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We study binary state dynamics on a network where each node acts in responseto the average state of its neighborhood. Allowing varying amounts ofstochasticity in both the network and node responses, we find differentoutcomes in random and deterministic versions of the model. In the limit of alarge, dense network, however, we show that these dynamics coincide. Weconstruct a general mean field theory for random networks and show thispredicts that the dynamics on the network are a smoothed version of the averageresponse function dynamics. Thus, the behavior of the system can range fromsteady state to chaotic depending on the response functions, networkconnectivity, and update synchronicity. As a specific example, we model thecompeting tendencies of imitation and non-conformity by incorporating anoff-threshold into standard threshold models of social contagion. In this waywe attempt to capture important aspects of fashions and societal trends. Wecompare our theory to extensive simulations of this "limited imitationcontagion" model on Poisson random graphs, finding agreement between themean-field theory and stochastic simulations.

Dynamical influence processes on networks: General theory and applications to social contagion