Multi-dimensional extensions of the Hegselmann-Krause model

In this paper we consider two multi-dimensional Hagselmann-Krause (HK) models for opinion dynamics. The two models describe how individuals adjust their opinions on multiple topics, based on the influence of their peers. The models differ in the criterion according to which individuals decide whom they want to be influenced from. In the average-based model individuals compare their average opinions on the various topics with those of the other individuals, and interact only with those individuals whose average opinions lie within a confidence interval. For this model we show that the agents’ opinions reach consensus/clustering if and only if their average opinions do so. In the uniform affinity model agents compare their opinions on each single topic and influence each other only if, topic-wise, such opinions do not differ more than a given tolerance. We identify conditions under which the uniform affinity model enjoys the order-preservation property topic-wise and we prove that the global range of opinions (and hence the range of opinions on each single topic) are non-increasing.