Opinion dynamics in communities with major influencers and implicit social influence via mean-field approximation

TL;DR

This paper investigates binary opinion formation in large communities influenced by leaders and social factors, using mean-field approximation to analyze opinion dynamics, long-term behavior, and phenomena like echo chambers and social inertia.

Contribution

It introduces a mean-field approximation framework for opinion dynamics with influencers and incomplete information, providing theoretical analysis and numerical illustrations of complex social phenomena.

Findings

Mean-field approximation accurately models opinion dynamics.

Propagation of chaos established under common noise influence.

Long-term behavior and fluctuations analyzed for linear models.

Abstract

We study binary opinion formation in a large population where individuals are influenced by the opinions of other individuals. The population is characterised by the existence of (i) communities where individuals share some similar features, (ii) opinion leaders that may trigger unpredictable opinion shifts in the short term (iii) some degree of incomplete information in the observation of the individual or public opinion processes. In this setting, we study three different approximate mechanisms: common sampling approximation, independent sampling approximation, and, what will be our main focus in this paper, McKean-Vlasov (or mean-field) approximation. We show that all three approximations perform well in terms of different metrics that we introduce for measuring population level and individual level errors. In the presence of a common noise represented by the major influencers opinions processes, and despite the absence of idiosyncratic noises, we derive a propagation of chaos type result. For the particular case of a linear model and particular specifications of the major influencers opinion dynamics, we provide additional analysis, including long term behavior and fluctuations of the public opinion. The theoretical results are complemented by some concrete examples and numerical analysis, illustrating the formation of echo-chambers, the propagation of chaos, and phenomena such as snowball effect and social inertia.42 pages