Frog model wakeup time on the complete graph

TL;DR

This paper analyzes the frog model on complete graphs, providing an elementary proof that the expected wake-up time scales as Theta(log n) and deriving an explicit distributional equation for it.

Contribution

It offers a new elementary proof for the wake-up time scaling and presents an explicit distributional equation for the wake-up time in the frog model.

Findings

Expected wake-up time is Theta(log n)

Derived an explicit distributional equation for wake-up time

Simplified understanding of the frog model on complete graphs

Abstract

The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wake-up time, i.e. the expected time for every particle to be activated, is Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a weighted sum of geometric random variables. This project was part of the University of Washington Research Experience for Undergraduates program.