## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 6 (2017), 3787-3844.

### Synchronization of reinforced stochastic processes with a network-based interaction

Giacomo Aletti, Irene Crimaldi, and Andrea Ghiglietti

#### Abstract

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*.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 6 (2017), 3787-3844.

**Dates**

Received: July 2016

Revised: March 2017

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1513328714

**Digital Object Identifier**

doi:10.1214/17-AAP1296

**Mathematical Reviews number (MathSciNet)**

MR3737938

**Zentralblatt MATH identifier**

1382.60046

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 62P35: Applications to physics 91D30: Social networks

**Keywords**

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

#### Citation

Aletti, Giacomo; Crimaldi, Irene; Ghiglietti, Andrea. Synchronization of reinforced stochastic processes with a network-based interaction. Ann. Appl. Probab. 27 (2017), no. 6, 3787--3844. doi:10.1214/17-AAP1296. https://projecteuclid.org/euclid.aoap/1513328714