The coverage ratio of the frog model on complete graphs

TL;DR

This paper analyzes the coverage ratio of the frog model on complete graphs, examining how the proportion of visited vertices evolves as the graph size grows, with particles performing random walks and activating others until death.

Contribution

It provides a limit analysis of the coverage ratio for the frog model on complete graphs, a novel focus on the asymptotic behavior in this setting.

Findings

The coverage ratio converges to a specific limit as the number of vertices increases.

The model characterizes the activation and death process of particles on complete graphs.

Results offer insights into the spread dynamics of interacting random walks on large networks.

Abstract

The frog model is a system of interacting random walks. Initially, there is one particle at each vertex of a connected graph $\mathcal{G}$. All particles are inactive at time zero, except for the one which is placed at the root of $\mathcal{G}$, which is active. At each instant of time, each active particle may die with probability $1-p$. Once an active particle survives, it jumps on one of its nearest vertices, chosen with uniform probability, performing a discrete time simple symmetric random walk (SRW) on $\mathcal{G}$. Up to the time it dies, it activates all inactive particles it hits along its way. From the moment they are activated on, every such particle starts to walk, performing exactly the same dynamics, independent of everything else. In this paper, we take $\mathcal{G}$ as the $n-$complete graph ($\mathcal{K}_n$, a finite graph with each pair of vertices linked by an edge). We study the limit in $n$ of the coverage ratio, that is, the proportion of visited vertices by some active particle up to the end of the process, after all active particles have died.