On-off Threshold Models of Social Behavior

We study binary state dynamics on a social network, where nodes act according to individual response functions of the average state of their neighborhood. These response functions 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 response functions, we find different outcomes in the random and deterministic versions of the model. In the limit of a large, dense network, however, these dynamics coincide. The dynamical behavior of the system ranges from steady state to chaotic depending on network connectivity and update synchronicity. A mean field theory is laid out in detail 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 response function dynamics. The theory is compared to simulations on Poisson random graphs with response functions that average to the chaotic tent map. Table of