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Does there exist a definition for weighted degrees of multidimensional networks?

I understand that the basic definition would be:

Let $v\in V$ be a node of a network $G=(N,V,L)$. The function Weighted Degree: $$V \times P(L) \longrightarrow \mathbb{N}$$ is defined as

$${weighted\_degree}(v,D)=\sum_{(u,v,d)\in G, d\in D}\ weight(u,v)$$

where $D\subseteq L$ and $weight(u,v)$ is the weight of the edge $(u,v)$.

I cannot find this definition anywhere on the Internet. But would that be right? Is a multidimensional version of any interest?

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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.

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