# American Institute of Mathematical Sciences

September  2013, 6(3): 459-479. doi: 10.3934/krm.2013.6.459

## On the dynamics of social conflicts: Looking for the black swan

 1 Department of Mathematica Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino 2 Department of Applied Mathematics, Universidad Complutense, Plaza de Ciencias 3, Ciudad Universitaria, 28040 Madrid, Spain 3 Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy

Received  January 2013 Revised  January 2013 Published  May 2013

This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical principles.
Citation: Nicola Bellomo, Miguel A. Herrero, Andrea Tosin. On the dynamics of social conflicts: Looking for the black swan. Kinetic and Related Models, 2013, 6 (3) : 459-479. doi: 10.3934/krm.2013.6.459
##### References:
 [1] W. B. Arthur, S. N. Durlauf and D. A. Lane, eds., "The Economy as an Evolving Complex System II," Studies in the Sciences of Complexity, Vol. XXVII, Addison-Wesley, 1997. [2] D. Acemoglu and J. A. Robinson, "Economic Origins of Dictatorship and Democracy," Cambridge University Press, 2005. doi: 10.1017/CBO9780511510809. [3] D. Acemoglu, D. Ticchi and A. Vindigni, Emergence and persistence of inefficient states, J. Eur. Econ. Assoc., 9 (2011), 177-208. [4] G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinet. Relat. Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249. [5] G. Ajmone Marsan, On the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems decomposition, Comput. Math. Appl., 57 (2009), 710-728. doi: 10.1016/j.camwa.2008.09.003. [6] A. Alesina and E. Spolaore, War, peace, and the size of countries, J. Public Econ., 89 (2005), 1333-1354. [7] L. Arlotti and N. Bellomo, Solution of a new class of nonlinear kinetic models of population dynamics, Appl. Math. Lett., 9 (1996), 65-70. doi: 10.1016/0893-9659(96)00014-6. [8] L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Math. Models Methods Appl. Sci., 12 (2002), 567-591. doi: 10.1142/S0218202502001799. [9] L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Lett., 25 (2012), 490-495. doi: 10.1016/j.aml.2011.09.043. [10] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105. [11] A.-L. Barabási, R. Albert and H. Jeong, Mean-field theory for scale-free random networks, Physica A, 272 (1999), 173-187. [12] U. Bastolla, M. A. Fortuna, A. Pascual-García, A. Ferrera, B. Luque and J. Bascompte, The architecture of mutualistic networks minimizes competition and increases biodiversity, Nature, 458 (2009), 1018-1020. doi: 10.1038/nature07950. [13] N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29 pp. doi: 10.1142/S0218202511400069. [14] A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), 1140003, 35 pp. doi: 10.1142/S0218202511400033. [15] M. L. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Anal. Real World Appl., 9 (2008), 183-196. doi: 10.1016/j.nonrwa.2006.09.012. [16] M. L. Bertotti and M. Delitala, On a discrete generalized kinetic approach for modelling persuader's influence in opinion formation processes, Math. Comput. Modelling, 48 (2008), 1107-1121. doi: 10.1016/j.mcm.2007.12.021. [17] C. F. Camerer, "Behavioral Game Theory: Experiments in Strategic Interaction," Princeton University Press, 2003. [18] V. Coscia, L. Fermo and N. Bellomo, On the mathematical theory of living systems II: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011), 3902-3911. doi: 10.1016/j.camwa.2011.09.043. [19] E. Cristiani, P. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588. doi: 10.1007/s00285-010-0347-7. [20] M. Delitala, P. Pucci and M. C. Salvatori, From methods of the mathematical kinetic theory for active particles to modeling virus mutations, Math. Models Methods Appl. Sci., 21 (2011), 843-870. doi: 10.1142/S0218202511005398. [21] B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, P. R. Soc. London Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. [22] D. Helbing, "Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes," Second edition, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11546-2. [23] J. C. Nuño, M. A. Herrero and M. Primicerio, A mathematical model of criminal-prone society, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 193-207. doi: 10.3934/dcdss.2011.4.193. [24] D. G. Rand, S. Arbesman and N. A. Christakis, Dynamic social networks promote cooperation in experiments with humans, Proc. Natl. Acad. Sci. USA, 108 (2011), 19193-19198. [25] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59. doi: 10.1038/nature08227. [26] H. A. Simon, Theories of decision-making in economics and behavioral science, Am. Econ. Rev., 49 (1959), 253-283. [27] H. A. Simon, "Models of Bounded Rationality: Economic Analysis and Public Policy," Vol. 1, MIT Press, Cambridge, MA, 1982. [28] H. A. Simon, "Models of Bounded Rationality: Empirically Grounded Economic Reason," Vol. 3, MIT Press, Cambridge, MA, 1997. [29] N. N. Taleb, "The Black Swan: The Impact of the Highly Improbable," Random House, New York City, 2007. [30] G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. [31] F. Vega-Redondo, "Complex Social Networks," Econometric Society Monographs, 44, Cambridge University Press, Cambridge, 2007. [32] G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.

