Brief paperOn sector stability of opinion dynamics with stubborn extremists☆
Introduction
Recent years have witnessed an increasing research interest in opinion dynamics where individuals interact with each other according to some social psychology theories. Many mathematical models have been presented to model the behaviour of opinion dynamics (DeGroot, 1974, Friedkin, 2015, Friedkin and Johnsen, 1999, Ghaderi and Srikant, 2014, Groeber et al., 2014, Jia et al., 2015, Pirani and Sundaram, 2014, Proskurnikov and Cao, 2017, Xu et al., 2015), among which the basic model is DeGroot model dating back to Abelson (1964) and DeGroot (1974). But the DeGroot model cannot model social behaviour of stubborn agents, which is a widespread phenomenon in society. To make up for this limitation, Friedkin–Johnsen model was proposed in Friedkin (2015) and Friedkin and Johnsen (1999). In Friedkin–Johnsen model, agents are allowed to have different susceptibilities to persuasion, and the state of the agents do not depend on the current opinion or attitude. Based on reflected appraisal mechanism (Friedkin, 2011), DeGroot–Friedkin model with inclusion of social power and interpersonal influences for a group of individuals was presented in Jia et al. (2015). Many subsequent extensions of the above-mentioned basic models can be seen in the survey paper (Proskurnikov & Tempo, 2017).
According to different theories of social psychology, three different types of polar opinion dynamics with cooperative interactions were presented and studied in Amelkin, Bullo, and Singh (2017). Opinion dynamics with stubborn extremists, based on social-psychological considerations (Bassili, 1996, Eagly and Chaiken, 1993, Miller et al., 1993), assumes that the individuals with extreme opinions are resistant to change. Opinion dynamics with stubborn positives, according to Friedkin (2015), is a special version of model with stubborn extremists. Opinion dynamics with stubborn neutrals, based on social comparison theory (Festinger, 1954) and social norms (Friedkin, 2015), assumes that the individuals with neutral opinions are resistant to change. It is well known that cooperative and antagonistic interactions usually coexist in opinion forming (Altafini, 2012, Easley and Kleinberg, 2010, Wasserman and Faust, 1994). To overcome the limitations of the work in Amelkin et al. (2017), our recent paper (Zhai & Zheng, 2021) investigated generalized opinion dynamics with cooperative and antagonistic interactions, which can describe the degree of an agent’s opinion and can be seen as a generalization of the works in Altafini and Lini (2015) and Amelkin et al. (2017).
Our generalized model in Zhai and Zheng (2021) uses the state-dependent susceptibility to persuasion, which was introduced in Amelkin et al. (2017). The main difference is that the weights of interaction graph in our generalized model can be positive or negative, while the weights of interaction graph in Amelkin et al. (2017) are all positive. Compared with the models in the works (Altafini and Lini, 2015, Proskurnikov and Cao, 2017, Proskurnikov and Tempo, 2017), our generalized model in Zhai and Zheng (2021) considered the state-dependent susceptibility to persuasion in modelling opinion dynamics for the first time in the literature. Since some criteria given in Zhai and Zheng (2021) depend on the stability of equilibrium points lying on the boundary, the stability of equilibrium points on the boundary is very important for the dynamic behaviour of opinions. However, the stability of these equilibrium points is a rather complicated issue. For example, even if all eigenvalues of the interaction matrix have non-positive real parts, some equilibrium points on the boundary can still be unstable (Zhai & Zheng, 2021). If one of these equilibrium points is asymptotically stable, then some individuals will have extreme opinions while the other part does not have extreme opinions. Specifically, there may exist some local stable equilibrium points (corresponding to local mainstream opinions) and unstable equilibrium points (corresponding to unacceptable opinions). Thus, two important questions naturally arise: how big is the attraction region of the stable equilibrium point lying on the boundary and how to get such an attraction region? If these two questions can be answered, then an appropriate control scheme can be designed such that the opinions of the individuals will approach some specific states.
This paper will answer the aforementioned two questions. The stability problem of opinion dynamics with stubborn extremists and antagonistic interactions is studied. The stability of equilibrium points lying on the boundary is defined as the sector stability. First of all, a sufficient condition for the local sector stability of an equilibrium point is established. In order to obtain the attraction region of a sector stable equilibrium point, another sufficient condition is presented in terms of polynomial inequality, followed by developing an algorithm to estimate the attraction region. Moreover, some necessary conditions for the sector stability of an equilibrium point are derived. Additionally, further discussion is made about under what conditions there does not exist an equilibrium point that is sector stable. Finally, three illustrative examples are utilized to validate the efficiency of the theoretical findings.
