How Complex Contagions Spread Quickly in Preferential Attachment Models and Other Time-Evolving Networks

Abstract

The $k$ -complex contagion model is a social contagion model which describes the diffusion of behaviors in networks where the successful adoption of a behavior requires influence from multiple contacts. It has been argued that complex contagions better model behavioral changes such as adoption of new beliefs, fashion trends or expensive technology innovations. A contagion in this model starts from a set of initially infected seeds and progresses in rounds. In any round any node with at least $k>1$ infected neighbors becomes infected. Previous work on $k$ -complex contagions was focused on networks with uniform degree distributions. However, many real-world network topologies have non-uniform degree distribution and evolve over time. We analyze the spreading rate of a $k$ -complex contagion in a general family of time-evolving networks which includes the preferential attachment (PA) model. We prove that if the initial seeds are chosen as the $k$ earliest nodes in a network of this family, a $k$ -complex contagion covers the entire network of $n$ nodes in $O(\log n)$ rounds with high probability (w.h.p). We prove that the choice of the seeds is crucial: in the PA model, even if a much larger number of seeds are chosen uniformly randomly , the contagion stops prematurely w.h.p. Although the earliest nodes in a PA model are likely to have high degrees, it is actually the evolutionary graph structure of such models that facilitates fast spreading of complex contagions. The general family of time-evolving graphs with this property even contains networks without a power law degree distribution. Finally, we prove that when a $k$ -complex contagion starts from an arbitrary set of initial seeds on a general graph, determining if the number of infected vertices is above a given threshold is ${\mathbf {P}}$ -complete. Thus, one cannot hope to categorize all the settings in which $k$ -complex contagions percolate in a graph.