Random Walks and the Meeting Time for Trees

TL;DR

This paper analyzes the meeting time of random walks on trees, characterizing extremal structures that maximize or minimize the expected meeting time based on tree shape, order, and diameter.

Contribution

It provides a characterization of extremal tree structures for the meeting time of random walks, identifying the broom and double broom graphs as extremal cases.

Findings

Broom graph maximizes meeting time for fixed order and diameter.

Balanced double broom minimizes meeting time under certain conditions.

Explicit extremal structures depend on tree parameters.

Abstract

Consider a random walk on a tree $G=(V,E)$. For $v,w \in V$, let the hitting time $H(v,w)$ denote the expected number of steps required for the random walk started at $v$ to reach $w$, and let $\pi_v = \mathrm{deg}(v)/2|E|$ denote the stationary distribution for the random walk. We characterize the extremal tree structures for the meeting time $T_{\mathrm{meet}}(G) = \max_{w \in V} \sum_{v \in V} \pi_v H(v,w)$. For fixed order $n$ and diameter $d$, the meeting time is maximized by the broom graph. The meeting time is minimized by the balanced double broom graph, or a slight variant, depending on the relative parities of $n$ and $d$.