Consensus in the Hegselmann-Krause model

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 $Δ$ 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 $Δ$.

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