Interacting non-linear reinforced stochastic processes: synchronization or non-synchronization

Rich get richer rule comforts previously often chosen actions. What is happening to the evolution of individual inclinations to choose an action when agents do interact? Interaction tends to homogenize while each individual dynamics tends to reinforce its own position. Interacting stochastic systems of reinforced processes were recently considered in many papers, where the asymptotic behavior was proven to exhibit almost sure synchronization. We consider in this paper models where, even if interaction among agents is present, absence of synchronization may happen due to the choice of an individual non- linear reinforcement. We show how these systems can naturally be considered as models for coordination games, technological or opinion dynamics.