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}
}
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… 
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