# Laws of large numbers for the frog model on the complete graph

## TL;DR

This paper investigates the frog model on complete graphs, proving that as the number of vertices grows large, the process's behavior can be approximated by a deterministic dynamical system, revealing insights into epidemic spreading dynamics.

## Contribution

The paper establishes laws of large numbers for the frog model on complete graphs, connecting stochastic processes to deterministic dynamical systems for large network sizes.

## Key findings

- Process approximated by 3D dynamical system as N→∞
- Long-term behavior of the deterministic system analyzed
- Different lifetime models for active particles studied

## Abstract

The frog model is a stochastic model for the spreading of an epidemic on a graph, in which a dormant particle starts to perform a simple random walk on the graph and to awake other particles, once it becomes active. We study two versions of the frog model on the complete graph with $N + 1$ vertices. In the first version we consider, active particles have geometrically distributed lifetimes. In the second version, the displacement of each awakened particle lasts until it hits a vertex already visited by the process. For each model, we prove that as $N \to \infty$, the trajectory of the process is well approximated by a three-dimensional discrete-time dynamical system. We also study the long-term behavior of the corresponding deterministic systems.