  A great deal of effort has gone into trying 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 the adoption --- i.e., change of state --- of each agent, which in
turn delays the adoptions of its neighbors. With a homogeneous-distributed
timer, in which all nodes exhibit the same amount of delay, adoption delays are
also homogeneous, so 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 timer
distribution and network structure. We derive an analytical approximation for
the temporal evolution of the fraction of adopters by modifying a pair
approximation of the Watts threshold model, and we find good agreement with
numerical computations. We also examine our new timer model on networks
constructed from empirical data.
