Interacting generalized Friedman’s urn systems

Abstract

Abstract. We consider systems of interacting Generalized Friedman's Urns (GFUs) having irreducible mean replacement matrices. The interaction is modeled through the probability to sample the colors from each urn, that is defined as convex combination of the urn proportions in the system. From the weights of these combinations we individuate subsystems of urns evolving with different behaviors. We provide a complete description of the asymptotic properties of urn proportions in each subsystem by establishing limiting proportions, convergence rates and Central Limit Theorems. The main proofs are based on a detailed eigenanalysis and stochastic approximation techniques.Keywords. Interacting systems; Urn models; Strong Consistency; Central Limit Theorems; Stochastic approximation.2010 MSC classification: 60K35; 62L20; 60F05; 60F15. IntroductionThe stochastic evolution of systems composed by elements which interact among each other has always been of great interest in several areas of application, e.g. in medicine a tumor growth is the evolution of a system of interacting cells [35], in socio-economics and life sciences a collective phenomenon reflects the result of the interactions among the individuals [27], in physics the concentration of certain molecules within cells varies over time due to interactions between different cells [31]. In the last decade several models have been proposed in which the elements of the system are represented by urns containing balls of different colors, in which the urn proportions reflect the status of the elements, and the evolution of the system is established by studying the dynamics at discrete times of this collection of dependent urn processes. The main reason of this popularity is concerned with the urn dynamics, which is (i) suitable to describe random phenomena in different scientific fields (see e.g. [21]), (ii) flexible to cover a wide range of possible asymptotic behaviors, (iii) intuitive and easy to be implemented in several fields of application.The dynamics of a single urn typically consists in a sequential repetition of a sampling phase, when a ball is sampled from the urn, and a replacement phase, when a certain quantity of balls is replaced in the urn. The basic model is the Pólya's urn proposed in [16]: from an urn containing balls of two colors, balls are sequentially sampled and then replaced in the urn with a new ball of the same color. This updating scheme is then iterated generating a sequence of urn proportions whose almost sure limit is random and Beta distributed. Starting from this simple model, several interesting variations have been suggested by considering different distributions in the sampling phase, e.g. [19,20], or in the replacement phase, e.g. [3,18,30]. In a general K-colors urn model, the color sampled at time n is usually represented by a vector X n such that X i,n = 1 when the sampled color is i ∈ {1, . . ., K}, X i,n = 0 otherwise; the quantities of balls replaced in the urn at time n are typically defined by a matrix D n such that D ki,n i...

Synchronization of reinforced stochastic processes with a network-based interaction

“…The present work have some issues in common with [12,13] and [2], but at the same time some significant differences can be pointed out. In particular, we share with [2] a general interacting framework driven by the interacting matrix (here called weighted adjacency matrix). However, here we mainly consider irreducible interacting matrices and hence the decomposition of the system in sub-groups is only sketched.…”

“…To this purpose, here we highlight that for the Eggenberger-Póya urn we have lim n nr n = 1. We refer to [12,Example 1.2] for a meaningful case of reinforced stochastic process of the type (1)- (2) where lim n n γ r n = c with γ < 1 and c ∈ (0, +∞). This example concerns an opinion dynamics in an evolving population, modeled by a graph evolving according to preferential attachment [1,19,28].…”

“…In [12] the same mean-field interaction is adopted, but the analysis has been extended to the general class of reinforced stochastic processes, providing central limit theorems also in functional form. Differently from these works, the model proposed in [2] concerns with a system of generalized Friedman urns with irreducible mean replacement matrices based on a general interaction structure, which includes the mean-field interaction as a special case. In particular, this interaction acts as follows: the probability to sample a certain color in each urn is a convex combination of the urn proportions of the entire system, and the weights of such combinations are gathered in the interacting matrix.…”

“…However, here we mainly consider irreducible interacting matrices and hence the decomposition of the system in sub-groups is only sketched. Moreover, with respect to [2], we study a class of stochastic processes for which we obtain synchronization. This class does not include the generalized Friedman urns studied in [2]; while it includes urn models with not-irreducible mean replacement matrices, as Pólya urns.…”

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

“…The irreducibility of W reflects a situation in which all the vertices are connected among each others and hence there are no sub-systems with independent dynamics (see [2,3] for further details). The diagonalizability of W allows us to find a non-singular matrix U such that U ⊤ W ( U ⊤ ) −1 is diagonal with complex elements λ j ∈ Sp(W ).…”