The Energy Landscape of the Kuramoto Model in Random Geometric Graphs in a Circle

Abstract

We study the energy function of the Kuramoto model in random geometric graphs defined in the unit circle as the number of nodes diverges. We prove the existence of at least one local minimum for each winding number with high probability, hence providing a large family of graphs that support patterns that are generic. These states are in correspondence with the explicit twisted states found in WSG and other highly symmetric networks, but in our situation there is no explicit formula due to the lack of symmetry. The method of proof is simple and robust. It allows other types of graphs like nn graphs or the boolean model and holds also for graphs defined in any simple closed curve or even a small neighborhood of the curve and for weighted graphs. It seems plausible that the method can be extended also to higher dimensions, but a more careful analysis is required.