Gaussian consensus processes and their Lyapunov exponents

TL;DR

This paper introduces a mathematical model of opinion dynamics with Gaussian perturbations, analyzing conditions for consensus and deriving Lyapunov exponents using random matrix theory.

Contribution

It provides a rigorous analysis of opinion convergence conditions and computes the Lyapunov exponent for a novel Gaussian consensus model.

Findings

Opinions converge under specific parameter conditions.

Asymptotic correlation between opinions on different topics.

Explicit Lyapunov exponent expression derived.

Abstract

We introduce a simple dynamic model of opinion formation, in which a finite population of individuals hold vector-valued opinions. At each time step, each individual's opinion moves towards the mean opinion but is then perturbed independently by a centred multivariate Gaussian random variable, with covariance proportional to the covariance matrix of the opinions of the population. We establish precise necessary and sufficient conditions on the parameters of the model, under which all opinions converge to a common limiting value. Asymptotically perfect correlation emerges between opinions on different topics. Our results are rigorous and based on properties of the partial products of an i.i.d. sequence of random matrices. Each matrix is a fixed linear combination of the identity matrix and a real Ginibre matrix. We derive an analytic expression for the maximal Lyapunov exponent of this product sequence. We also analyze a continuous-time analogue of our model.