This reference could be what you are looking for:

<A HREF="https://www-users.cs.umn.edu/~zhang089/Papers/Li-Yanhua-Digraph-Laplacian-special%20issue%20of%20Internet%20Mathematics.pdf">Digraph Laplacian and the Degree of Asymmetry</A>:

> We introduce a metric – the largest singular value $\delta$ 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.