# On resistance distance of Markov chain and its sum rules

## TL;DR

This paper introduces a new resistance distance measure for finite ergodic Markov chains, explores its properties, and establishes sum rules analogous to those in graph theory, linking it to classical Markov chain parameters.

## Contribution

It defines a novel resistance distance for Markov chains, derives equivalent formulations, and develops sum rules similar to those in graph resistance theory.

## Key findings

- New resistance distance relates to fundamental matrix and eigenvalues.
- Sum rules for resistance distance mirror classical graph formulas.
- Connections established with hitting times, stationary distribution, and group inverse.

## Abstract

Motivated by the notion of resistance distance on graph, we define a new resistance distance between two states on a given finite ergodic Markov chain based on its fundamental matrix. We prove a few equivalent formulations and discuss its relation with other parameters of the Markov chain such as its group inverse, stationary distribution, eigenvalues or hitting time. In addition, building upon existing sum rules for the hitting time of Markov chain, we give sum rules of this new resistance distance of Markov chains that resembles the sum rules of the resistance distance on graph. This yields Markov chain counterparts of various classical formulae such as Foster's first formula or the Kirchhoff index formulae.