# American Institute of Mathematical Sciences

September  2015, 10(3): 477-509. doi: 10.3934/nhm.2015.10.477

## Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model

 1 Bayreuth University, Universitaetsstrasse 30, 95440 Bayreuth, Germany 2 Bremen University, Bibliotheksstrasse 1, 28359 Bremen, Germany

Received  December 2014 Revised  March 2015 Published  July 2015

By a simple extension of the bounded confidence model, it is possible to model the influence of a radical group, or a charismatic leader on the opinion dynamics of normal' agents that update their opinions under both, the influence of their normal peers, and the additional influence of the radical group or a charismatic leader. From a more abstract point of view, we model the influence of a signal, that is constant, may have different intensities, and is heard' only by agents with opinions, that are not too far away. For such a dynamic a Constant Signal Theorem is proven. In the model we get a lot of surprising effects. For instance, the more intensive signal may have less effect; more radicals may lead to less radicalization of normal agents. The model is an extremely simple conceptual model. Under some assumptions the whole parameter space can be analyzed. The model inspires new possible explanations, new perspectives for empirical studies, and new ideas for prevention or intervention policies.
Citation: Rainer Hegselmann, Ulrich Krause. Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model. Networks and Heterogeneous Media, 2015, 10 (3) : 477-509. doi: 10.3934/nhm.2015.10.477
##### References:
 [1] D. Acemoglu and A. Ozdaglar, Opinion dynamics and learning in social networks, Dynamic Games and Applications, 1 (2011), 3-49. doi: 10.1007/s13235-010-0004-1. [2] M. Baurmann, G. Betz and R. Cramm, Meinungsdynamiken in fundamentalistischen Gruppen - Erklärungshypothesen auf der Basis von Simulationsmodellen, Analyse and Kritik, 36 (2014), 61-102. [3] V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM Journal on Control and Optimization, 48 (2010), 5214-5240, doi: 10.1137/090766188. [4] B. Chazelle, The total s-energy of a multiagent system, SIAM Journal of Control and Optimization, 49 (2011), 1680-1706. doi: 10.1137/100791671. [5] G. Deffuant, F. Amblard, G. Weisbuch and T. Faure, How can extremism prevail? A study based on the relative agreement interaction model, Journal of Artificial Societies and Social Simulation, 5 (2002). Available from: http://jasss.soc.surrey.ac.uk/5/4/1.html. [6] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. doi: 10.1142/S0219525900000078. [7] M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121. doi: 10.1080/01621459.1974.10480137. [8] R. Hegselmann, Bounded confidence, radical groups, and charismatic leaders, in Advances in Computational Social Science and Social Simulation. Proceedings of the Social Simulation Conference 2014 Barcelona, Catalunya (Spain), September 15 (eds. F. J. Miguel, F. Amblard, J. A. Barceló and M. Madella), Autònoma University of Barcelona, (DDD repository http://ddd.uab.cat/record/125597), Barcelona, 2014, 217-219. [9] R. Hegselmann, S. König, S. Kurz, C. Niemann and J. Rambau, Optimal opinion control: The campaign problem, Journal of Artificial Societies and Social Simulation (JASSS), (2015), 47pp. doi: 10.2139/ssrn.2516866. [10] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002). Available from: http://jasss.soc.surrey.ac.uk/5/3/2.html. [11] R. Hegselmann and U. Krause, Truth and cognitive division of labour: First steps towards a computer aided social epistemology, Journal of Artificial Societies and Social Simulation, 9 (2006). Available from: http://jasss.soc.surrey.ac.uk/9/3/10.html. [12] S. Huet, G. Deffuant and W. Jager, Rejection mechanism in 2d bounded confidence provides more conformity, Advances in Complex Systems, 11 (2008), 529-549. doi: 10.1142/S0219525908001799. [13] U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models, and Applications, De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696. [14] S. Kurz and J. Rambau, On the Hegselmann-Krause conjecture in opinion dynamics, Journal of Difference Equations and Applications, 17 (2011), 859-876. doi: 10.1080/10236190903443129. [15] J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789. [16] N. Oreskes and E. M. Conway, Merchants of Doubt - How a Handful of Scientists Obscured the Truth on Issues from Tobacco Smoke to Global Warming, Bloomsbury Press, 2010. [17] G. G. Polhill, L. R. Izquierdo and N. M. Gotts, The ghost in the model (and other effects of floating point arithmetic), Journal of Artificial Societies and Social Simulation, 8 (2005). Available from: http://jasss.soc.surrey.ac.uk/8/1/5.html. [18] S. Wongkaew, M. Caponigro and A. Borzí, On the control through leadership of the Hegselmann-Krause opinion formation model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 565-585. doi: 10.1142/S0218202515400060. [19] H. Xia, H. Wang and Z. Xuan, Opinion dynamics: A multidisciplinary review and perspective on future research, International Journal of Knowledge and Systems Science, 2 (2011), 72-91. doi: 10.4018/978-1-4666-3998-0.ch021.

