The distance distribution of graphs

Q: Why is the degree distribution of vertices more widely studied than the distance distribution of links?

A: Vertices are more fundamental than links, in the following sense:
A vertex is associated with a scale, its degree (= number of nodes connected to it), which is determined only by the topology of the network. A link, on the other hand, has no intrinsic scale. One would need to attribute a weight to each link, such as its length, to define a scale, but there is no unique way to do that for a given network topology.

The lack of an intrinsic scale diminishes the fundamental interest of the link distance distribution. For that reason, Zhou, Meng, and Stanley [1] have recently studied an alternative distance distribution, the degree distance distribution, which does have an intrinsic scale. The degree distance $d$ of a link connecting two vertices of degree $k$ and $k'$ is defined as $d=\log|k-k'|$.

The authors argue that a power law degree distance distribution better represents the scale-free property of a network than a power law degree distribution.

[1] Bin Zhou, Xiangyi Meng, and H. Eugene Stanley, Power-law distribution of degree–degree distance: A better representation of the scale-free property of complex networks (2020)