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Synchronization of reinforced stochastic processes with a network-based interaction. (English) Zbl 1382.60046
Summary: Randomly evolving systems composed by elements which interact 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 different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement that recently has been introduced in [I. Crimaldi et al., “Synchronization and functional central limit theorems for interacting reinforced random walks”, Preprint, arXiv:1602.06217]. Generally speaking, by reinforcement we mean any mechanism for which the probability that a given 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 that synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus on the relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After proving the almost sure synchronization, we provide some CLTs in the sense of stable convergence that establish the convergence rates and the asymptotic 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 and of statistical tests to make inference on the topology of the network.

MSC:
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
60K35 Interacting random processes; statistical mechanics type models; percolation theory
62P35 Applications of statistics to physics
91D30 Social networks; opinion dynamics
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