Finding Non-Zero Stable Fixed Points of the Weighted Kuramoto model is NP-hard

The Kuramoto model when considered over the full space of phase angles [$0,2\pi$) can have multiple stable fixed points which form basins of attraction in the solution space. In this paper we illustrate the fundamentally complex relationship between the network topology and the solution space by showing that determining the possibility of multiple stable fixed points from the network topology is NP-hard for the weighted Kuramoto Model. In the case of the unweighted model this problem is shown to be at least as difficult as a number partition problem, which we conjecture to be NP-hard. We conclude that it is unlikely that stable fixed points of the Kuramoto model can be characterized in terms of easily computable network invariants.