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 df(d)∑d′f(d′). 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.
1 Answer
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∑ddf(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∑d′d′f(d′)).
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3In 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. Commented Nov 20, 2015 at 15:06