Opinion Dynamics on a General Compact Riemannian Manifold

Aylin Aydoğdu; Sean T. McQuade; Nastassia Pouradier Duteil; Aylin Aydoğdu; Sean T. McQuade; Nastassia Pouradier Duteil

doi:10.3934/nhm.2017021

Networks and Heterogeneous Media

2017, Volume 12, Issue 3: 489-523. doi: 10.3934/nhm.2017021

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Opinion Dynamics on a General Compact Riemannian Manifold

Center for Computational and Integrative Biology, Rutgers University -Camden, 303 Cooper Street, Camden, NJ 08102, USA

Received: 01 March 2017 Revised: 01 July 2017
Primary: 34C40; Secondary: 37N40, 91B14

This work formulates the problem of defining a model for opinion dynamics on a general compact Riemannian manifold. Two approaches to modeling opinions on a manifold are explored. The first defines the distance between two points using the projection in the ambient Euclidean space. The second approach defines the distance as the length of the geodesic between two agents. Our analysis focuses on features such as equilibria, the long term behavior, and the energy of the system, as well as the interactions between agents that lead to these features. Simulations for specific manifolds, $\mathbb{S}^1, \mathbb{S}^2,$ and $\mathbb{T}^2$ , accompany the analysis. Trajectories given by opinion dynamics may resemble $n-$ body Choreography and are called "social choreography". Conditions leading to various types of social choreography are investigated in $\mathbb{R}^2$ .

Keywords:

Citation: Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold[J]. Networks and Heterogeneous Media, 2017, 12(3): 489-523. doi: 10.3934/nhm.2017021

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Abstract

References

[1]	A. Aydoğdu, M. Caponigro, S. McQuade, B. Piccoli, N. Pouradier Duteil, F. Rossi and E. Trélat, Interaction network, state space and control in social dynamics, in Active Particles Volume 1, Theory, Methods, and Applications (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser-Springer, 2017.
[2]	The geodesic lines on the anchor ring. Annals of Mathematics (1902) 4: 1-21.
[3]	A nonlinear model of opinion formation on the sphere. Discrete and Continuous Dynamical Systems -Series A (2014) 35: 4241-4268.
[4]	J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, Vol. 365, AMS Chelsea Publishing, 1975.
[5]	D. Chi, S. -H. Choi and S. -Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, Journal of Mathematical Physics, 55 (2014), 052703, 18pp. doi: 10.1063/1.4878117
[6]	Effects of anisotropic interactions on the structure of animal groups. Journal of Mathematical Biology (2011) 62: 569-588.
[7]	Synchronization in complex oscillator networks and smart grids. Proceedings of the National Academy of Sciences (2013) 110: 2005-2010.
[8]	J. Gravesen, S. Markvorsen, R. Sinclair and M. Tanaka, The Cut Locus of a Torus of Revolution, Technical University of Denmark, Department of Mathematics, 2003.
[9]	Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings. IEEE Transactions on Automatic Control (2010) 55: 1679-1683.
[10]	Collective synchronization of classical and quantum oscillators. EMS Surveys in Mathematical Sciences (2016) 3: 209-267.
[11]	R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation 5 (2002).
[12]	Cooperative dynamics of oscillator community a study based on lattice of rings. Progress of Theoretical Physics Supplement (1984) 79: 223-240.
[13]	Braids in classical dynamics. Physical Review Letters (1993) 70: 3675-3679.
[14]	New periodic orbits for the n-body problem. ASME. J. Comput. Nonlinear Dynam. (2006) 1: 307-311.
[15]	Heterophilious dynamics enhances consensus. SIAM Review (2014) 56: 577-621.
[16]	Coordinated motion design on lie groups. Automatic Control, IEEE Transactions on (2010) 55: 1047-1058.
[17]	Consensus optimization on manifolds. SIAM Journal on Control and Optimization (2009) 48: 56-76.
[18]	Synchronization and balancing on the N-torus. Sys. Cont. Let. (2007) 56: 335-341.
[19]	Stabilization of planar collective motion: All-to-all communication. Automatic Control, IEEE Transactions on (2007) 52: 811-824.
[20]	Stabilization of planar collective motion with limited communication. Automatic Control, IEEE Transactions on (2008) 53: 706-719.
[21]	P. Sobkowicz, Modelling opinion formation with physics tools: Call for closer link with reality, Journal of Artificial Societies and Social Simulation, 12 (2009), p11.
[22]	From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena (2000) 143: 1-20.

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