q, &\text{if }\;\; i=j, \end{cases} $$ where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $\Gamma$ that have one tree containing $i$ and the other tree converging to $j$, $q_j$ is the total weight of spanning trees converging to $j$ in $\Gamma,$ and $q=\sum_{j=1}^nq_j$ is the total weight of all spanning trees in $\Gamma.$ Moreover, $f_{ij}$ and $q_j$ can be calculated by an algebraic recurrent procedure. A forest expression for Kemeny's constant is an immediate consequence of this result. Further, we discuss the properties of the hitting time quasi-metric $m$ on the set of vertices of $\Gamma$: $m(i,j)=\hat m_{ij}$, $i\neq j$, and $m(i,i)=0$. We also consider a number of other metric structures on the set of graph vertices related to the hitting time quasi-metric $m$---along with various connections between them. The notions and relationships under study are illustrated by two examples.