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December 2017 Synchronization of reinforced stochastic processes with a network-based interaction
Giacomo Aletti, Irene Crimaldi, Andrea Ghiglietti
Ann. Appl. Probab. 27(6): 3787-3844 (December 2017). DOI: 10.1214/17-AAP1296


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 [Crimaldi et al. (2016)]. 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.


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Giacomo Aletti. Irene Crimaldi. Andrea Ghiglietti. "Synchronization of reinforced stochastic processes with a network-based interaction." Ann. Appl. Probab. 27 (6) 3787 - 3844, December 2017.


Received: 1 July 2016; Revised: 1 March 2017; Published: December 2017
First available in Project Euclid: 15 December 2017

zbMATH: 1382.60046
MathSciNet: MR3737938
Digital Object Identifier: 10.1214/17-AAP1296

Primary: 60F05 , 60F15 , 60K35
Secondary: 62P35 , 91D30

Keywords: asymptotic normality , complex networks , Interacting systems , reinforced stochastic processes , synchronization , urn models

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 6 • December 2017
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