Consensus in the Hegselmann-Krause model

TL;DR

This paper analyzes the probability of reaching consensus in a spatially explicit Hegselmann-Krause opinion dynamics model on social networks, providing bounds based on initial opinion distributions and network structure.

Contribution

It introduces a multivariate, graph-based version of the Hegselmann-Krause model and derives lower bounds for consensus probability considering initial opinion distributions.

Findings

Derived lower bounds for consensus probability.

Extended the model to multivariate opinions on social networks.

Analyzed the influence of initial opinion distribution on consensus.

Abstract

This paper is concerned with the probability of consensus in a multivariate, spatially explicit version of the Hegselmann-Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions $\Delta$ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold $\tau$, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold $\tau$. Each vertex $x$ updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at $x$ to be replaced by a convex combination of the opinion at $x$ and the nearby opinions: $\alpha$ times the opinion at $x$ plus $(1 - \alpha)$ times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set $\Delta$.