Critical Conditions for the Coverage of Complete Graphs with the Frog Model

TL;DR

This paper analyzes the frog model on complete graphs, identifying conditions under which the entire graph is visited, especially as the activation probability approaches one, and determines the critical growth rate of this probability.

Contribution

It establishes the critical threshold for activation probability ensuring complete coverage of the graph in the frog model.

Findings

For $p_n o 1$, almost all vertices are visited with high probability.

The critical growth rate $p_n=1-rac{eta}{ ext{log} n}$ determines full coverage based on $eta$ and $E( ext{eta})$.

Complete graph coverage occurs with high probability when $eta < E( ext{eta})$, and not when $eta > E( ext{eta})$.

Abstract

We consider a system of interacting random walks known as the frog model. Let $\mathcal{K}_n=(\mathcal{V}_n,\mathcal{E}_n)$ be the complete graph with $n$ vertices and $o\in\mathcal{V}_n$ be a special vertex called the root. Initially, $1+\eta_o$ active particles are placed at the root and $\eta_v$ inactive particles are placed at each other vertex $v\in\mathcal{V}_n\setminus\{o\}$, where $\{\eta_v\}_{v\in \mathcal{V}_n}$ are i.i.d. random variables. At each instant of time, each active particle may die with probability $1-p$. Every active particle performs a simple random walk on $\mathcal{K}_n$ until the moment it dies, activating all inactive particles it hits along its path. Let $V_\infty(\mathcal{K}_n,p)$ be the total number of visited vertices by some active particle up to the end of the process, after all active particles have died. In this paper, we show that $V_\infty(\mathcal{K}_n,p_n)\geq (1-\epsilon)n$ with high probability for any fixed $\epsilon>0$ whenever $p_n\rightarrow 1$. Furthermore, we establish the critical growth rate of $p_n$ so that all vertices are visited. Specifically, we show that if $p_n=1-\frac{\alpha}{\log n}$, then $V_\infty(\mathcal{K}_n,p_n)=n$ with high probability whenever $0<\alpha<E(\eta)$ and $V_\infty(\mathcal{K}_n,p_n)<n$ with high probability whenever $\alpha>E(\eta)$.