Bootstrap percolation on the random graph $G_{n,p}$

TL;DR

This paper analyzes the bootstrap percolation process on random graphs, revealing a sharp phase transition in the final active set size and providing detailed asymptotic behavior and a phase diagram.

Contribution

It offers a complete phase diagram, asymptotic formulas, and a central limit theorem for the bootstrap percolation process on $G_{n,p}$.

Findings

Identifies a sharp phase transition in the final active set size.

Provides asymptotic formulas for the size of the active set.

Proves a central limit theorem for the active set size in certain regimes.

Abstract

Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r\geq2$ active neighbors become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either $n-o(n)$ or it is $o(n)$. We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for $A^*$; we also prove a central limit theorem for $A^*$ in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops.