When social networks polarize: On the number of clusters in the Hegselmann–Krause model

In the present paper, we study opinion dynamics in a social network, where individuals only listen to those with opinion not farther away than a given threshold from their own opinion (known as bounded confidence models, proposed by Hegselmann and Krause). It is well known that in bounded confidence models consensus does not always exist, and that agents split in clusters (polarization), with convergence to consensus in each cluster. We are precisely interested in the effect of bounded confidence on polarization in the network. Our main focus concerns the formation of clusters and their number, as well as its non-monotonicity with respect to the value of the threshold. First, a framework with a finite number of agents is considered. We study analytically disintegration of various types of opinion chains (clusters), and investigate by simulation the likelihood of chains of a certain length and their disintegration. Next, we examine the (non-)monotonicity of the number of clusters with respect to the threshold for a given initial vector of opinions and in expectation. Finally, we analyze in a formal way the formation of clusters in a model with a continuum of agents. • Polarization in bounded confidence models is not well understood. • We give necessary and sufficient conditions for polarization with 5 and 6 agents. • The techniques can be readily extended to longer chains of communicating agents. • Non-monotonicity of the number of clusters w.r.t. the confidence bound is clarified. • Moreover, we analyze cluster formation in a model with a continuum of agents.