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December  2015, 10(4): 877-896. doi: 10.3934/nhm.2015.10.877

## Modeling opinion dynamics: How the network enhances consensus

 1 Dep. of Civil, Computer, Construction, Environmental Engineering and of Applied Mathematics (DICIEAMA), University of Messina, Contrada Di Dio Vill. S. Agata, Messina, Italy 2 Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa

Received  November 2014 Revised  May 2015 Published  October 2015

In this paper we analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical solutions addressed in this direction, i.e. comparing the emergent behaviors of a consensus-dissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered. We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of specific models. A specific model is derived from the general mathematical structure by introducing a consensus-dissent dynamics of interactions and a qualitative analysis is given.
Citation: Marina Dolfin, Mirosław Lachowicz. Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 2015, 10 (4) : 877-896. doi: 10.3934/nhm.2015.10.877
##### References:
 [1] D. Acemoglu and J. A. Robinson, Economic Backwardness in Political Perspective, Am. Pol. Sci. Rev., 100 (2006), 115-131. [2] G. Ajmone Marsan, N. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation, Springer Briefs in Mathematics, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-7242-1. [3] J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment, Math. Models Methods Appl. Sci., 23 (2013), 2647-2670. doi: 10.1142/S0218202513500425. [4] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser, 2014. doi: 10.1007/978-3-319-05140-6. [5] N. Bellomo, F. Colasuonno, D. Knopoff and J. Soler, From systems theory of sociology to modeling the onset and evolution of criminality, NHM, 10 (2015), 421-441. doi: 10.3934/nhm.2015.10.421. [6] N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts looking for the Black Swan, Kinet. Relat. Models, 6 (2013), 459-479. doi: 10.3934/krm.2013.6.459. [7] N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X. [8] 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. [9] M. L. Bertotti and M. Delitala, Cluster formation in opinion dynamics: A qualitative analysis, ZAMP, 61 (2010), 583-602. doi: 10.1007/s00033-009-0040-0. [10] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, Kinetic, and Hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, (G. Naldi, L. Pareschi, and G. Toscani) Birkhäuser, (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12. [11] V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation, Kinet. Relat. Models, 2 (2009), 135-149. doi: 10.3934/krm.2009.2.135. [12] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. of Statist.l Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0. [13] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Complex Syst., 3 (2000), 87-98. doi: 10.1142/S0219525900000078. [14] M. Dolfin and M. Lachowicz, Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions, Math. Models . Methods Appl. Sci., 24 (2014), 2361-2381. doi: 10.1142/S0218202514500237. [15] M. Dolfin, M. Lachowicz and Z. Szymanska, A general framework for multiscale modeling of tumor - immune system interactions, in Mathematical Oncology 2013 - Modeling and simulation in science, engineering and technology, (A. d'Onofrio, A. Gandolfi) Birkhäuser, (2014), 151-180. [16] M. Dolfin, L. Leonida, D. Maimone Ansaldo Patti and P. Navarra, Escaping the trap of blocking: A kinetic model linking economic development and political perspectives,, work in progress., (). [17] I. Down and C. J. Wilson, From "Permissive Consensus" to "Constraining Dissensus": A Polarizing Union?, Acta Politica, 43 (2008), 26-49. doi: 10.1057/palgrave.ap.5500206. [18] B. During, D. Matthes and G. Toscani, Kinetic equations modelling Wealth Redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12pp. doi: 10.1103/PhysRevE.78.056103. [19] D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24 (2014), 405-426. doi: 10.1142/S0218202513400137. [20] M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Anal. Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014. [21] S. Motsch and E. Tadmor, Heterophilius dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866. [22] L. Pareschi and G. Toscani, Interacting Multiagent Systems - Kinetic equations and Monte Carlo methods, Oxford Univ. Press, Oxford, 2014. [23] R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic population, J. Math. Biol., 70 (2015), 1295-1321. doi: 10.1007/s00285-014-0798-3. [24] H. A. Simon, Models of Bounded Rationality: Economic Analysis and Public Policy, MIT Press, Cambridge, MA, 1982. [25] G. Toscani, Kinetic models of opinion formation, Comm. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. [26] G. Weisbuch and D. Stauffer, Adjustment and social choice, Physica A, 323 (2003), 651-662. doi: 10.1016/S0378-4371(03)00010-4. [27] G. Weisbuch, G. Deffuant, F. Amblard and J. P. Nadal, Meet, discuss and segregate!, Complexity, 7 (2002), 55-63. doi: 10.1002/cplx.10031.

