The analogue of Von Neumann entropy for graphs appears to have been constructed here: https://arxiv.org/abs/0812.2597

If you can compute the Laplacian for the graph in the way they define there, you get a density matrix. Therefore, finding the entropy will just involve computing the trace of this matrix, which you can do by performing a spectral decomposition of this matrix and finding its eigenvalues. Evaluating the entropy is then just the negative sum of each eigenvalue times its logarithm:

S = -\mathrm{Tr}(\rho \log \rho) = -\sum_i \lambda_i \log \lambda_i

The localized structures would have reduced density matrices in this way as well, so if you find the Von Neumann entropy for each you could have a measure of mutual information:

I(A,B) = S(A) + S(B) - S(A B)