From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we generalise to networks over n-dimensional spheres. We show that, for almost all initial conditions, the sphere model converges to a set with small diameter if the model parameters satisfy a given bound. Moreover, for even n, a special case of the generalized model can achieve phase synchronization with nonidentical frequency parameters. These results contrast with the standard n = 1 Kuramoto model, which is multistable (i.e., has multiple equilibria), and converges to phase synchronization only if the frequency parameters are identical. Hence, this paper shows that the generalized network Kuramoto models for n ≥ 2 displays more coherent and predictable behavior than the standard n = 1 model, a desirable property both in flocks of animals and for robot control.

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Multi-agent systems are known to exhibit stable emergent behaviors, including polarization, over R n or highly symmetric nonlinear spaces. In this article, we eschew linearity and symmetry of the underlying spaces, and study the stability of polarized equilibria of multi-agent gradient flows evolving on general hypermanifolds. The agents attract or repel each other according to the partition of the communication graph that is connected but otherwise arbitrary. The manifolds are outfitted with geometric features styled "dimples" and "pimples" that characterize the absence of flatness. The signs of inter-agent couplings together with these geometric features give rise to stable polarization under various sufficient conditions. We propose tangible interpretation of the system in the context of opinion dynamics, and highlight throughout the text its versatility in modeling diverse aspects of the polarization phenomenon.

No abstract

Multi-agent systems are known to exhibit stable emergent behaviors, including polarization, over R n or highly symmetric nonlinear spaces. In this article, we eschew linearity and symmetry of the underlying spaces, and study the stability of polarized equilibria of multi-agent gradient flows evolving on general hypermanifolds. The agents attract or repel each other according to the partition of the communication graph that is connected but otherwise arbitrary. The manifolds are outfitted with geometric features styled "dimples" and "pimples" that characterize the absence of flatness. The signs of inter-agent couplings together with these geometric features give rise to stable polarization under various sufficient conditions. We propose tangible interpretation of the system in the context of opinion dynamics, and highlight throughout the text its versatility in modeling various aspects of the polarization phenomenon.