Anti-conformism in the threshold model of collective behavior

We provide a detailed study of the threshold model, where both conformist and anti-conformist agents coexist. Our study bears essentially on the convergence of the opinion dynamics in the society of agents, i.e., finding absorbing classes, cycles, etc. Also, we are interested in the existence of cascade effects, as this may constitute a undesirable phenomenon in collective behavior. We divide our study into two parts. In the first one, we basically study the threshold model supposing a fixed complete network, where every one is connected to every one, like in the seminal work of Granovetter. We study the case of a uniform distribution of the threshold, of a Gaussian distribution, and finally give a result for arbitrary distributions, supposing there is one type of anti-conformist. In a second-part, the graph is no more complete and we suppose that the neighborhood of an agent is random, drawn at each time step from a distribution. We distinguish the case where the degree (number of links) of an agent is fixed, and where there is an arbitrary degree distribution. We show the existence of cascades and that for most societies, the opinion converges to a chaotic situation.