Generalized threshold-based epidemics in random graphs: the power of extreme values

TL;DR

This paper introduces a generalized framework for studying phase transitions in epidemic processes on various random graph models, highlighting the critical role of extreme values in determining outbreak vulnerability.

Contribution

It extends bootstrap percolation analysis to broader graph models with random thresholds and influences, revealing the impact of extreme values on epidemic thresholds.

Findings

Critical seed size is highly sensitive to distribution extremes.

Large-scale outbreaks can occur with sub-linear seeds due to extreme values.

Simulation confirms theoretical predictions on synthetic and real graphs.

Abstract

Bootstrap percolation is a well-known activation process in a graph, in which a node becomes active when it has at least $r$ active neighbors. Such process, originally studied on regular structures, has been recently investigated also in the context of random graphs, where it can serve as a simple model for a wide variety of cascades, such as the spreading of ideas, trends, viral contents, etc. over large social networks. In particular, it has been shown that in $G(n,p)$ the final active set can exhibit a phase transition for a sub-linear number of seeds. In this paper, we propose a unique framework to study similar sub-linear phase transitions for a much broader class of graph models and epidemic processes. Specifically, we consider i) a generalized version of bootstrap percolation in $G(n,p)$ with random activation thresholds and random node-to-node influences; ii) different random graph models, including graphs with given degree sequence and graphs with community structure (block model). The common thread of our work is to show the surprising sensitivity of the critical seed set size to extreme values of distributions, which makes some systems dramatically vulnerable to large-scale outbreaks. We validate our results running simulation on both synthetic and real graphs.