Metastability in systems of interacting point processes with memory of variable length modeling social and neuronal networks - Laxa, Kádmo de Souza (2022)
We study the metastable behavior of two systems of interacting point processes with memory of variable length. One of the systems is a new model for a highly polarized social network. In this system, the point processes are marked and indicate the successive times in which a social actor express a favorable or contrary opinion on a certain subject. For this model, we prove that when the polarization coefficient diverges, the social network reaches instantaneous consensus and this consensus has a metastable behavior. This means that the direction of the social pressures on the actors globally changes after a long and unpredictable random time. The second system we consider models a network of spiking neurons. In this model, associated to each neuron there are two point processes, describing its successive spiking and leakage times. We prove that this system has a metastable behaviour when the population size diverges. This means that the time at which the system gets trapped by the list of null membrane potentials suitably re-scaled converges to a mean one exponential random time.