Synchronization and functional central limit theorems for interacting reinforced random walks

Abstract

We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic algorithms. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.

Networks of reinforced stochastic processes: Asymptotics for the empirical means

“…At the contrary, when γ = 1 the stochastic processes N j = (N n,j ) n synchronize and converge almost surely to Z ∞ at the same velocity. The same asymptotic behaviors characterize the stochastic processes Z j = (Z n,j ) n , as proved also in [2,21]. However, while it is somehow guessable from (8) that the velocities of synchronization and convergence for the processes Z j = (Z n,j ) n depend on the parameter γ, it could be somehow unexpected that, although the discount factor of the increments (N n − N n−1 ) is always n −1 , the corresponding velocities for the processes N j = (N n,j ) n also depend on γ and, in general, also these processes do not synchronize and converge to Z ∞ at the same velocity.…”

“…The previous quoted papers [2,21,22,25] are all focused on the asymptotic behavior of the stochastic processes of the "personal inclinations" {Z j = (Z n,j ) n : j ∈ V } of the agents. On the contrary, in this work we focus on the average of times in which the agents adopt "action 1", i.e.…”

“…The interaction among the evolving dynamics of these processes depends on the weighted adjacency matrix W associated to the underlying graph: indeed, the probability that an agent j chooses a certain action depends on its personal "inclination" Zn,j and on the inclinations Z n,h , with h = j, of the other agents according to the elements of W .Asymptotic results for the stochastic processes of the personal inclinations Z j = (Zn,j )n have been subject of studies in recent papers (e.g. [2,21]); while the asymptotic behavior of quantities based on the stochastic processes X j of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means Nn,j = n k=1 X k,j /n, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence.…”

“…Asymptotic results for the stochastic processes of the personal inclinations Z j = (Zn,j )n have been subject of studies in recent papers (e.g. [2,21]); while the asymptotic behavior of quantities based on the stochastic processes X j of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means Nn,j = n k=1 X k,j /n, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence.…”

Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means

“…(We refer to [18] for a discussion on the case 0 < γ ≤ 1/2, for which there is a different asymptotic behavior of the model that is out of the scope of this research work.) The process X describes the sequence of actions along the time-steps and, if at time-step n, the "action 1" has taken place, that is X n = 1, then for "action 1" the probability of occurrence at time-step (n + 1) increases.…”