Networks of reinforced stochastic processes: estimation of the probability of asymptotic polarization

In a network of reinforced stochastic processes [arXiv:2206.07514, arXiv:1607.08514], for certain values of the parameters, there exists a strictly positive probability that all the agents asymptotically polarize, i.e. they almost surely converge toward one (the same for all the agents) of the two extreme inclinations. This work aims to provide a suitable technique in order to estimate this probability, given the observation of the network until a certain time-step. The problem has been framed in the more general setting of a class of martingales taking values in [0, 1] and following a specific dynamics: the above probability of asymptotic polarization corresponds to the probability of touching the barriers {0, 1} in the limit. Therefore, the illustrated technique can also be applied in different contexts.