Synchronization of reinforced Stochastic processes with a network-based interaction

Randomly evolving systems composed by elements whichinteract among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of differentcomponents to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in [Crimaldi, Dai Pra, Louis, Minelli]. Generally speaking, by reinforcement we mean any mechanism for which the probability that agiven event occurs has an increasing dependence on the number of times that events of the same type occurred in the past. The particularity of systems of such interacting stochastic processes is thatsynchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus onthe relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After provingthe almost sure synchronization, we provide some CLTs in the senseof stable convergence that establish the convergence rates and theasymptotic distributions for both convergence to the common limit and synchronization. The obtained results lead to the construction of asymptotic confidence intervals for the limit random variable andof statistical tests to make inference on the topology of the network.