The dispersion time of random walks on finite graphs

TL;DR

This paper analyzes the dispersion time of two IDLA-inspired random walk processes on graphs, providing bounds and comparisons between sequential and parallel versions across various graph classes.

Contribution

It introduces a coupling method to compare dispersion times of sequential and parallel IDLA processes and derives tight bounds for multiple graph types.

Findings

Parallel-IDLA dispersion time stochastically dominates Sequential-IDLA

Expected dispersion time of Parallel-IDLA is bounded by Sequential-IDLA up to a log n factor

Tight asymptotic bounds on dispersion time for various graph classes

Abstract

We study two random processes on an $n$-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. In both processes $n$ particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until first encountering an unoccupied vertex, and at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called \textit{Sequential-IDLA}, only one particle moves until settling and only then does the next particle start whereas in the second process, called \textit{Parallel-IDLA}, all unsettled particles move simultaneously. Our main goal is to analyze the so-called dispersion time of these processes, which is the maximum number of steps performed by any of the $n$ particles. In order to compare the two processes, we develop a coupling which shows the dispersion time of the Parallel-IDLA stochastically dominates that of the Sequential-IDLA; however, the total number of steps performed by all particles has the same distribution in both processes. This coupling also gives us that dispersion time of Parallel-IDLA is bounded in expectation by dispersion time of the Sequential-IDLA up to a multiplicative $\log n$ factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, $d$-dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.