Consensus in the Hegselmann–Krause Model

This paper is concerned with the probability of consensus in a multivariate, socially structured 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 Δ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold τ, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold τ. 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: α times the opinion at x plus (1 - α) 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 Δ.