Imagine a huge graph that is planar apart from a few lines. (like mini wormholes).
Is there a formula to get the approximate dimension of the graph from the adjacency matrix?
My thoughts were to get the number of paths from a point in the center to a certain radius and calculate the number of walks back to a distance of n steps away or back to the center.
i.e. for a 1D graph this is a lot different to a 2D grid-like graph.
$$ \sum_i \langle 0 | M^n |i\rangle \approx f(n,D) $$
Or for example the local approximate dimension if the graph is like a huge spherical mesh. The approximate local dimension is 2. Can you somehow calculate this from the adjacency matrix?
The reason I'm thinking about this is given a random adjacency matrix, how close is it to D-dimensional flat space.