I was reading this paper 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.
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$\begingroup$ The more general name for this is "size-biased distribution". $\endgroup$– James MartinNov 21, 2015 at 11:47
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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 $N\sum_{d'} d' f(d')$).
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3$\begingroup$ In other words, a random endpoint of a random edge is not a uniformly random vertex: there is a bias towards the vertices of higher degree. The similar fact that a random neighbour of a random vertex is not a uniformly random vertex leads to the observation that your friends will tend to have more friends than you on average. $\endgroup$ Nov 20, 2015 at 15:06