Complex Contagions and hybrid phase transitions in unclustered and clustered random networks

A complex contagion is an infectious process in which an individual may require multiple transmissions. We typically think of individuals as beginning inactive and becoming active once they are contacted by sufficient numbers of active partners. These have been studied in a number of contexts, but the analytic models for dynamic spread of complex contagions are typically complex. Here we study the dynamics of a generalized Watts Threshold Model (gWTM). We first show that a wide range of other processes can be thought of as a special case of this gWTM. Then we adapt an “edge-based compartmental modeling” approach used for infectious diseases in networks to develop and analyze analytic models for the dynamics the gWTM in configuration model and a class of random clustered (triangle-based) networks. The resulting model is relatively simple and compact, and we use this model to gain insights into the dynamics. Under some conditions a cascade can happen with an arbitrarily small initial proportion active, and we derive conditions for this to happen. We also sometimes observe hybrid phase transitions when cascades are not possible for small initial conditions, but do occur for large enough initial conditions. We derive some simple new sufficient conditions for the hybrid phase transition to occur. We show that in many cases, above the hybrid phase transition, all individuals will eventually become active. Finally, we discuss the role clustering plays in facilitating or impeding the spread. This approach allows us to unify many existing disparate results and derive some new results.