Frogs on trees?

TL;DR

This paper analyzes the spread of activation in a frog model on d-ary trees, establishing bounds on the infection time and cover time, revealing how the initial particle density influences the epidemic's progression.

Contribution

It provides probabilistic bounds on the infection and cover times in the frog model on trees, a novel analysis of the epidemic spread dynamics in this setting.

Findings

High probability bounds on the infection time scale with respect to initial density and tree size.

Upper bound on cover time showing it grows sub-exponentially with the number of vertices.

Dependence of the spread speed on initial particle density and tree parameters.

Abstract

We study a system of simple random walks on $\mathcal{T}_{d,n} = \mathcal{V}_{d,n}, \mathcal{E}_{d,n})$, the $d$-ary tree of depth $n$, known as the frog model. Initially there are Pois($\lambda$) particles at each site, independently, with one additional particle planted at some vertex $\mathbf{o}$. Initially all particles are inactive, except for the ones which are placed at $\mathbf{o}$. Active particles perform (independent) $ t \in \mathbb{N} \cup \{\infty \} $ steps of simple random walk on the tree. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let $\mathcal{R}_t$ be the set of vertices which are visited by the process. Let $\mathcal{S}(\mathcal{T}_{d,n}) := \inf \{t:\mathcal{R}_t = \mathcal{V}_{d,n} \} $. Let the cover time $\mathrm{CT}(\mathcal{T}_{d,n})$ be the first time by which every vertex was visited at least once, when we take $t=\infty$. We show that there exist absolute constants, $c,C>0$ such that for all $d \ge 2$ and all $\lambda = \lambda_n $ which does not diverge nor vanish too rapidly, with high probability $c \le \lambda \mathcal{S}(\mathcal{T}_{d,n}) /n \log (n/\lambda) \le C$ and $\mathrm{CT}(\mathcal{T}_{d,n}) \le 3^{4 \sqrt{ \log |\mathcal{V}_{d,n}| }}$.