Analytical formulation for multidimensional continuous opinion models

TL;DR

This paper develops an analytical framework for multidimensional continuous opinion models, revealing how opinions evolve and converge under different interaction mechanisms and correlations between opinion dimensions.

Contribution

It introduces an analytical approach for multidimensional opinion models, analyzing the effects of correlations and interaction types on opinion dynamics and convergence.

Findings

Mean opinion remains conserved under reciprocal interactions.

Opinion variance decreases over time with positive social influence.

Convergence time differs between correlated and uncorrelated opinion dimensions.

Abstract

Usually, opinion formation models assume that individuals have an opinion about a given topic which can change due to interactions with others. However, individuals can have different opinions in different topics and therefore n-dimensional models are best suited to deal with these cases. While there have been many efforts to develop analytical models for one dimensional opinion models, less attention has been paid to multidimensional ones. In this work, we develop an analytical approach for multidimensional models of continuous opinions where dimensions can be correlated or uncorrelated. We show that for any generic reciprocal interactions between agents, the mean value of initial opinion distribution is conserved. Moreover, for positive social influence interaction mechanisms, the variance of opinion distributions decreases with time and the system converges to a delta distributed function. In particular, we calculate the convergence time when agents get closer in a discrete quantity after interacting, showing a clear difference between correlated and uncorrelated cases.