Cover times for sequences of reversible Markov chains on random graphs

TL;DR

This paper classifies cover times of reversible Markov chains on random graphs into two types based on graph parameters, providing a framework to understand how graph structure influences random walk cover times.

Contribution

It introduces conditions based on resistance metrics and graph parameters to classify cover times into two distinct types, applied to various complex graph models.

Findings

Cover times are classified into two types based on resistance metrics.

Conditions involve graph volume, diameter, and coverings in the resistance metric.

Applications include Galton-Watson trees and Sierpinski gasket graphs.

Abstract

We provide conditions that classify cover times for sequences of random walks on random graphs into two types: One type (Type 1) is the class of cover times that are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (Type 2) is the class of cover times that are of the order of the maximal hitting times. The conditions are described by some parameters determined by the underlying graphs: the volumes, the diameters with respect to the resistance metric, the coverings or packings by balls in the resistance metric. We apply the conditions to and classify a number of examples, such as supercritical Galton-Watson trees, the incipient infinite cluster of a critical Galton-Watson tree and the Sierpinski gasket graph.