Complex Contagions with Timers

There has been a great deal of effort to try to model social influence-including the spread of behavior, norms, and ideas-on networks. Most models of social influence tend to assume that individuals react to changes in the states of their neighbors without any time delay, but this is often not true in social contexts, where (for various reasons) different agents can have different response times. To examine such situations, we introduce the idea of a timer into threshold models of social influence. The presence of timers on nodes delays adoptions-i.e., changes of state-by the agents, which in turn delays the adoptions of their neighbors. With a homogeneously-distributed timer, in which all nodes have the same amount of delay, the adoption order of nodes remains the same. However, heterogeneously-distributed timers can change the adoption order of nodes and hence the "adoption paths" through which state changes spread in a network. Using a threshold model of social contagions, we illustrate that heterogeneous timers can either accelerate or decelerate the spread of adoptions compared to an analogous situation with homogeneous timers, and we investigate the relationship of such acceleration or deceleration with respect to the timer distribution and network structure. We derive an analytical approximation for the temporal evolution of the fraction of adopters by modifying a pair approximation for the Watts threshold model, and we find good agreement with numerical simulations. We also examine our new timer model on networks constructed from empirical data.