Synchronization of reinforced stochastic processes with a network-based interaction

Abstract

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 [12]. 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 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 given the observation of the reinforced stochastic processes positioned at the vertices.

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

“…Proof. We observe that Y 1 (2) can be written as the general sum (58) with J = {1} and I 1 = {2}. Therefore case ν > γ coincides with the case (v) of Lemma 5.1, taking into account the value d 1(2),1 (2) computed in Section A.4 for this case and equality (61).…”

“…Remark 5.2. Note that, when ν = γ the convergence rates of the first and the second component of Y n are always different: indeed from [2], we know that, under our assumptions, the convergence rate of Z n is always n γ/2 , while the above theorem shows that the convergence rate of N n changes according to the pair (γ, ν).…”

“…When ν = 1, we assume q > 1/2. Note that in Assumption 2.2 condition (15) for the sequence (r n ) n is slightly more restrictive than the one assumed in [1,2]. However, it is always verified in the applicative contexts we have in mind.…”

“…We assume the weights to be normalized so that N h=1 w h,j = 1 for each j ∈ V . The interaction between the processes {X j : j ∈ V } is explicitly inserted in Equation (1) and it is modeled as follows: for any n ≥ 0, the random variables {X n+1,j : j ∈ V } are conditionally independent given F n with (4) P (X n+1,j = 1 | F n ) = N h=1 w h,j Z n,h = w jj Z n,j + h =j w h,j Z n,h , where F n := σ(Z 0,h : h ∈ V ) ∨ σ(X k,j : 1 ≤ k ≤ n, j ∈ V ) and, for each h ∈ V , the evolution dynamics of the single process (Z n,h ) n≥0 is the same as in (2), that is (5) Z n,h = (1 − r n−1 )Z n−1,h + r n−1 X n,h , with Z 0,h a random variable taking values in [0, 1] and (r n ) n≥0 a sequence of real numbers in (0, 1) such that condition (3) holds true.…”

“…The convergence rate and the second-order asymptotic distribution of (Y n − Z ∞ 1) will be obtained by studying separately and then combining together the second-order convergence of Y n to Z ∞ 1 and the second-order convergence of Y n to 0. To this regards, we recall that, by [2,Theorem 4.2], under Assumptions 2.1 and 2.2, we have for 1/2 < γ ≤ 1 that…”