Convergence to Fixed Points in One Model of Opinion Dynamics
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Convergence to Fixed Points in One Model of Opinion Dynamics Nikolai A. Bodunov1 · Sergei Yu. Pilyugin2 Revised: 21 January 2020 / Accepted: 6 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study the limit behavior of trajectories of a nonlinear and discontinuous model of opinion dynamics based on the notion of bounded confidence. This model was previously studied in the case where the influence function has the form i(v) = v. It was shown that, under a particular condition on parameters of the system, any its trajectory tends to a fixed point. In this paper, we prove a similar result under weaker conditions on the influence function: it is assumed that i(v) is continuous, nonstrictly increasing, and i(v) = 0 if and only if v = 0. Keywords Opinion dynamics · Dynamical system · Bounded confidence · Fixed point Mathematics Subject Classification (2010) 90B10 · 91D30 · 91B12
1 Introduction Studies in opinion dynamics are devoted to modeling processes of developing opinions in society. Such problems are intensively investigated (see, for example, the recent monographs [1] and [2]). At present, a lot of researchers study models motivated by the notion of bounded confidence. In such models, first introduced in [3, 4] and investigated in detail by Hegselmann and Krause in [5], one considers the dynamics of voters which have to choose between two alternatives; at every step of the iterative process which determines the result, any agent changes his/her opinion being influenced by agents with similar opinions. The recent survey [6] is devoted to the study of the Hegselmann–Krause (HK) model. It is worth to note that mostly, the HK model was studied using computer simulations, and the authors of [5] mentioned that “rigorious analytical results are difficult to obtain.” Sergei Yu. Pilyugin
[email protected] Nikolai A. Bodunov [email protected] 1
St. Petersburg Electrotechnical University (LETI), Prof. Popova st., 5, St. Petersburg, Russia, 197376
2
St. Petersburg State University, Universitetskaya nab., 7-9, St. Petersburg, Russia, 199034
Nikolai A. Bodunov and Sergei Yu. Pilyugin
One of the variants of a model using the concept of bounded confidence has been suggested by M. Campi. The main difference between this model and the HK model is as follows: in the HK model, the agent takes into account the average of the differences between the opinions in his “group of influence” and his/her own opinion, while in the Campi model, one takes into account the averages of the opinions in the group including his/her opinion. The reader is referred to the joint paper [7] of M. Campi and the second author of this paper for a detailed explanation of the model. The first variant of the Campi model was formulated as the following system of ordinary differential equations. A society consisting of N agents has to choose between two alternatives, 1 and −1. Two functions are fixed: the affinity function a(., .) and the influence function i(.). In addition, a number
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