Exact computation for meeting times and infection times of random walks on graphs

Takuya Ohwa

doi:10.1186/s40736-015-0016-2

Abstract

We consider independent multiple random walks on graphs and study comparison results of meeting times and infection times between many conditions of the random walks by obtaining the exact density functions or expectations.

Highlights

Let G = (V, E) be a finite connected graph and Xt(0), . . . , Xt(k) independent continuous time or discrete time k + 1-multiple random walks on G
We suppose |V | ≥ k + 1 and regard Xt(0) as an infected particle and consider an infection time tinfe: it is the first time that Xt(0) meets any other particles. It is given by tinfe max i∈{1,...,k}
We investigate the exact distributions or expectations for meeting times and infection times

Summary

Introduction

Ohwa Pacific Journal of Mathematics for Industry (2015) 7:5 et al [3], Cooper et al [4] and Coppersmith et al [5] investigated the expected meeting time of two independent Markov chains on a finite graph. Using their results, Draief and Ganesh [7] derived the upper bound for the expected time that all particles are infected for complete graphs and regular graphs. Draief and Ganesh [7] derived the upper bound for the expected time that all particles are infected for complete graphs and regular graphs In their model, the infected probability varies by the coincidence time with infected particles and the infected rate. Kurkova et al [9] investigated the model that the infection rule is same as ours for an infinite square grid

Models and notations

Cycle graphs