  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.
