How Clustering Affects Epidemics in Random Networks

TL;DR

This paper analyzes how clustering in random networks influences the spread of epidemics and contagion processes, revealing that clustering can both inhibit and promote diffusion depending on network connectivity.

Contribution

It provides a novel analysis of the effects of clustering on epidemic and contagion models, with explicit conditions and formulas for cascade sizes based on network properties.

Findings

Clustering inhibits diffusion in general.

In high connectivity regimes, clustering promotes global cascades.

Explicit formulas for cascade sizes as functions of degree distribution and clustering.

Abstract

Motivated by the analysis of social networks, we study a model of random networks that has both a given degree distribution and a tunable clustering coefficient. We consider two types of growth processes on these graphs: diffusion and symmetric threshold model. The diffusion process is inspired from epidemic models. It is characterized by an infection probability, each neighbor transmitting the epidemic independently. In the symmetric threshold process, the interactions are still local but the propagation rule is governed by a threshold (that might vary among the different nodes). An interesting example of symmetric threshold process is the contagion process, which is inspired by a simple coordination game played on the network. Both types of processes have been used to model spread of new ideas, technologies, viruses or worms and results have been obtained for random graphs with no clustering. In this paper, we are able to analyze the impact of clustering on the growth processes. While clustering inhibits the diffusion process, its impact for the contagion process is more subtle and depends on the connectivity of the graph: in a low connectivity regime, clustering also inhibits the contagion, while in a high connectivity regime, clustering favors the appearance of global cascades but reduces their size. For both diffusion and symmetric threshold models, we characterize conditions under which global cascades are possible and compute their size explicitly, as a function of the degree distribution and the clustering coefficient. Our results are applied to regular or power-law graphs with exponential cutoff and shed new light on the impact of clustering.