Weighted degree in Multidimensional networks
I think your weight function should depend on $d$ as well. 

The contribution from a single dimension is just the sum of incoming weights. Even for simple graphs this does not seem to have a standard name, although it is quite commonly encountered.
In the context of random walks it is sometimes called simply degree, in the context of empirical networks, I found node strength. 

I think this is a simple and natural statistic to look at if you wish to model something as a multidimensional network. However, being a linear sum of rather trivial functionals over simple graphs, I doubt it is of much fundamental interest.