I believe you're referring to something related to multi-layer networks. Here you can find an interesting reading about this kind of networks:
Also you can try with this one:
As a matter of fact, I'm not an expert of multi-layer networks but, as far as I can tell, the main problem (as you mentioned) is to find correlations, or connections, between the different layers in order to be able to draw some meaningful conclusion.
I am not sure you can call this a multi-layer network but this can be an example of what you're looking for: suppose you have a protein-protein interaction network. You can enrich your network using biological information. You can associate biological functions, or cellular locations or whatever you can find in GeneOntology (just to make an example of a biological database), as a further layer of information regarding your protein-protein interaction network.
This will help you in elucidating the role a node (or more nodes) is playing with respect to its neighbours: let's say five nodes form a clique and you also know they share the same cellular location...then you're able to say something about these nodes (maybe they form a complex?) that has a dual nature: biological and graph-based.
Hope it helps!