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We consider a finite collection of reinforced stochastic processes with ageneral network-based interaction among them. We provide sufficient andnecessary conditions in order to have some form of almost sure asymptoticsynchronization, which could be roughly defined as the almost sure long-rununiformization of the behavior of interacting processes. Specifically, wedetect a regime of complete synchronization, where all the processes convergetoward the same random variable, a second regime where the system almost surelyconverges, but there exists no form of almost sure asymptotic synchronization,and another regime where the system does not converge with a strictly positiveprobability. In this latter case, partitioning the system in cyclic classesaccording to the period of the interaction matrix, we have an almost sureasymptotic synchronization within the cyclic classes, and, with a strictlypositive probability, an asymptotic periodic behavior of these classes.

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