A nonlinear model of opinion formation on the sphere

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September 2015, 35(9): 4241-4268. doi: 10.3934/dcds.2015.35.4241

A nonlinear model of opinion formation on the sphere

Marco Caponigro ^1, , Anna Chiara Lai ^2, and Benedetto Piccoli ^3,

1.	Équipe M2N - EA 7340, Conservatoire National des Arts et Métiers, Paris, France
2.	Dipartimento di Matematica e Fisica, Università degli studi di Roma Tre, Rome, Italy
3.	Department of Mathematical Sciences & Center for Computational and Integrative Biology, Rutgers University, Camden, NJ

Received June 2014 Revised October 2014 Published April 2015

Abstract
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In this paper we present a model for opinion dynamics on the $d$-dimensional sphere based on classical consensus algorithms. The choice of the model is motivated by the analysis of the comprehensive literature on the subject, both from the mathematical and the sociological point of views. The resulting dynamics is highly nonlinear and therefore presents a rich structure. Equilibria and asymptotic behavior are then analysed and sufficient condition for consensus are established. Finally we address global stabilization and controllability.

Keywords: multiagent dynamics, nonlinear dynamical systems, control., Opinion formation, modeling.

Mathematics Subject Classification: Primary: 91D10, 34A34; Secondary: 34H05, 91B6.

Citation: Marco Caponigro, Anna Chiara Lai, Benedetto Piccoli. A nonlinear model of opinion formation on the sphere. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4241-4268. doi: 10.3934/dcds.2015.35.4241

References:

[1]	L. Behera and F. Schweitzer, On spatial consensus formation: Is the sznajd model different from a voter model?, International Journal of Modern Physics C, 14 (2003), 1331-1354. doi: 10.1142/S0129183103005467.
[2]	V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM Journal on Control and Optimization, 48 (2010), 5214-5240. doi: 10.1137/090766188.
[3]	G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. doi: 10.1142/S0219525900000078.
[4]	R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation,, Journal of Artificial Societies and Social Simulation, 5 ().
[5]	J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proceedings of the national academy of sciences, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.
[6]	A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control, IEEE Transactions on, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.
[7]	Y. Kuramoto, Cooperative dynamics of oscillator community a study based on lattice of rings, Progress of Theoretical Physics Supplement, 79 (1984), 223-240. doi: 10.1143/PTPS.79.223.
[8]	J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789.
[9]	L. Moreau, Stability of continuous-time distributed consensus algorithms, in Decision and Control, 2004. CDC. 43rd IEEE Conference on, vol. 4, IEEE, (2004), 3998-4003. doi: 10.1109/CDC.2004.1429377.
[10]	L. Moreau, Stability of multiagent systems with time-dependent communication links, Automatic Control, IEEE Transactions on, 50 (2005), 169-182. doi: 10.1109/TAC.2004.841888.
[11]	M. C. Nisbet, The competition for worldviews: Values, information, and public support for stem cell research, International Journal of Public Opinion Research, 17 (2005), 90-112. doi: 10.1093/ijpor/edh058.
[12]	H. Norpohth, M. S. L. Beck and J. D. Lafay, Economics and Politics: The Calculus of Support, University of Michigan Press, 1991.
[13]	R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.
[14]	R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.
[15]	A. Sarlette, Geometry and Symmetries in Coordination Control, PhD thesis, Université de Liège, 2009.
[16]	A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76. doi: 10.1137/060673400.
[17]	D. A. Scheufele and B. V. Lewenstein, The public and nanotechnology: How citizens make sense of emerging technologies, Journal of Nanoparticle Research, 7 (2005), 659-667. doi: 10.1007/s11051-005-7526-2.
[18]	R. Sepulchre, Consensus on nonlinear spaces, Annual reviews in control, 35 (2011), 56-64. doi: 10.1016/j.arcontrol.2011.03.003.
[19]	P. Sobkowicz, Modelling opinion formation with physics tools: Call for closer link with reality, Journal of Artificial Societies and Social Simulation, 12 (2009), 11pp, URL http://jasss.soc.surrey.ac.uk/12/1/11.html.
[20]	S. H. Strogatz, From kuramoto to crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D: Nonlinear Phenomena, 143 (2000), 1-20, URL http://www.sciencedirect.com/science/article/pii/S0167278900000944. doi: 10.1016/S0167-2789(00)00094-4.
[21]	K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1157-1165. doi: 10.1142/S0129183100000936.
[22]	J. N. Tsitsiklis, Problems in Decentralized Decision making and Computation., PhD thesis, MIT, 1984.
[23]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical review letters, 75 (1995), p1226. doi: 10.1103/PhysRevLett.75.1226.

