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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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