Extend the global synchronization result to the dynamical definition

Extend the global synchronization result for dense graphs from the absence of nonsynchronized local minima of the Kuramoto energy to the dynamical setting in which the negative gradient flow synchronizes from almost every initial condition, including the exactly clustered case.

Background

The paper defines global synchronization as the property that the Kuramoto energy has no spurious local minima, whereas other work uses the stronger dynamical notion that the negative gradient flow converges to full synchronization from almost every initial condition. The authors explain that their arguments already imply that, outside the exactly clustered case, every nonsynchronized second-order critical point is a strict saddle, so only a measure-zero set of initial conditions can converge to such points.

The unresolved part is the exactly clustered case. Establishing the dynamical conclusion there would extend the paper’s dense-graph theorem to the alternative gradient-flow formulation of global synchronization. The authors explicitly defer this extension to future work.