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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 & 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. Google Scholar

[2]

G. Ajmone Marsan, N. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation,, Springer Briefs in Mathematics, (2013). doi: 10.1007/978-1-4614-7242-1. Google Scholar

[3]

J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment,, Math. Models Methods Appl. Sci., 23 (2013), 2647. doi: 10.1142/S0218202513500425. Google Scholar

[4]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology,, Birkhäuser, (2014). doi: 10.1007/978-3-319-05140-6. Google Scholar

[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. doi: 10.3934/nhm.2015.10.421. Google Scholar

[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. doi: 10.3934/krm.2013.6.459. Google Scholar

[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. doi: 10.1142/S021820251350053X. Google Scholar

[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. doi: 10.1016/j.nonrwa.2006.09.012. Google Scholar

[9]

M. L. Bertotti and M. Delitala, Cluster formation in opinion dynamics: A qualitative analysis,, ZAMP, 61 (2010), 583. doi: 10.1007/s00033-009-0040-0. Google Scholar

[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, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar

[11]

V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation,, Kinet. Relat. Models, 2 (2009), 135. doi: 10.3934/krm.2009.2.135. Google Scholar

[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. doi: 10.1007/s10955-005-5456-0. Google Scholar

[13]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents,, Adv. Complex Syst., 3 (2000), 87. doi: 10.1142/S0219525900000078. Google Scholar

[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. doi: 10.1142/S0218202514500237. Google Scholar

[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, (2014), 151. Google Scholar

[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., (). Google Scholar

[17]

I. Down and C. J. Wilson, From "Permissive Consensus" to "Constraining Dissensus": A Polarizing Union?,, Acta Politica, 43 (2008), 26. doi: 10.1057/palgrave.ap.5500206. Google Scholar

[18]

B. During, D. Matthes and G. Toscani, Kinetic equations modelling Wealth Redistribution: A comparison of approaches,, Phys. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.056103. Google Scholar

[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. doi: 10.1142/S0218202513400137. Google Scholar

[20]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Anal. Real World Appl., 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014. Google Scholar

[21]

S. Motsch and E. Tadmor, Heterophilius dynamics enhances consensus,, SIAM Rev., 56 (2014), 577. doi: 10.1137/120901866. Google Scholar

[22]

L. Pareschi and G. Toscani, Interacting Multiagent Systems - Kinetic equations and Monte Carlo methods,, Oxford Univ. Press, (2014). Google Scholar

[23]

R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic population,, J. Math. Biol., 70 (2015), 1295. doi: 10.1007/s00285-014-0798-3. Google Scholar

[24]

H. A. Simon, Models of Bounded Rationality: Economic Analysis and Public Policy,, MIT Press, (1982). Google Scholar

[25]

G. Toscani, Kinetic models of opinion formation,, Comm. Math. Sci., 4 (2006), 481. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[26]

G. Weisbuch and D. Stauffer, Adjustment and social choice,, Physica A, 323 (2003), 651. doi: 10.1016/S0378-4371(03)00010-4. Google Scholar

[27]

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]

D. Acemoglu and J. A. Robinson, Economic Backwardness in Political Perspective,, Am. Pol. Sci. Rev., 100 (2006), 115. Google Scholar

[2]

G. Ajmone Marsan, N. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation,, Springer Briefs in Mathematics, (2013). doi: 10.1007/978-1-4614-7242-1. Google Scholar

[3]

J. Banasiak and M. Lachowicz, On a macroscopic limit of a kinetic model of alignment,, Math. Models Methods Appl. Sci., 23 (2013), 2647. doi: 10.1142/S0218202513500425. Google Scholar

[4]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology,, Birkhäuser, (2014). doi: 10.1007/978-3-319-05140-6. Google Scholar

[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. doi: 10.3934/nhm.2015.10.421. Google Scholar

[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. doi: 10.3934/krm.2013.6.459. Google Scholar

[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. doi: 10.1142/S021820251350053X. Google Scholar

[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. doi: 10.1016/j.nonrwa.2006.09.012. Google Scholar

[9]

M. L. Bertotti and M. Delitala, Cluster formation in opinion dynamics: A qualitative analysis,, ZAMP, 61 (2010), 583. doi: 10.1007/s00033-009-0040-0. Google Scholar

[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, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar

[11]

V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation,, Kinet. Relat. Models, 2 (2009), 135. doi: 10.3934/krm.2009.2.135. Google Scholar

[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. doi: 10.1007/s10955-005-5456-0. Google Scholar

[13]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents,, Adv. Complex Syst., 3 (2000), 87. doi: 10.1142/S0219525900000078. Google Scholar

[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. doi: 10.1142/S0218202514500237. Google Scholar

[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, (2014), 151. Google Scholar

[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., (). Google Scholar

[17]

I. Down and C. J. Wilson, From "Permissive Consensus" to "Constraining Dissensus": A Polarizing Union?,, Acta Politica, 43 (2008), 26. doi: 10.1057/palgrave.ap.5500206. Google Scholar

[18]

B. During, D. Matthes and G. Toscani, Kinetic equations modelling Wealth Redistribution: A comparison of approaches,, Phys. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.056103. Google Scholar

[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. doi: 10.1142/S0218202513400137. Google Scholar

[20]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Anal. Real World Appl., 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014. Google Scholar

[21]

S. Motsch and E. Tadmor, Heterophilius dynamics enhances consensus,, SIAM Rev., 56 (2014), 577. doi: 10.1137/120901866. Google Scholar

[22]

L. Pareschi and G. Toscani, Interacting Multiagent Systems - Kinetic equations and Monte Carlo methods,, Oxford Univ. Press, (2014). Google Scholar

[23]

R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic population,, J. Math. Biol., 70 (2015), 1295. doi: 10.1007/s00285-014-0798-3. Google Scholar

[24]

H. A. Simon, Models of Bounded Rationality: Economic Analysis and Public Policy,, MIT Press, (1982). Google Scholar

[25]

G. Toscani, Kinetic models of opinion formation,, Comm. Math. Sci., 4 (2006), 481. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[26]

G. Weisbuch and D. Stauffer, Adjustment and social choice,, Physica A, 323 (2003), 651. doi: 10.1016/S0378-4371(03)00010-4. Google Scholar

[27]

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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