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##### References:
 [1] W. B. Arthur, S. N. Durlauf and D. A. Lane, eds., "The Economy as an Evolving Complex System II," Studies in the Sciences of Complexity, Vol. XXVII, Addison-Wesley, 1997. [2] D. Acemoglu and J. A. Robinson, "Economic Origins of Dictatorship and Democracy," Cambridge University Press, 2005. doi: 10.1017/CBO9780511510809. [3] D. Acemoglu, D. Ticchi and A. Vindigni, Emergence and persistence of inefficient states, J. Eur. Econ. Assoc., 9 (2011), 177-208. [4] G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinet. Relat. Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249. [5] G. Ajmone Marsan, On the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems decomposition, Comput. Math. Appl., 57 (2009), 710-728. doi: 10.1016/j.camwa.2008.09.003. [6] A. Alesina and E. Spolaore, War, peace, and the size of countries, J. Public Econ., 89 (2005), 1333-1354. [7] L. Arlotti and N. Bellomo, Solution of a new class of nonlinear kinetic models of population dynamics, Appl. Math. Lett., 9 (1996), 65-70. doi: 10.1016/0893-9659(96)00014-6. [8] L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Math. Models Methods Appl. Sci., 12 (2002), 567-591. doi: 10.1142/S0218202502001799. [9] L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Lett., 25 (2012), 490-495. doi: 10.1016/j.aml.2011.09.043. [10] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105. [11] A.-L. Barabási, R. Albert and H. Jeong, Mean-field theory for scale-free random networks, Physica A, 272 (1999), 173-187. [12] U. Bastolla, M. A. Fortuna, A. Pascual-García, A. Ferrera, B. Luque and J. Bascompte, The architecture of mutualistic networks minimizes competition and increases biodiversity, Nature, 458 (2009), 1018-1020. doi: 10.1038/nature07950. [13] N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29 pp. doi: 10.1142/S0218202511400069. [14] A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), 1140003, 35 pp. doi: 10.1142/S0218202511400033. [15] M. L. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Anal. Real World Appl., 9 (2008), 183-196. doi: 10.1016/j.nonrwa.2006.09.012. [16] M. L. Bertotti and M. Delitala, On a discrete generalized kinetic approach for modelling persuader's influence in opinion formation processes, Math. Comput. Modelling, 48 (2008), 1107-1121. doi: 10.1016/j.mcm.2007.12.021. [17] C. F. Camerer, "Behavioral Game Theory: Experiments in Strategic Interaction," Princeton University Press, 2003. [18] V. Coscia, L. Fermo and N. Bellomo, On the mathematical theory of living systems II: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011), 3902-3911. doi: 10.1016/j.camwa.2011.09.043. [19] E. Cristiani, P. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, J. Math. Biol., 62 (2011), 569-588. doi: 10.1007/s00285-010-0347-7. [20] M. Delitala, P. Pucci and M. C. Salvatori, From methods of the mathematical kinetic theory for active particles to modeling virus mutations, Math. Models Methods Appl. Sci., 21 (2011), 843-870. doi: 10.1142/S0218202511005398. [21] B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, P. R. Soc. London Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. [22] D. Helbing, "Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes," Second edition, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-11546-2. [23] J. C. Nuño, M. A. Herrero and M. Primicerio, A mathematical model of criminal-prone society, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 193-207. doi: 10.3934/dcdss.2011.4.193. [24] D. G. Rand, S. Arbesman and N. A. Christakis, Dynamic social networks promote cooperation in experiments with humans, Proc. Natl. Acad. Sci. USA, 108 (2011), 19193-19198. [25] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59. doi: 10.1038/nature08227. [26] H. A. Simon, Theories of decision-making in economics and behavioral science, Am. Econ. Rev., 49 (1959), 253-283. [27] H. A. Simon, "Models of Bounded Rationality: Economic Analysis and Public Policy," Vol. 1, MIT Press, Cambridge, MA, 1982. [28] H. A. Simon, "Models of Bounded Rationality: Empirically Grounded Economic Reason," Vol. 3, MIT Press, Cambridge, MA, 1997. [29] N. N. Taleb, "The Black Swan: The Impact of the Highly Improbable," Random House, New York City, 2007. [30] G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. [31] F. Vega-Redondo, "Complex Social Networks," Econometric Society Monographs, 44, Cambridge University Press, Cambridge, 2007. [32] G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.
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