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Digraph Laplacian and the Degree of Asymmetry:
We introduce a metric – the largest singular value $\lambda$ of $(\Gamma − \Gamma^T )/2$, where $\Gamma$ is the Laplacian of a directed graph – to quantify and measure the degree of asymmetry in the graph. The degree of asymmetry captures the overall "directedness" of the graph.
One could normalise $\lambda$ by the largest singular value of $\Gamma$ itself, for a relative degree of asymmetry.