= Weighted collective oscillator frequency
{title2=$\Omega=\pi^{\mathsf T}\xi,\quad\pi^{\mathsf T}L=0$}
Linearizing weakly coupled locked phases gives $0=\xi-\Omega\mathbf1-kL\theta$, 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, $\pi=\mathbf1/N$ 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 $(q,p)/(p+q)$.
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