  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. Specifically, we
detect a regime of complete synchronization, where all the processes converge
toward the same random variable, a second 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.
