Bootstrap percolation and the geometry of complex networks

TL;DR

This paper studies bootstrap percolation on a hyperbolic geometric network model, identifying a critical infection probability that determines whether the infection spreads widely or not, and shows this behavior is robust to edge deletions.

Contribution

It introduces a critical infection threshold for bootstrap percolation on hyperbolic geometric graphs and demonstrates the robustness of this threshold under random edge deletions.

Findings

Existence of a critical infection probability p_c(N) for widespread infection.

When p >> p_c(N), infection spreads to a positive fraction of vertices.

When p << p_c(N), infection remains localized.

Abstract

On a geometric model for complex networks (introduced by Krioukov et al.) we investigate the bootstrap percolation process. This model consists of random geometric graphs on the hyperbolic plane having $N$ vertices, a dependent version of the Chung-Lu model. The process starts with infection rate $p=p(N)$. Each uninfected vertex with at least $\mathbf{r}\geq 1$ infected neighbors becomes infected, remaining so forever. We identify a function $p_c(N)=o(1)$ such that a.a.s.\ when $p\gg p_c(N)$ the infection spreads to a positive fraction of vertices, whereas when $p\ll p_c(N)$ the process cannot evolve. Moreover, this behavior is "robust" under random deletions of edges.