Sampling from naturally truncated power laws: The matchmaking paradox

Abstract

Consider a network of M >> 1 nodes connected by N >> 1 links, in which the distribution of the number of links per node follows a power law with exponent 0<\alpha <1. The power law is naturally truncated due to the fact that N is finite. A subset of m << M nodes is sampled arbitrarily, yielding the 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 population mean \nu =N/M are extremely broad and strongly skewed -- yielding typical values which are systematically and significantly smaller than the population mean \nu. Applying these results to the case of bipartite networks, we show that the sample means of the two parts of these networks generally differ -- the fact we call "matchmaking paradox" in the title.