Complex contagions and hybrid phase transitions

A complex contagion is an infectious process in which individuals may require multiple transmissions before changing state. These are used to model behaviours if an individual only adopts a particular behaviour after perceiving a consensus among others. We may think of individuals as beginning inactive and becoming active once they are contacted by a sufficient number of active partners. These have been studied in a number of cases, but analytic models for the dynamic spread of complex contagions are typically complex. Here we study the dynamics of the Watts threshold model (WTM) assuming that transmission occurs in continuous time as a Poisson process, or in discrete time where individuals transmit to all partners in the time step following their infection. We adapt techniques developed for infectious disease modelling to develop an analyse analytic models for the dynamics of the WTM in configuration model networks and a class of random clustered (triangle-based) networks. The resulting model is relatively simple and compact. We use it to gain insights into the dynamics of the contagion. Taking the infinite population limit, we derive conditions under which cascades happen with an arbitrarily small initial proportion active, confirming a hypothesis of Watts for this case. We also observe hybrid phase transitions when cascades are not possible for small initial conditions, but occur for large enough initial conditions. We derive sufficient conditions for this hybrid phase transition to occur. We show that in many cases, if the hybrid phase transition occurs, then all individuals eventually become active. Finally, we discuss the role clustering plays in facilitating or impeding the spread and find that the hypothesis of Watts that was confirmed in configuration model networks does not hold in general. This approach allows us to unify many existing disparate observations and derive new results.