September  2017, 12(3): 489-523. doi: 10.3934/nhm.2017021

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  March 2017 Revised  July 2017 Published  September 2017

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 $\mathbb{R}^2$.

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

G. Bliss, The geodesic lines on the anchor ring, Annals of Mathematics, 4 (1902), 1-21.  doi: 10.2307/1967147.

[3]

M. CaponigroA. C. Lai and B. Piccoli, A nonlinear model of opinion formation on the sphere, Discrete and Continuous Dynamical Systems -Series A, 35 (2014), 4241-4268.  doi: 10.3934/dcds.2015.35.4241.

[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]

E. CristianiP. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, Journal of Mathematical Biology, 62 (2011), 569-588.  doi: 10.1007/s00285-010-0347-7.

[7]

F. DörflerM. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010.  doi: 10.1073/pnas.1212134110.

[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]

S.-Y. HaT. Ha and J. H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Transactions on Automatic Control, 55 (2010), 1679-1683.  doi: 10.1109/TAC.2010.2046113.

[10]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[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]

Y. Kuramoto, Cooperative dynamics of oscillator community a study based on lattice of rings, Progress of Theoretical Physics Supplement, 79 (1984), 223-240. 

[13]

C. Moore, Braids in classical dynamics, Physical Review Letters, 70 (1993), 3675-3679.  doi: 10.1103/PhysRevLett.70.3675.

[14]

C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, ASME. J. Comput. Nonlinear Dynam., 1 (2006), 307-311.  doi: 10.1115/1.2338323.

[15]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[16]

A. SarletteS. Bonnabel and R. Sepulchre, Coordinated motion design on lie groups, Automatic Control, IEEE Transactions on, 55 (2010), 1047-1058.  doi: 10.1109/TAC.2010.2042003.

[17]

A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76.  doi: 10.1137/060673400.

[18]

L. ScardoviA. Sarlette and R. Sepulchre, Synchronization and balancing on the N-torus, Sys. Cont. Let., 56 (2007), 335-341.  doi: 10.1016/j.sysconle.2006.10.020.

[19]

R. SepulchreD. Paley and N. E. Leonard, Stabilization of planar collective motion: All-to-all communication, Automatic Control, IEEE Transactions on, 52 (2007), 811-824.  doi: 10.1109/TAC.2007.898077.

[20]

R. SepulchreD. Paley and N. E. Leonard, Stabilization of planar collective motion with limited communication, Automatic Control, IEEE Transactions on, 53 (2008), 706-719.  doi: 10.1109/TAC.2008.919857.

[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]

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.  doi: 10.1016/S0167-2789(00)00094-4.

show all references

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]

G. Bliss, The geodesic lines on the anchor ring, Annals of Mathematics, 4 (1902), 1-21.  doi: 10.2307/1967147.

[3]

M. CaponigroA. C. Lai and B. Piccoli, A nonlinear model of opinion formation on the sphere, Discrete and Continuous Dynamical Systems -Series A, 35 (2014), 4241-4268.  doi: 10.3934/dcds.2015.35.4241.

[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]

E. CristianiP. Frasca and B. Piccoli, Effects of anisotropic interactions on the structure of animal groups, Journal of Mathematical Biology, 62 (2011), 569-588.  doi: 10.1007/s00285-010-0347-7.

[7]

F. DörflerM. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010.  doi: 10.1073/pnas.1212134110.

[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]

S.-Y. HaT. Ha and J. H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Transactions on Automatic Control, 55 (2010), 1679-1683.  doi: 10.1109/TAC.2010.2046113.

[10]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[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]

Y. Kuramoto, Cooperative dynamics of oscillator community a study based on lattice of rings, Progress of Theoretical Physics Supplement, 79 (1984), 223-240. 

[13]

C. Moore, Braids in classical dynamics, Physical Review Letters, 70 (1993), 3675-3679.  doi: 10.1103/PhysRevLett.70.3675.

[14]

C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, ASME. J. Comput. Nonlinear Dynam., 1 (2006), 307-311.  doi: 10.1115/1.2338323.

[15]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.

[16]

A. SarletteS. Bonnabel and R. Sepulchre, Coordinated motion design on lie groups, Automatic Control, IEEE Transactions on, 55 (2010), 1047-1058.  doi: 10.1109/TAC.2010.2042003.

[17]

A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76.  doi: 10.1137/060673400.

[18]

L. ScardoviA. Sarlette and R. Sepulchre, Synchronization and balancing on the N-torus, Sys. Cont. Let., 56 (2007), 335-341.  doi: 10.1016/j.sysconle.2006.10.020.

