A large deviation approach to super-critical bootstrap percolation on the random graph $G_{n,p}$

TL;DR

This paper analyzes the final size of an epidemic process on Erdős–Rényi graphs using large deviation principles, providing detailed asymptotic results in a super-critical regime.

Contribution

It offers a quantitative, large deviation analysis of bootstrap percolation on $G_{n,p}$, extending previous results with explicit rate functions and broad scaling.

Findings

Established large deviation principles for the final active nodes

Derived explicit rate functions for the asymptotic behavior

Extended asymptotic analysis to a wide range of scaling functions

Abstract

We consider the Erd\"{o}s--R\'{e}nyi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^*$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible range.