Sensitivity of mixing times

TL;DR

This paper presents an example of bounded-degree graphs where adjusting edge conductances significantly reduces the total variation mixing time of the random walk, highlighting the sensitivity of mixing times to conductance modifications.

Contribution

It introduces a specific graph construction demonstrating how bounded conductance changes can substantially decrease mixing times, revealing their sensitivity.

Findings

Mixing time can be decreased by a factor of rac{\u2212log n}{\u2212log\u2217log n} through conductance adjustments.

Bounded-degree graphs can exhibit high sensitivity of mixing times to conductance modifications.

The result provides insight into how conductance influences the efficiency of random walks on graphs.

Abstract

In this note, we demonstrate an instance of bounded-degree graphs of size $n$, for which the total variation mixing time for the random walk is decreased by a factor of $\log n/ \log\log n$ if we multiply the edge-conductances by bounded factors in a certain way.