Networks of reinforced stochastic processes: a complete description of the first-order asymptotics

We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions in order to have some form of almost sure asymptotic synchronization, which could be roughly defined as the almost sure long-run uniformization of the behavior of interacting processes. In the case of a complete almost sure asymptotic synchronization, all the processes converge toward a certain random variable and, in order to complete the picture, we provide results about the distribution of this common limit, pointing out when it can take the extreme values, 0 or 1, (asymptotic polarization) with a strictly positive probability. Finally, we analyze the behavior of the system when such a form of synchronization is not guaranteed. Specifically, we detect a regime where the system almost surely converges, but there exists no form of almost sure asymptotic synchronization, and another regime where the system does not converge with a strictly positive probability. In this latter case, partitioning the system in cyclic classes according to the period of the interaction matrix, we have an almost sure asymptotic synchronization within the cyclic classes, and, with a strictly positive probability, an asymptotic periodic behavior of these classes.