A modified Hegselmann-Krause model for interacting voters and political parties

TL;DR

This paper extends the Hegselmann-Krause opinion dynamics model by including political parties, analyzing how their interactions with voters influence consensus formation and opinion clustering through simulations and analytical methods.

Contribution

It introduces a modified model incorporating political parties into the opinion dynamics framework, providing analytical conditions for consensus and phase transition insights.

Findings

Opinion clusters form similarly to the original model.

A phase transition from disagreement to consensus is observed.

Analytical conditions for unanimous consensus are derived.

Abstract

The Hegselmann--Krause model is a prototypical model for opinion dynamics. It models the stochastic time evolution of an agent's or voter's opinion in response to the opinion of other like-minded agents. The Hegselmann--Krause model only considers the opinions of voters; we extend it here by incorporating the dynamics of political parties which influence and are influenced by the voters. We show in numerical simulations for $1$- and $2$-dimensional opinion spaces that, as for the original Hegselmann--Krause model, the modified model exhibits opinion cluster formation as well as a phase transition from disagreement to consensus. We provide an analytical sufficient condition for the formation of unanimous consensus in which voters and parties collapse to the same point in opinion space in the deterministic case. Using mean-field theory, we further derive an approximation for the critical noise strength delineating consensus from non-consensus in the stochastically driven modified Hegselmann--Krause model. We compare our analytical findings with simulations of the modified Hegselmann--Krause model.