I was reading [this paper](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2425490) when I came across something called the edge-perspective degree distribution in a network. Consider a graph $G$, the degree distribution of whose nodes is $f(d)$. They say the edge-perspective degree distribution is $\frac{d f(d)}{\sum{d^\prime} f(d^\prime) }$. What is the significance of this distribution? The paper does not seem to explain anywhere. I can see the denominator corresponds to mean degree, but the numerator is obscure to me. 



Edge-perspective degree distribution
Its meaning should be something like "degree distribution seen from a random edge". Indeed, let us first think of an unoriented edge as of two oriented (in opposite directions) ones. Assume that there are $N$ nodes, where $N$ is large. Then, the total number of (oriented) edges should be roughly $N\sum\_d d f(d)$. So, if you choose one edge at random and then ask "what's the probability that the degree of a node where this edge begins is $d$?", it is clear that this probability should be proportional to $df(d)$ (since there are $Ndf(d)$ such edges out of $N\sum\_{d'} d' f(d')$).


