Networks of reinforced stochastic processes: probability of asymptotic polarization and related general results

TL;DR

This paper investigates the conditions under which agents in a network of reinforced stochastic processes asymptotically polarize or synchronize, providing theoretical insights and estimation techniques for polarization probabilities.

Contribution

It clarifies when polarization occurs in reinforced stochastic networks and introduces a technique to estimate polarization probabilities within a martingale framework.

Findings

Conditions for asymptotic polarization identified

A technique for estimating polarization probability developed

Results framed within a martingale-based general setting

Abstract

In a network of reinforced stochastic processes, for certain values of the parameters, all the agents' inclinations synchronize and converge almost surely toward a certain random variable. The present work aims at clarifying when the agents can asymptotically polarize, i.e. when the common limit inclination can take the extreme values, 0 or 1, with probability zero, strictly positive, or equal to one. Moreover, we present a suitable technique to estimate this probability that, along with the theoretical results, has been framed in the more general setting of a class of martingales taking values in [0, 1] and following a specific dynamics.