[19]

R. SepulchreD. Paley and N. E. Leonard, Stabilization of planar collective motion: All-to-all communication, Automatic Control, IEEE Transactions on, 52 (2007), 811-824.  doi: 10.1109/TAC.2007.898077.

[20]

R. SepulchreD. Paley and N. E. Leonard, Stabilization of planar collective motion with limited communication, Automatic Control, IEEE Transactions on, 53 (2008), 706-719.  doi: 10.1109/TAC.2008.919857.

[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]

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.  doi: 10.1016/S0167-2789(00)00094-4.

Figure 1.  An example of a manifold $M$ such that $d_p(x_i,x_j) \neq d_p(x_j,x_i)$, Using system (4), an agent is subject to ''local visibility'', and movement of $x_i$ along $T_{x_{i}}M$ (dashed line through $x_i$) will not bring $x_i$ closer to $x_j$ in this local sense
Figure 2.  The set of agents that influence $x_1$ depends on how the interaction network is defined. In (a) and (b) the dashed lines show the projection of agents onto the tangent space of $x_i$, ($T_{x_{i}}\mathbb{S}^1)$. The agents depicted in red with larger dots influence $x_1$. With the same configuration on $\mathbb{S}^1$, four combinations are possible (approach {A, B} type {Metric, Topological}). Each combination implies $x_1$ interacts with a different set of agents
Figure 3.  The agent $x_1$ is influenced by different agents depending on how the interaction network is defined. These networks may change as the dynamics move the agents on $\mathbb{S}^1$. Each agent $x_j, j\in \{1,\ldots,6\}$ will have a network describing which other agents influence $x_j$. The interaction networks corresponding to systems from Figure 2
Figure 4.  Initial (empty circles) and final positions (filled circles) of 4 agents initially on the vertices of a rectangle with Approach B (left) and Approach A (right), with $A=\mathbb{1}$, $\Psi \equiv \mathrm{Id}$ (Approach A) and $\Psi = \Psi^{3\pi/4}$(Approach B) (see equation (26)). Notice that with Approach A, initial and final positions are identical since any rectangle configuration is an equilibrium. However, with Approach B, the system reaches a square configuration, the only possible equilibrium with pairwise distinct positions
Figure 5.  Evolution of the system (27) with Approach A (left) Approach B (center) when the interaction matrix satisfies condition (28) for the projection distance. Right: Kinetic energy
Figure 6.  Evolution of the system (27) with Approach A (left) Approach B (center) when the interaction matrix satisfies condition (28) for the geodesic distance. Right: Kinetic energy
Figure 7.  A dancing equilibrium for Approach A. The energy becomes constant in time after initial fluctuations
Figure 8.  The left side shows candidates for the choice of function $\Psi$. The right side shows how choice of function determines the energy of the system, for the case of $a= \frac{3\pi}{4}$ the system forms an antipodal equilibrium
Figure 9.  A comparison of the effect of the choice of influence function for Approach B. For $a = \frac{3\pi}{4}$ an antipodal equilibrium occurs (see Definition 2.2)
Figure 10.  Dynamics with Approach A on $\mathbb{S}^2$, using the interactions matrix (36). If the agents' initial positions are close enough to each other, the agents with will form trajectories that remain in a neighborhood of their initial position
Figure 11.  Trajectories of three agents interacting according to the matrix $A$ given in (36). Left: Dynamics in $\mathbb{R}^2$, with periodic trajectories on a unique orbit. Center: Dynamics on $M = \mathbb{T}^2$ with small initial mutual distances. Right: Dynamics on $M = \mathbb{T}^2$ with large initial distances
Figure 12.  Evolutions of the coordinates of the three agents evolving on $\mathbb{T}^2$ with interaction matrix $A$ from equation (36), with small initial mutual distances. Left: Evolution of $\phi$. Center: Evolution of $\theta$. Right: Evolution of the kinetic energy
Figure 13.  Left: Evolution of 12 agents with the conditions of Theorem 6.2, with $k=3$, resulting in diverging trajectories. Dark to light color scale indicates earlier to later time. Right: corresponding exploding kinetic energy. The interaction matrix $A$ and the initial positions were generated according to a random algorithm, with the conditions of Theorem 6.2
