Opinion Dynamics on a General Compact Riemannian Manifold

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