Weighted collective oscillator frequency ($\Omega ={\pi }^{\mathsf{T}}\xi ,{\pi }^{\mathsf{T}}L=0$)

Linearizing weakly coupled locked phases gives $0=\xi -\Omega 1-kL\theta$, where $L$ is the directed graph Laplacian. Multiplying by a normalized left null vector ${\pi }^{\mathsf{T}}$ eliminates coupling and gives the displayed weighted frequency. In a balanced network, including symmetric reciprocal coupling, $\pi =1/N$ and the frequency is the arithmetic mean. A zero diagonal in the coupling matrix alone does not imply balance. For two directed weights $p,q>0$, the weights are $(q,p)/(p+q)$.