show all references

##### References:
 [1] D. Acemoglu and J. A. Robinson, Economic Backwardness in Political Perspective, Am. Pol. Sci. Rev., 100 (2006), 115-131. [2] G. Ajmone Marsan, N. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation, Springer Briefs in Mathematics, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-7242-1. [3] J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment, Math. Models Methods Appl. Sci., 23 (2013), 2647-2670. doi: 10.1142/S0218202513500425. [4] J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser, 2014. doi: 10.1007/978-3-319-05140-6. [5] N. Bellomo, F. Colasuonno, D. Knopoff and J. Soler, From systems theory of sociology to modeling the onset and evolution of criminality, NHM, 10 (2015), 421-441. doi: 10.3934/nhm.2015.10.421. [6] N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts looking for the Black Swan, Kinet. Relat. Models, 6 (2013), 459-479. doi: 10.3934/krm.2013.6.459. [7] N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X. [8] 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. [9] M. L. Bertotti and M. Delitala, Cluster formation in opinion dynamics: A qualitative analysis, ZAMP, 61 (2010), 583-602. doi: 10.1007/s00033-009-0040-0. [10] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, Kinetic, and Hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, (G. Naldi, L. Pareschi, and G. Toscani) Birkhäuser, (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12. [11] V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation, Kinet. Relat. Models, 2 (2009), 135-149. doi: 10.3934/krm.2009.2.135. [12] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. of Statist.l Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0. [13] G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Adv. Complex Syst., 3 (2000), 87-98. doi: 10.1142/S0219525900000078. [14] M. Dolfin and M. Lachowicz, Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions, Math. Models . Methods Appl. Sci., 24 (2014), 2361-2381. doi: 10.1142/S0218202514500237. [15] M. Dolfin, M. Lachowicz and Z. Szymanska, A general framework for multiscale modeling of tumor - immune system interactions, in Mathematical Oncology 2013 - Modeling and simulation in science, engineering and technology, (A. d'Onofrio, A. Gandolfi) Birkhäuser, (2014), 151-180. [16] M. Dolfin, L. Leonida, D. Maimone Ansaldo Patti and P. Navarra, Escaping the trap of blocking: A kinetic model linking economic development and political perspectives,, work in progress., (). [17] I. Down and C. J. Wilson, From "Permissive Consensus" to "Constraining Dissensus": A Polarizing Union?, Acta Politica, 43 (2008), 26-49. doi: 10.1057/palgrave.ap.5500206. [18] B. During, D. Matthes and G. Toscani, Kinetic equations modelling Wealth Redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12pp. doi: 10.1103/PhysRevE.78.056103. [19] D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24 (2014), 405-426. doi: 10.1142/S0218202513400137. [20] M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Anal. Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014. [21] S. Motsch and E. Tadmor, Heterophilius dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866. [22] L. Pareschi and G. Toscani, Interacting Multiagent Systems - Kinetic equations and Monte Carlo methods, Oxford Univ. Press, Oxford, 2014. [23] R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic population, J. Math. Biol., 70 (2015), 1295-1321. doi: 10.1007/s00285-014-0798-3. [24] H. A. Simon, Models of Bounded Rationality: Economic Analysis and Public Policy, MIT Press, Cambridge, MA, 1982. [25] G. Toscani, Kinetic models of opinion formation, Comm. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. [26] G. Weisbuch and D. Stauffer, Adjustment and social choice, Physica A, 323 (2003), 651-662. doi: 10.1016/S0378-4371(03)00010-4. [27] G. Weisbuch, G. Deffuant, F. Amblard and J. P. Nadal, Meet, discuss and segregate!, Complexity, 7 (2002), 55-63. doi: 10.1002/cplx.10031.
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