Hitting time quasi-metric and its forest representation

Abstract

Let 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 Γ 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 thatwhere f ij is the total weight of 2-tree spanning converging forests in Γ 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 Γ, and q = n j=1 q j is the total weight of all spanning trees in Γ. 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 Γ: m(i, j) = m ij , i = 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.

On resistance distance of Markov chain and its sum rules

“…Let q j to be total weight of in-trees rooted at j and q := j∈X q j . According to the Markov chain tree theorem Anantharam and Tsoucas (1989) and recent results in Chebotarev (2007); Chebotarev and Deza (2018), one can express the stationary distribution and mean hitting time via these graph-theoretic parameters as, for i, j ∈ X ,…”

Families of graphs with twin pendent paths and the Braess edge

“…Moreover, the resistance distance is a metric on V (G) (see [2]). As proved in [7], the number f G i,j of 2-tree spanning forests of G having i and j in different trees is f G i,j = τ G Ω i,j . Therefore, we have the following properties endowed by the metric Ω i,j :…”

Families of graphs with twin pendent paths and the Braess edge

“…Moreover, the resistance distance is a metric on V (G) (see [2]). As proved in [7], the number f G i,j of 2-tree spanning forests of G having i and j in different trees is…”