Contagions in Random Networks with Overlapping Communities

TL;DR

This paper analyzes a threshold epidemic model on a clustered random graph with overlapping communities, identifying phase transitions for epidemic cascades using multi-type branching processes.

Contribution

It introduces a novel framework combining overlapping community structures with threshold epidemic dynamics and characterizes phase transitions via multi-type branching processes.

Findings

Identifies conditions for epidemic cascades in overlapping community networks.

Derives the phase transition threshold using eigenvalues of a matrix.

Models epidemic spread with a multi-type branching process.

Abstract

We consider a threshold epidemic model on a clustered random graph with overlapping communities. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the matrix whose largest eigenvalue gives the phase transition.