# American Institute of Mathematical Sciences

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 & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021
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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
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
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
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
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
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
A dancing equilibrium for Approach A. The energy becomes constant in time after initial fluctuations
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
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)
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
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
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
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
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
Evolution of 10 agents with initial conditions and interaction matrix given in (44). The agents have periodic trajectories along one shared circular orbit
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
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 $\tau/2$
Left: evolution of 9 agents with periodic trajectories, each orbit shared by 3 agents. Right: periodic kinetic energy
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)
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)
Evolution of the first and second coordinates of 4 agents with helical trajectories
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)
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
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
 Approach A B A 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 $\Psi$ $\Psi(0) = 0$ $\Psi(d) = 0 \text{ for all } d\geq \epsilon$ $\Psi(0) = 0$
 Approach A B A 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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