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This reference could be what you are looking for:

Digraph Laplacian and the Degree of Asymmetry:

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.

For the purpose of of normalisation (to arrive at a "percentage" of asymmetry) one might want to compare $\delta$ with the largest singular value $\sigma$ of the full matrix $\Gamma$, but the ratio $\delta/\sigma$ is not guaranteed to be $\leq 1$. (The cited reference gives as an upper bound for $\delta$ the second largest singular value of the transmission probability matrix.)

This reference could be what you are looking for:

Digraph Laplacian and the Degree of Asymmetry:

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.

This reference could be what you are looking for:

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.

This reference could be what you are looking for:

Digraph Laplacian and the Degree of Asymmetry:

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.

For the purpose of of normalisation (to arrive at a "percentage" of asymmetry) one might want to compare $\delta$ with the largest singular value $\sigma$ of the full matrix $\Gamma$, but the ratio $\delta/\sigma$ is not guaranteed to be $\leq 1$. (The cited reference gives as an upper bound for $\delta$ the second largest singular value of the transmission probability matrix.)

This reference could be what you are looking for:

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.

This reference could be what you are looking for:

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.