Cluster formation in opinion dynamics: a qualitative analysis

Abstract

In this paper we formulate a discrete version of the bounded confidence model (Deffuant et al. in Adv Complex Syst 3:87-98, 2000; Weisbuch et al. in Complexity 7:55-63, 2002), which is representable as a family of ordinary differential equation systems. Then, we analytically study these systems. We establish the existence of equilibria which correspond to opinion profiles displaying a finite number of isolated clusters. We prove the asymptotic stability of some of these equilibria and show that they represent the asymptotic trend of the solutions of the systems under consideration. For a particular case, we also characterize the initial profiles that lead to different cluster configurations.

On a class of dynamical systems with emerging cluster structure

“…with the meaning of the symbols given in the next lines, have been proposed in [7], towards modeling of opinion dynamics.…”

“…The goal of the present paper is to analytically study as a whole, namely for any admissible value of n and m, the family of systems introduced in [7]. In fact, we first characterize the equilibrium configurations and then investigate their attractivity properties.…”

“…In [7], the task was taken up to provide an analytical proof of this cluster emergence phenomenon. In view of that, a discrete version of the bounded confidence model was formulated, the term "discrete" referring to the possible values of the opinion variable.…”

“…The organization of this paper is as follows: in Section 2, the dynamical systems derived in [7] are recalled for the reader's convenience. Their equilibria are found in the first of three subsections of Section 3.…”

“…attention has been devoted to the celebrated "bounded confidence model", originally introduced in [9] and [18] (see also [12]), and then variously extended and analyzed in other papers (see, e.g., [19,8,13,14]). Such a model describes the evolution in time of the opinion, represented by a continuous scalar variable, of a finite set of agents.…”