Mesoscopic analytical approach in a three state opinion model with continuous internal variable

TL;DR

This paper introduces a mesoscopic analytical approach to a three-state opinion model combining discrete and continuous variables, accurately predicting opinion dynamics and phase transitions despite complex nonlinear equations.

Contribution

It develops an approximation method for a nonlinear coupled system of master equations in a mixed opinion model, enabling analytical insights into opinion polarization.

Findings

Accurately predicts transition from neutral to polarized states.

Provides an excellent approximation for average leaning dynamics.

Validates the approximation even outside strict time scale separation.

Abstract

Analytical approaches in models of opinion formation have been extensively studied either for an opinion represented as a discrete or a continuous variable. In this paper, we analyze a model which combines both approaches. The state of an agent is represented with an internal continuous variable (the leaning or propensity), that leads to a discrete public opinion: pro, against or neutral. This model can be described by a set of master equations which are a nonlinear coupled system of first order differential equations of hyperbolic type including non-local terms and non-local boundary conditions, which can't be solved analytically. We developed an approximation to tackle this difficulty by deriving a set of master equations for the dynamics of the average leaning of agents with the same opinion, under the hypothesis of a time scale separation in the dynamics of the variables. We show that this simplified model accurately predicts the expected transition between a neutral consensus and a bi-polarized state, and also gives an excellent approximation for the dynamics of the average leaning of agents with the same opinion, even when the time separation scale hypothesis is not completely fulfilled.