Consensus in the Hegselmann-Krause model. (English) Zbl 1490.60276
Summary: 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 \(\Delta\) 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 \(\Delta \).
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
Keywords:
interacting particle systems; Hegselmann-Krause model; opinion dynamics; martingale; martingale convergence theorem; optional stopping theorem; confidence threshold; consensusReferences:
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