The rest of the paper is arranged with the following sections. Some notations and the problem statement are given in Section 2. Section 3 first shows some sufficient conditions for the sector stability together with an algorithm for estimating the attraction region, and then presents some necessary conditions for the sector stability as well as further discussion. Three numerical examples are provided in Section 4 to verify the obtained results. Finally, the paper is concluded in Section 5, with an outlook at future work.
Section snippets
Preliminaries
This section will provide some notations used in this paper and the problem statement.
Main results
This section will first present two types of sufficient conditions for the sector stability of an equilibrium point of the opinion dynamics (1) and a numerical method to obtain the attraction region of a sector stable equilibrium point. Then we will show some necessary conditions for the sector stability of an equilibrium point. Finally, we will further discuss about under what conditions there does not exist an equilibrium point that is sector stable.
Illustrative examples
This section will present three numerical examples to illustrate the effectiveness of the derived results.
Example 1 Consider the opinion dynamics (1) with three agents This opinion dynamics involves some antagonistic interactions and can be expressed in the matrix form as (2) with . By using Matlab, one can obtain a diagonal positive matrix such that . Hence, one can know from Theorem 5 that
Conclusion
For opinion dynamics with stubborn extremists, this paper has addressed the stability problem of the equilibrium points lying on the boundary. First, the corresponding stability problem has been defined as the sector stability. Next, two types of sufficient conditions have been provided for the sector stability, and an algorithm has been developed to estimate the region of sector stability. Then some necessary conditions have been derived for the sector stability of an equilibrium point lying
Acknowledgements
This work was supported in part by the Natural Science Foundation of Chongqing of China under Grant cstc2019jcyj-msxmX0109, in part by the Scientific and Technological Research Program of Chongqing Municipal Education Commission, China under Grant KJQN202000608, and in part by the Science and Technology Innovation Project of “Construction of Chengdu-Chongqing Double City Economic Circle” under GrantKJCX2020029.
Shidong Zhai received the B.Sc. degree in Applied Mathematics in 2008 from China Three Gorges University, Yichang, China. He received the M.Sc. degree in Operations Research and Cybernetic in 2010, and the Ph.D. degree in Circuit and Systems in 2014 all from Huazhong University of Science and Technology, Wuhan, China. He was a visiting scholar at Western Sydney University, Sydney, Australia from April 2017 to April 2018. He is currently an Associate Professor at the School of Automation,
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Cited by (0)
Shidong Zhai received the B.Sc. degree in Applied Mathematics in 2008 from China Three Gorges University, Yichang, China. He received the M.Sc. degree in Operations Research and Cybernetic in 2010, and the Ph.D. degree in Circuit and Systems in 2014 all from Huazhong University of Science and Technology, Wuhan, China. He was a visiting scholar at Western Sydney University, Sydney, Australia from April 2017 to April 2018. He is currently an Associate Professor at the School of Automation, Chongqing University of Posts and Telecommunications, Chongqing, China. His current research interests include nonlinear control and dynamics, complex network and control, and collective behaviour in animal groups.
Wei Xing Zheng received the B.Sc. degree in Applied Mathematics in 1982, the M.Sc. degree in Electrical Engineering in 1984, and the Ph.D. degree in Electrical Engineering in 1989, all from Southeast University, Nanjing, China. He is currently a University Distinguished Professor at Western Sydney University, Sydney, Australia. Over the years he has also held various faculty/research/visiting positions at Southeast University, Nanjing, China; Imperial College of Science, Technology and Medicine, London, UK; University of Western Australia, Perth, Australia; Curtin University of Technology, Perth, Australia; Munich University of Technology, Munich, Germany; University of Virginia, Charlottesville, VA, USA; University of California-Davis, Davis, CA, USA; etc. His research interests are in the areas of systems and controls, signal processing, and communications.
Prof. Zheng is a Fellow of IEEE. He received the 2017 Vice-Chancellor’s Award for Excellence in Research (Researcher of the Year) at Western Sydney University, Sydney, Australia. Previously, he served as an Associate Editor for IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, IEEE Transactions on Automatic Control, IEEE Signal Processing Letters, IEEE Transactions on Circuits and Systems-II: Express Briefs, IEEE Transactions on Fuzzy Systems, and IEEE Transactions on Neural Networks and Learning Systems, and as a Guest Editor for IEEE Transactions on Circuits and Systems-I: Regular Papers. Currently, he is an Associate Editor of Automatica, IEEE Transactions on Cybernetics, IEEE Transactions on Control of Network Systems, IEEE Transactions on Circuits and Systems-I: Regular Papers, and other flagship journals. He is also an Associate Editor of IEEE Control Systems Society’s Conference Editorial Board. He was the Publication Co-Chair of the 56th IEEE Conference on Decision and Control in Melbourne, Australia in December 2017. He has been an IEEE Distinguished Lecturer of IEEE Control Systems Society.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Claudio Altafini under the direction of Editor Christos G. Cassandras.