Hitting time, access time and optimal transport on graphs

TL;DR

This paper investigates the optimal transport time between distributions on graphs using Markov chains, providing bounds related to hitting times and analyzing specific graph structures like complete graphs and winning streak chains.

Contribution

It introduces bounds for optimal transport time on graphs in terms of hitting times and distribution parameters, and compares transport efficiency across different Markov chain models.

Findings

Random walks on complete graphs have linear growth in transport time with respect to state space size.

Winning streak Markov chains exhibit exponential growth in transport time.

Bounds relate transport time to hitting times and distribution moments.

Abstract

Given a discrete source distribution $\mu$ and discrete target distribution $\nu$ on a common finite state space $\mathcal{X}$, we are tasked with transporting $\mu$ to $\nu$ using a given discrete-time Markov chain $X$ with the quickest possible time on average. We define the optimal transport time $H(\mu,\nu)$ as stopping rule of $X$ that gives the minimial expected transport time. This is also known as the access time from $\mu$ to $\nu$ of $X$ in [L. Lov\'{a}sz and P. Winkler. Efficient Stopping Rules for Markov Chains. Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing (STOC '95) 76-82.]. We study bounds of $H(\mu,\nu)$ in various special graphs, which are expressed in terms of the mean hitting times of $X$ as well as parameters of $\mu$ and $\nu$ such as their moments. Among the Markov chains that we study, random walks on complete graphs is a good choice for transport as $H(\mu,\nu)$ grows linearly in $n$, the size of the state space, while that of the winning streak Markov chain exhibits exponential dependence in $n$.