On Computing the Critical Coupling Coefficient for the Kuramoto Model on a Complete Bipartite Graph

We extend recent results on the existence of global phase-locked states (GPLS) in the Kuramoto model on a complete graph to the case of a complete bipartite graph. In particular, we prove that, for the Kuramoto model on a complete bipartite graph, the value of the critical coupling coefficient, i.e., the smallest coupling coefficient for which a GPLS can exist, can be determined by solving a system of two nonlinear equations that do not depend on the coupling coefficient itself. We show that said system of equations can be solved using an efficient algorithm described in the paper.