  A complex contagion is an infectious process in which individuals may require
multiple transmissions before changing state. These are used to model behaviors
if an individual only adopts a particular behavior after perceiving a consensus
among others. We may think of individuals as beginning inactive and becoming
active after contact with 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 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 activation. We adapt techniques
developed for infectious disease modeling to develop and analyze 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 contagion
dynamics. In 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, 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.
