Ranking graphs through Markov chains and hitting times

TL;DR

This paper demonstrates how to construct a Markov chain on any directed graph such that the ordering of hitting times reflects the graph's edges, connecting to paradoxes like non-transitive dice.

Contribution

It introduces a method to encode directed graph relations into hitting times of a Markov chain, revealing new links between graph theory, Markov processes, and probability paradoxes.

Findings

Hitting times can be designed to encode graph edges

Probability of one hitting time exceeding another reflects graph direction

Connects Markov chain properties with non-transitive probability paradoxes

Abstract

In the present paper we show that for any given digraph $\mathbb{G} =([n], \vec{E})$, i.e. an oriented graph without self-loops and 2-cycles, one can construct a 1-dependent Markov chain and $n$ identically distributed hitting times $T_1, \ldots , T_n $ on this chain such that the probability of the event $T_i > T_j $, for any $i, j = 1, \ldots n$, is larger than $\frac{1}{2}$ if and only if $(i,j)\in \vec{E}$. This result is related to various paradoxes in probability theory, concerning in particular non-transitive dice.