• DOI:10.1007/s11590-018-1314-2
• Corpus ID: 119128691

Hitting time quasi-metric and its forest representation

@article{Chebotarev2020HittingTQ,
  title={Hitting time quasi-metric and its forest representation},
  author={Pavel Yu. Chebotarev and Elena Deza},
  journal={Optimization Letters},
  year={2020},
  volume={14},
  pages={291-307}
}

• P. Chebotarev, E. Deza
• Published 1 January 2018
• Mathematics
• Optimization Letters

Let $$\widehat{m}_{ij}$$ 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 $$\Gamma $$ 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 that $$\begin{aligned} \widehat{m}_{ij}= q_j^{-1}\cdot {\left\{ \begin{array}{ll} f_{ij},&{}\text {if }\;\; i\ne j,\\ q, &{}\text {if }\;\; i=j, \end{array}\right. } \end{aligned}$$ m ^ ij = q j - 1 · f ij , if i ≠ j , q , if i = j , where $$f_{ij}$$ f ij is the total weight of 2-tree spanning converging forests in $$\Gamma $$ Γ that have one tree containing  i and the other tree converging to j , $$q_j$$ q j is the total weight of spanning trees converging to  j in  $$\Gamma ,$$ Γ , and $$q=\sum _{j=1}^nq_j$$ q = ∑ j = 1 n q j is the total weight of all spanning trees in  $$\Gamma .$$ Γ . Moreover, $$f_{ij}$$ f ij and $$q_j$$ 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 $$\Gamma $$ Γ : $$m(i,j)=\widehat{m}_{ij}$$ m ( i , j ) = m ^ ij , $$i\ne j$$ i ≠ j , and $$m(i,i)=0$$ 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.