Power laws, the Price model, and the Pareto type-2 distribution

Abstract

We consider a version of D. Price’s model for the growth of a bibliographic network, where in each iteration, a constant number of citations is randomly allocated according to a weighted combination of the accidental (uniformly distributed) and the preferential (rich-get-richer) rule. Instead of relying on the typical master equation approach, we formulate and solve this problem in terms of the rank–size distribution. We show that, asymptotically, such a process leads to a Pareto-type 2 distribution with a new, appealingly interpretable parametrisation. We prove that the solution to the Price model expressed in terms of the rank–size distribution coincides with the expected values of order statistics in an independent Paretian sample. An empirical analysis of a large repository of academic papers yields a good fit not only in the tail of the distribution (as it is usually the case in the power law-like framework), but also across a significantly larger fraction of the data domain.