Analysis of a threshold model of social contagion on degree-correlated networks

TL;DR

This paper analytically explores the conditions under which social contagion can spread globally on degree-correlated networks, providing formulas for critical thresholds, vulnerable components, and final spread size.

Contribution

It introduces a unified analytical framework for social contagion on degree-correlated networks, linking it to disease models and giant component conditions.

Findings

Derived thresholds for global spreading

Calculated expected size of vulnerable components

Provided exact expressions for spreading quantities

Abstract

We analytically determine when a range of abstract social contagion models permit global spreading from a single seed on degree-correlated random networks. We deduce the expected size of the largest vulnerable component, a network's tinderbox-like critical mass, as well as the probability that infecting a randomly chosen individual seed will trigger global spreading. In the appropriate limits, our results naturally reduce to standard ones for models of disease spreading and to the condition for the existence of a giant component. Recent advances in the distributed, infinite seed case allow us to further determine the final size of global spreading events, when they occur. To provide support for our results, we derive exact expressions for key spreading quantities for a simple yet rich family of random networks with bimodal degree distributions.