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May  2013, 12(3): 1487-1499. doi: 10.3934/cpaa.2013.12.1487

## Asymptotic analysis of continuous opinion dynamics models under bounded confidence

 1 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, Italy

Received  May 2012 Revised  July 2012 Published  September 2012

This paper deals with the asymptotic behavior of mathematical models for opinion dynamics under bounded confidence of Deffuant-Weisbuch type. Focusing on the Cauchy Problem related to compromise models with homogeneous bound of confidence, a general well-posedness result is provided and a systematic study of the asymptotic behavior in time of the solution is developed. More in detail, we prove a theorem that establishes the weak convergence of the solution to a sum of Dirac masses and characterizes the concentration points for different values of the model parameters. Analytical results are illustrated by means of numerical simulations.
Citation: Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487
##### References:
 [1] G. Aletti, G. Naldi and G. Toscani, First-order continuous models of opinion formation,, SIAM J. Appl. Math., 67 (2007), 837. doi: 10.1137/060658679. Google Scholar [2] A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi, Kinetic models of inelastic gases,, Mat. Mod. Meth. Appl. Sci., 12 (2002), 965. doi: 10.1142/S0218202502001982. Google Scholar [3] E. Ben-Naim, P. L. Krapivsky and S. Redner, Bifurcation and patterns in compromise processes,, Phys. D, 183 (2003), 190. doi: 10.1016/S0167-2789(03)00171-4. Google Scholar [4] M. L. Bertotti, On a class of dynamical systems with emerging cluster structure,, Jour. Diff. Eq., 249 (2010), 2757. doi: 10.1016/j.jde.2010.03.014. Google Scholar [5] V. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On the $2R$ conjecture for multiagent systems,, Proc. Europ. Control Conf., (2007), 874. Google Scholar [6] L. Boudin and F. Salvarani, Modelling opinion formation by means of kinetic equations,, in, (2010), 245. doi: 10.1007/978-0-8176-4946-3_10. Google Scholar [7] L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior,, Kinet. Relat. Models, 2 (2009), 433. doi: 10.3934/krm.2009.2.433. Google Scholar [8] L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation,, M2AN Math. Model. Numer. Anal., 43 (2009), 507. doi: 10.1051/m2an/2009004. Google Scholar [9] C. Canuto, F. Fagnani and P. Tilli, A Eulerian approach to the analysis of rendez-vous algorithms,, Proc. of 2008 IFAC Conf., (2008), 9039. Google Scholar [10] G. Como and F. Fagnani, Scaling limits for continuous opinion dynamics systems,, Ann. Appl. Probab., 21 (2011), 1537. doi: 10.1214/10-AAP739. Google Scholar [11] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy,, Journal of Statistical Physics, 120 (2005), 253. doi: 10.1007/s10955-005-5456-0. Google Scholar [12] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents,, Adv. Comp. Sys., 3 (2001), 87. doi: 10.1142/S0219525900000078. Google Scholar [13] 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,, Proc. R. Soc. Lond. A, 465 (2009), 3687. doi: 10.1098/rspa.2009.0239. Google Scholar [14] S. Galam, A review of Galam models,, Int. J. Mod. Phys. C, 409 (2008), 3687. Google Scholar [15] U. Krause, A discrete nonlinear and non-autonomous model of consensus formation,, in, (2000), 227. Google Scholar [16] J. Lorenz, A stabilization theorem for continuous opinion dynamics,, Phys. A, 355 (2005), 217. doi: 10.1016/j.physa.2005.02.086. Google Scholar [17] J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey,, Internat. J. Modern Phys. C, 18 (2007), 1819. doi: 10.1142/S0129183107011789. Google Scholar [18] S. McNamara and W. R. Young, Kinetics of a one dimensional granular medium in the quasi elastic limit,, Phys. Fluids A, 5 (1993), 34. doi: 10.1063/1.858896. Google Scholar [19] L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models,, J. Stat. Phys., 124 (2006), 747. doi: 10.1007/s10955-006-9025-y. Google Scholar [20] G. Toscani, Kinetic models of opinion formation,, Commun. Math. Sci., 4 (2006), 481. Google Scholar [21] G. Weisbuch, G. Deffuant, F. Amblard and J. P. Nadal, Meet, discuss, and segregate!,, Complexity, 7 (2002), 55. doi: 10.1002/cplx.10031. Google Scholar

