Sampling from scale-free networks and the matchmaking paradox.

Consider a large finite scale-free network consisting of M>>1 nodes and N>>1 links, in which the degree distribution of links per bond is governed by a power-law P(n) approximately n(-1-alpha) with exponent 0<alpha<1. A subset of m<<M nodes is sampled arbitrarily, yielding the empirical sample mean eta : the average number of links per node, within the sampled subset. We explore the statistics of the sample mean eta and show that its fluctuations around the network mean nu=N/M are extremely broad and strongly skewed--yielding typical values, which are systematically and significantly smaller than the network mean nu. Applying these results to the case of bipartite scale-free networks, we show that the sample means of the two parts of these networks generally differ--a fact we refer to as the "matchmaking paradox."