Hitting Time Quasi-metric and Its Forest Representation

TL;DR

This paper introduces a forest-based representation of hitting times in Markov chains, revealing new metric structures and algebraic procedures for their computation, with implications for understanding Markov chain properties.

Contribution

It provides a novel forest representation of hitting times and explores related metric structures, offering algebraic methods for efficient calculation and deeper insight into Markov chain metrics.

Findings

Hitting times are expressed via forest weights in a digraph.

A recurrent algebraic procedure computes forest-based quantities.

The paper discusses properties and relationships of the hitting time quasi-metric.

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.