Sensitivity of mixing times in Eulerian digraphs

TL;DR

This paper investigates how the mixing times of lazy random walks behave on Eulerian digraphs, revealing bounds, sensitivities, and universal properties that extend understanding from undirected graphs to directed Eulerian structures.

Contribution

It establishes new bounds for mixing, exploration, and cover times on Eulerian digraphs and uncovers the sensitivity of mixing times to laziness parameters in directed cases.

Findings

Mixing time on Eulerian digraphs is O(mn).

Mixing times are robust to laziness changes in reversible cases.

In directed graphs, mixing times can be sensitive to laziness parameters.

Abstract

Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cover times for random walks on Eulerian digraphs and prove universal upper bounds in analogy to the undirected case.