Models of Opinion Dynamics with Random Parametrisation

TL;DR

This paper investigates a generalized opinion dynamics model with random deviations, analyzing how the distribution of flip probabilities influences the stability and number of fixed points in small and large discussion groups.

Contribution

It introduces a probabilistic framework for opinion dynamics with random flip parameters, providing exact numerical results and limit theorems for large groups.

Findings

Large majorities can be stable or unstable depending on flip parameters.

Certain fixed points are stable across various distributions.

The model captures diverse behaviors in social and political group settings.

Abstract

We analyse a generalisation of the Galam model of binary opinion dynamics in which iterative discussions take place in local groups of individuals and study the effects of random deviations from the group majority. The probability of a deviation or flip depends on the magnitude of the majority. Depending on the values of the flip parameters which give the probability of a deviation, the model shows a wide variety of behaviour. We are interested in the characteristics of the model when the flip parameters are themselves randomly selected, following some probability distribution. Examples of these characteristics are whether large majorities and ties are attractors or repulsors, or the number of fixed points in the dynamics of the model. Which of the features of the model are likely to appear? Which ones are unlikely because they only present as events of low probability with respect to the distribution of the flip parameters? Answers to such questions allow us to distinguish mathematical properties which are stable under a variety of assumptions on the distribution of the flip parameters from features which are very rare and thus more of theoretical than practical interest. In this article, we present both exact numerical results for specific distributions of the flip parameters and small discussion groups and rigorous results in the form of limit theorems for large discussion groups. Small discussion groups model friend or work groups -- people that personally know each other and frequently spend time together. Large groups represent scenarios such as social media or political entities such as cities, states, or countries.