Steering Opinions over n-Dimensional Spherical Manifolds

This paper considers opinion dynamics on an n-sphere, where leaders steer their followers’ opinions to a desirable region on the sphere. In the process of doing so, the leaders may encounter other leaders who influence the opinion of these followers and entice them away. To prevent this, the leader will need to exert additional effort to keep the followers together and away from the influence of the adversarial leaders. This problem is formulated as an optimal control problem of choosing a piecewise geodesic path to the opinion goal region while optimizing the cost incurred due to the additional effort required to negate the influence of the adversarial leaders. The paper builds upon several well-known concepts, such as the representation of opinions as vectors on an n-sphere, the bounded confidence model that defines the interaction of proximal opinions, and the methods by which leaders try to influence the opinion of their followers and strengthen their hold on them. However, instead of looking at consensus and related concepts, the paper produces novel results on how leaders can successfully steer the opinion of their followers over the spherical manifold. Simulation results are given to illustrate the concepts presented in the paper.