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##### References:
 [1] D. Acemoglu and A. Ozdaglar, Opinion dynamics and learning in social networks, Dynamic Games and Applications, 1 (2011), 3-49. doi: 10.1007/s13235-010-0004-1. [2] M. Baurmann, G. Betz and R. Cramm, Meinungsdynamiken in fundamentalistischen Gruppen - Erklärungshypothesen auf der Basis von Simulationsmodellen, Analyse and Kritik, 36 (2014), 61-102. [3] V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM Journal on Control and Optimization, 48 (2010), 5214-5240, doi: 10.1137/090766188. [4] B. Chazelle, The total s-energy of a multiagent system, SIAM Journal of Control and Optimization, 49 (2011), 1680-1706. doi: 10.1137/100791671. [5] G. Deffuant, F. Amblard, G. Weisbuch and T. Faure, How can extremism prevail? A study based on the relative agreement interaction model, Journal of Artificial Societies and Social Simulation, 5 (2002). Available from: http://jasss.soc.surrey.ac.uk/5/4/1.html. [6] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. doi: 10.1142/S0219525900000078. [7] M. H. DeGroot, Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121. doi: 10.1080/01621459.1974.10480137. [8] R. Hegselmann, Bounded confidence, radical groups, and charismatic leaders, in Advances in Computational Social Science and Social Simulation. Proceedings of the Social Simulation Conference 2014 Barcelona, Catalunya (Spain), September 15 (eds. F. J. Miguel, F. Amblard, J. A. Barceló and M. Madella), Autònoma University of Barcelona, (DDD repository http://ddd.uab.cat/record/125597), Barcelona, 2014, 217-219. [9] R. Hegselmann, S. König, S. Kurz, C. Niemann and J. Rambau, Optimal opinion control: The campaign problem, Journal of Artificial Societies and Social Simulation (JASSS), (2015), 47pp. doi: 10.2139/ssrn.2516866. [10] R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002). Available from: http://jasss.soc.surrey.ac.uk/5/3/2.html. [11] R. Hegselmann and U. Krause, Truth and cognitive division of labour: First steps towards a computer aided social epistemology, Journal of Artificial Societies and Social Simulation, 9 (2006). Available from: http://jasss.soc.surrey.ac.uk/9/3/10.html. [12] S. Huet, G. Deffuant and W. Jager, Rejection mechanism in 2d bounded confidence provides more conformity, Advances in Complex Systems, 11 (2008), 529-549. doi: 10.1142/S0219525908001799. [13] U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models, and Applications, De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696. [14] S. Kurz and J. Rambau, On the Hegselmann-Krause conjecture in opinion dynamics, Journal of Difference Equations and Applications, 17 (2011), 859-876. doi: 10.1080/10236190903443129. [15] J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789. [16] N. Oreskes and E. M. Conway, Merchants of Doubt - How a Handful of Scientists Obscured the Truth on Issues from Tobacco Smoke to Global Warming, Bloomsbury Press, 2010. [17] G. G. Polhill, L. R. Izquierdo and N. M. Gotts, The ghost in the model (and other effects of floating point arithmetic), Journal of Artificial Societies and Social Simulation, 8 (2005). Available from: http://jasss.soc.surrey.ac.uk/8/1/5.html. [18] S. Wongkaew, M. Caponigro and A. Borzí, On the control through leadership of the Hegselmann-Krause opinion formation model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 565-585. doi: 10.1142/S0218202515400060. [19] H. Xia, H. Wang and Z. Xuan, Opinion dynamics: A multidisciplinary review and perspective on future research, International Journal of Knowledge and Systems Science, 2 (2011), 72-91. doi: 10.4018/978-1-4666-3998-0.ch021.
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