How to quantify social network similarity/distance?

Question

Context

In the context of animal social networks analysis, I wish to compare two empirical sampling methods, M1 and M2, through simulations.

Simulation workflow is as follow:

1. I generate a theoretical weighted, potentially directed adjacency matrix (A0)
2. I simulate the process of sampling via each methods and end up with an adjacency matrix for each (A1 and A2)
3. I want to quantify how
   • A1 is (dis)similar to A0
   • A2 is (dis)similar to A0
   • and thus assess how acceptable and similar/different are both methods

Steps 1-3 are repeated a large number of time, so I will generate a whole distribution of the desired (dis)similarity measure(s) to base my analysis on.

Adjacency matrices are always of the same dimensions, of the same vertices.

Because of the field of application, networks are usually not as large as the one from social media or the world wide web for instance (they range from ~5 to few hundreds nodes maximum, but probably on average ~20-50). Basic pairwise association/interaction are quantitatively important of course, but so are some network aspects like node centrality, graph connectedness, modularity, and other such "higher order" emerging properties.

What I considered so far

In the corresponding animal social network literature, it seems common to extract a few network statistics, such as node degree, node centrality rank given a network metric (e.g. eigenvector centrality), graph modularity, etc., and run summary statistics on them (e.g. node centrality rank correlation between two graphs of the same nodes). But while it captures the impact of differences between networks on those specific aspects, I feel it lacks the "general picture" (like only looking at a 2D projection of a 3D object).

I also thought about comparing adjacency matrices directly, but failed to be confident with which approach I need. Here's a few leads I had:

• "flattening" adjacency matrices and doing a pairwise comparison of vector from A0, A1 and A2 (possibly only considering upper/lower triangles in the case of undirected networks)
• I came across the Mantel test, but its ground of comparing node "distances" feels different in assumptions and makes me wonder about its full relevance in social networks in my case
• I looked at matrix-/network- distances, considering networks as whole objects, i.e. calculating the Frobenius norm of A0-A1 and A0-A2, but I don't know if this approach is much different from the matrix "flattening" I mentioned (it being based on pariwise square differences). Or if I should calculate distances on higher order/spectral aspects of the adjacency matrix (involving graph's Laplacian? Eigenvectors/values?). Or a combination of both.

My background is mostly in biology, so my knowledge of linear algebra, set and graph theory details involved is often lacking when looking through the more specialized literature. Hence my question to hear well-informed opinion on this.

Question

How can I quantify how (dis)similar are A1 and A2 relatively to A0, in a "network-holistic" way that accounts for both basic pairwise weights differences and other higher order/structural aspects of the networks?