 Kuramoto order parameter

The complex centroid of the oscillators' unit phase vectors has magnitude $r$ and argument $\psi$. A magnitude near one indicates phase coherence; zero indicates a balanced phase distribution, which need not be uniquely uniform at finite population size. Taking the imaginary part after multiplication by $e^{-i{\theta }_i}$ rewrites all-to-all sine coupling exactly in terms of these two collective variables.