Extreme values of the stationary distribution of random walks on directed graphs

TL;DR

This paper investigates the extremal behavior of the stationary distribution of random walks on directed graphs, focusing on the principal ratio, providing bounds, characterizations, and constructions of extremal graphs.

Contribution

It establishes an upper bound for the principal ratio on strongly connected graphs, characterizes extremal graphs, and discusses bounds under various conditions.

Findings

Upper bound for principal ratio on strongly connected graphs.

Characterization of graphs achieving the extremal ratio.

Counterexamples showing limitations of bounds under weaker conditions.

Abstract

We examine the stationary distribution of random walks on directed graphs. In particular, we focus on the {\em principal ratio}, which is the ratio of maximum to minimum values of vertices in the stationary distribution. We give an upper bound for this ratio over all strongly connected graphs on $n$ vertices. We characterize all graphs achieving the upper bound and we give explicit constructions for these extremal graphs. Additionally, we show that under certain conditions, the principal ratio is tightly bounded. We also provide counterexamples to show the principal ratio cannot be tightly bounded under weaker conditions.