Figure 14.  Left: Evolution of 12 agents with the conditions of Theorem 6.2, with $k=3$, resulting in convergence to consensus. Dark to light color scale indicates earlier to later time. Right: corresponding kinetic energy converging to zero. The interaction matrix $A$ and the initial positions were generated according to a random algorithm, with the conditions of Theorem 6.2
Figure 15.  Evolution of 10 agents with initial conditions and interaction matrix given in (44). The agents have periodic trajectories along one shared circular orbit
Figure 16.  Left: Directed graph corresponding to the matrix $A$ given in (44). Full arrows represent positive coefficients ($a_{ij}>0$) while dashed ones represent negative coefficients ($a_{ij}<0$). Right: Weighted directed graph corresponding to the matrix $A$ given in (46). Thin arrows represent the weighted edges $|a_{ij}|=a$ while bold ones represent the weight $|a_{ij}|=b$. Nodes with the same color and symbol share orbits but are not directly connected in the graph
Figure 17.  Left: Periodic trajectories of 8 agents sharing orbits two by two, in the situation of Theorem 6.5. Matrix $A$ from (46) was constructed with $(a,b)=(1,3)$. The initial positions $x_1(0)$ and $x_2(0)$ were randomly generated and the other 6 were obtained by rotation. The period is $\tau = 2\pi/\sqrt{6}$. Right: Corresponding kinetic energy, of period $\tau/2$
Figure 18.  Left: evolution of 9 agents with periodic trajectories, each orbit shared by 3 agents. Right: periodic kinetic energy
Figure 19.  Isolated orbits of the evolution shown in Figure 18. Left: trajectories of agents 3, 6, 9. Middle: trajectories of agents 1, 4, 7. Right: trajectories of agents 2, 5, 8)
Figure 20.  Left: Trajectories of 4 agents with helical trajectories. Parameters for matrix $A$ (48) chosen to be $(a,b,c,d) = (1,2,3,4)$. Dark to light color indicates earlier to later time. Right: Corresponding kinetic energy. The period is $\tau = 2\pi((a+c)(b+d))^{-1/2} = \pi/\sqrt{6}$ (see proof of Theorem 6.7)
Figure 21.  Evolution of the first and second coordinates of 4 agents with helical trajectories
Figure 22.  Energy of the system using Approach A, 15 agents, and a general interaction matrix (left). A snapshot of the energy oscillations to match with trajectories in Figure 23 (right)
Figure 23.  An agent's trajectory simulated with Approach A, 15 agents, and a general interaction matrix. The trajectory in shown the top right is of a second agent. The agents oscillate with amplitudes that increase with time, eventually the trajectory approximates a great circle, after which the oscillations resume with smaller amplitudes
Table 1.  Possible discontinuities of the right-hand side of (1). The bottom row of the table show conditions for $\Psi$ so that the system is continuous
ApproachABA and B
Critical points$x_j \in \mathcal{N}_i$ $x_j \in \mathcal{CL}(x_i)$ $ x_j = x_i$
Discontinuities$ \displaystyle \lim_{x_j \rightarrow \mathcal{N}_i}\| \nu_{ij} \| = 1 $
$ \|\nu_{ij}\| = 0 \text{ for } x_j\in\mathcal{N}_i $
$ \displaystyle \lim_{x_j \rightarrow \mathcal{CL}(x_i)}\| \nu_{ij} \| = 1 $
$ \|\nu_{ij}\| = 0 \text{ for } x_j\in\mathcal{CL}(x_i) $
$ \displaystyle \lim_{x_j \rightarrow x_i}\| \nu_{ij} \| = 1 $
$ \|\nu_{ii}\| = 0 $
Condition on $\Psi$$\Psi(0) = 0$$\Psi(d) = 0 \text{ for all } d\geq \epsilon$$\Psi(0) = 0$
ApproachABA and B
Critical points$x_j \in \mathcal{N}_i$ $x_j \in \mathcal{CL}(x_i)$ $ x_j = x_i$
Discontinuities$ \displaystyle \lim_{x_j \rightarrow \mathcal{N}_i}\| \nu_{ij} \| = 1 $
$ \|\nu_{ij}\| = 0 \text{ for } x_j\in\mathcal{N}_i $
$ \displaystyle \lim_{x_j \rightarrow \mathcal{CL}(x_i)}\| \nu_{ij} \| = 1 $
$ \|\nu_{ij}\| = 0 \text{ for } x_j\in\mathcal{CL}(x_i) $
$ \displaystyle \lim_{x_j \rightarrow x_i}\| \nu_{ij} \| = 1 $
$ \|\nu_{ii}\| = 0 $
Condition on $\Psi$$\Psi(0) = 0$$\Psi(d) = 0 \text{ for all } d\geq \epsilon$$\Psi(0) = 0$
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