Propagation of chaos and phase transition in a stochastic model for a social network

TL;DR

This paper models social opinion dynamics using interacting point processes, proves propagation of chaos as the number of actors grows, and identifies a phase transition related to social pressure strength affecting opinion consensus.

Contribution

It introduces a stochastic model for social networks with opinion influence, proves propagation of chaos, and characterizes phase transition phenomena in the system.

Findings

Propagation of chaos established as N approaches infinity.

Existence of a phase transition depending on social pressure parameter h.

Presence of multiple invariant measures indicating different social states.

Abstract

We consider a model for a social network with N interacting social actors. This model is a system of interacting marked point processes in which each point process indicates the successive times in which a social actor expresses a "favorable" (+1) or "contrary" (-1) opinion. The orientation and the rate at which an actor expresses an opinion is influenced by the social pressure exerted on this actor. The social pressure of an actor is reset to 0 when the actor expresses an opinion, and simultaneously the social pressures on all the other actors change by h/N in the direction of the opinion that was just expressed. We prove propagation of chaos of the system, as N diverges to infinity, to a limit nonlinear jumping stochastic differential equation. Moreover, we prove that under certain conditions the limit system exhibits a phase transition described as follows. If h is smaller or equal than a certain threshold, the limit system has only the null Dirac measure as an invariant probability measure, corresponding to a vanishing social pressure on all actors. However, if h is greater than the threshold, the system has two additional non-trivial invariant probability measures. One of these measures has support on the positive real numbers and the other is obtained by symmetrization with respect to 0, having thus support on the negative real numbers.