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References:

[1]	L. Behera and F. Schweitzer, On spatial consensus formation: Is the sznajd model different from a voter model?, International Journal of Modern Physics C, 14 (2003), 1331-1354. doi: 10.1142/S0129183103005467.
[2]	V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM Journal on Control and Optimization, 48 (2010), 5214-5240. doi: 10.1137/090766188.
[3]	G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. doi: 10.1142/S0219525900000078.
[4]	R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence models, analysis, and simulation,, Journal of Artificial Societies and Social Simulation, 5 ().
[5]	J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proceedings of the national academy of sciences, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.
[6]	A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, Automatic Control, IEEE Transactions on, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.
[7]	Y. Kuramoto, Cooperative dynamics of oscillator community a study based on lattice of rings, Progress of Theoretical Physics Supplement, 79 (1984), 223-240. doi: 10.1143/PTPS.79.223.
[8]	J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789.
[9]	L. Moreau, Stability of continuous-time distributed consensus algorithms, in Decision and Control, 2004. CDC. 43rd IEEE Conference on, vol. 4, IEEE, (2004), 3998-4003. doi: 10.1109/CDC.2004.1429377.
[10]	L. Moreau, Stability of multiagent systems with time-dependent communication links, Automatic Control, IEEE Transactions on, 50 (2005), 169-182. doi: 10.1109/TAC.2004.841888.
[11]	M. C. Nisbet, The competition for worldviews: Values, information, and public support for stem cell research, International Journal of Public Opinion Research, 17 (2005), 90-112. doi: 10.1093/ijpor/edh058.
[12]	H. Norpohth, M. S. L. Beck and J. D. Lafay, Economics and Politics: The Calculus of Support, University of Michigan Press, 1991.
[13]	R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.
[14]	R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.
[15]	A. Sarlette, Geometry and Symmetries in Coordination Control, PhD thesis, Université de Liège, 2009.
[16]	A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76. doi: 10.1137/060673400.
[17]	D. A. Scheufele and B. V. Lewenstein, The public and nanotechnology: How citizens make sense of emerging technologies, Journal of Nanoparticle Research, 7 (2005), 659-667. doi: 10.1007/s11051-005-7526-2.
[18]	R. Sepulchre, Consensus on nonlinear spaces, Annual reviews in control, 35 (2011), 56-64. doi: 10.1016/j.arcontrol.2011.03.003.
[19]	P. Sobkowicz, Modelling opinion formation with physics tools: Call for closer link with reality, Journal of Artificial Societies and Social Simulation, 12 (2009), 11pp, URL http://jasss.soc.surrey.ac.uk/12/1/11.html.
[20]	S. H. Strogatz, From kuramoto to crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D: Nonlinear Phenomena, 143 (2000), 1-20, URL http://www.sciencedirect.com/science/article/pii/S0167278900000944. doi: 10.1016/S0167-2789(00)00094-4.
[21]	K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics C, 11 (2000), 1157-1165. doi: 10.1142/S0129183100000936.
[22]	J. N. Tsitsiklis, Problems in Decentralized Decision making and Computation., PhD thesis, MIT, 1984.
[23]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical review letters, 75 (1995), p1226. doi: 10.1103/PhysRevLett.75.1226.

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