The Best Mixing Time for Random Walks on Trees

TL;DR

This paper identifies extremal tree structures that optimize the mixing time of random walks starting from the best vertex, revealing that stars minimize and certain paths maximize the mixing time depending on the number of vertices.

Contribution

It characterizes the trees that minimize and maximize the best mixing time for random walks, providing a complete extremal analysis for trees.

Findings

Stars uniquely minimize the best mixing time among all trees.

Paths uniquely maximize the best mixing time for even number of vertices.

A modified path with a leaf maximizes the best mixing time for odd number of vertices.

Abstract

We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let $G=(V,E)$ be a tree with stationary distribution $\pi$. For a vertex $v \in V$, let $H(v,\pi)$ denote the expected length of an optimal stopping rule from $v$ to $\pi$. The \emph{best mixing time} for $G$ is $\min_{v \in V} H(v,\pi)$. We show that among all trees with $|V|=n$, the best mixing time is minimized uniquely by the star. For even $n$, the best mixing time is maximized by the uniquely path. Surprising, for odd $n$, the best mixing time is maximized uniquely by a path of length $n-1$ with a single leaf adjacent to one central vertex.