show all references

##### References:
 [1] G. Aletti, G. Naldi and G. Toscani, First-order continuous models of opinion formation,, SIAM J. Appl. Math., 67 (2007), 837. doi: 10.1137/060658679. Google Scholar [2] A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi, Kinetic models of inelastic gases,, Mat. Mod. Meth. Appl. Sci., 12 (2002), 965. doi: 10.1142/S0218202502001982. Google Scholar [3] E. Ben-Naim, P. L. Krapivsky and S. Redner, Bifurcation and patterns in compromise processes,, Phys. D, 183 (2003), 190. doi: 10.1016/S0167-2789(03)00171-4. Google Scholar [4] M. L. Bertotti, On a class of dynamical systems with emerging cluster structure,, Jour. Diff. Eq., 249 (2010), 2757. doi: 10.1016/j.jde.2010.03.014. Google Scholar [5] V. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On the $2R$ conjecture for multiagent systems,, Proc. Europ. Control Conf., (2007), 874. Google Scholar [6] L. Boudin and F. Salvarani, Modelling opinion formation by means of kinetic equations,, in, (2010), 245. doi: 10.1007/978-0-8176-4946-3_10. Google Scholar [7] L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior,, Kinet. Relat. Models, 2 (2009), 433. doi: 10.3934/krm.2009.2.433. Google Scholar [8] L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation,, M2AN Math. Model. Numer. Anal., 43 (2009), 507. doi: 10.1051/m2an/2009004. Google Scholar [9] C. Canuto, F. Fagnani and P. Tilli, A Eulerian approach to the analysis of rendez-vous algorithms,, Proc. of 2008 IFAC Conf., (2008), 9039. Google Scholar [10] G. Como and F. Fagnani, Scaling limits for continuous opinion dynamics systems,, Ann. Appl. Probab., 21 (2011), 1537. doi: 10.1214/10-AAP739. Google Scholar [11] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy,, Journal of Statistical Physics, 120 (2005), 253. doi: 10.1007/s10955-005-5456-0. Google Scholar [12] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents,, Adv. Comp. Sys., 3 (2001), 87. doi: 10.1142/S0219525900000078. Google Scholar [13] 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,, Proc. R. Soc. Lond. A, 465 (2009), 3687. doi: 10.1098/rspa.2009.0239. Google Scholar [14] S. Galam, A review of Galam models,, Int. J. Mod. Phys. C, 409 (2008), 3687. Google Scholar [15] U. Krause, A discrete nonlinear and non-autonomous model of consensus formation,, in, (2000), 227. Google Scholar [16] J. Lorenz, A stabilization theorem for continuous opinion dynamics,, Phys. A, 355 (2005), 217. doi: 10.1016/j.physa.2005.02.086. Google Scholar [17] J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey,, Internat. J. Modern Phys. C, 18 (2007), 1819. doi: 10.1142/S0129183107011789. Google Scholar [18] S. McNamara and W. R. Young, Kinetics of a one dimensional granular medium in the quasi elastic limit,, Phys. Fluids A, 5 (1993), 34. doi: 10.1063/1.858896. Google Scholar [19] L. Pareschi and G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models,, J. Stat. Phys., 124 (2006), 747. doi: 10.1007/s10955-006-9025-y. Google Scholar [20] G. Toscani, Kinetic models of opinion formation,, Commun. Math. Sci., 4 (2006), 481. Google Scholar [21] G. Weisbuch, G. Deffuant, F. Amblard and J. P. Nadal, Meet, discuss, and segregate!,, Complexity, 7 (2002), 55. doi: 10.1002/cplx.10031. Google Scholar
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