  We study binary state contagion dynamics on a social network where nodes act
in response to the average state of their neighborhood. We model the competing
tendencies of imitation and non-conformity by incorporating an off-threshold
into standard threshold models of behavior. In this way, we attempt to capture
important aspects of fashions and general societal trends. Allowing varying
amounts of stochasticity in both the network and node responses, we find
different outcomes in the random and deterministic versions of the model. In
the limit of a large, dense network, however, we show that these dynamics
coincide. The dynamical behavior of the system ranges from steady state to
chaotic depending on network connectivity and update synchronicity. We
construct a mean field theory for general random networks. In the undirected
case, the mean field theory predicts that the dynamics on the network are a
smoothed version of the average node response dynamics. We compare our theory
to extensive simulations on Poisson random graphs with node responses that
average to the chaotic tent map.
