This answers my question. Thanks!

I would contest the statement that the ego-nets tend to be disconnected in general. In a graph with high local clustering coefficient, I would expect then to be connected more frequently than not.

This is very helpful! I take it the other way around though: knowing only $v$ and $u$ can say something about $v$ and $W$. Now I'm wondering whether $v$ being close to $W$ affects the convergence rate of power methods for computing eigenvalues...

Woops I was thinking of permutation matrices.

@RobertIsrael - I just saw your latest edit - perhaps this ratio being equal to $1$ is an indication of normality rather than symmetricity.

I didn't say it was a surprise, I said it was interesting in light of my conjecture (see original post) that this ratio measures how far $A$ is from being symmetric.

Interestingly, the ratio equals $1$ exactly when the matrix is symmetric.

@KevinCasto, yes, studying that alternative would also be of interest!

@AnthonyQuas the ratio is still uniquely defined as stated (with $v$ having unit length). In any case, Kevin Casto proposes an alternative.