Sampling from scale-free networks and the matchmaking paradox

Consider a large finite scale-free network consisting of M1 nodes and N1 links, in which the degree distribution of links per bond is governed by a power-law P(n)similar or equal to n(-1-alpha) with exponent 0 < 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..