Stationary States in a Model of Position Selection by Individuals
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ATION SCIENCE
Stationary States in a Model of Position Selection by Individuals A. P. Petrova,* and O. G. Pronchevab,** a
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia b Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141701 Russia *e-mail: [email protected] **e-mail: [email protected] Received May 7, 2020; revised May 7, 2020; accepted June 9, 2020
Abstract—A model of position selection by individuals in the propaganda battle of two parties is considered. The position selection is based on a neurological decision-making model the input of which is the information stimuli arriving to the individual from the opposing parties and which produces as its output the support of one of these parties. In this version of the model, assortativity and the incomplete coverage of the population by mass media are also taken into account. The number and stability of equilibriums are investigated and a meaningful interpretation is proposed. Keywords: ordinary differential equations, stability, neurological decision-making model, propaganda battle, echo chamber DOI: 10.1134/S0965542520100115
1. INTRODUCTION Real-life societies rarely reach consensus even though its existence is predicted by the class of mathematical models of opinion dynamics. The well-known early paper (which actually shaped the whole research area) by DeGroot is called Reaching a Consensus [1]. In a simplified form, the mechanics of such models is as follows. The individual’s attitude to a certain subject at a certain time is represented by a scalar quantity. The individual discusses this subject with other individuals, and they modify their attitudes to this topic making steps in each other direction. Thus, the hypothesis about the individual’s behavior is that he or she determines his position at the time t + 1 as the weighted average value of the positions of different individuals (including his own) at the time t . The weights are specified by the influence matrix the entries of which indicate the degree of influence of the opinion of individual j on the opinion of individual i . Under this assumption, a consensus is always reached; i.e., all the individuals come to the same opinion after certain time (except for a special case when the influence matrix decomposes into separate blocks). Presently, there are a lot of publications concerning models of opinion dynamics (e.g., see the references in [2–5]). However, real-life societies (and large groups) rarely reach consensus; furthermore, the dynamics is often directed to the polarization of opinions rather than to reaching the consensus. Sometimes, interacting left-wing and right-wing communities that hardly communicate with each other are formed. Often, groups of ideological minorities called echo chambers (see [6–8]) occur that are able to maintain their opinions (or get more radical) in spite of the domination of the majority party. Polarization, rather than reaching a consensus, is registered in increasingly growing p
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