Power Laws, the Price Model, and the Pareto type-2 Distribution

TL;DR

This paper analyzes a modified Price model for bibliographic network growth, showing it leads to a Pareto-2 distribution, and demonstrates its empirical fit to academic citation data across the entire distribution spectrum.

Contribution

It introduces a rank-size distribution approach to the Price model, providing new insights into its asymptotic behavior and empirical applicability.

Findings

The model asymptotically yields a Pareto-2 distribution.

Empirical data fits well across the entire distribution, not just the tail.

Results suggest higher preferential attachment and less randomness than previous studies.

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 accidental (uniformly distributed) and preferential (rich-get-richer) rules. 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 an 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. We study the bias and the mean squared error of three well-behaving estimators of the underlying model parameters. 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 the whole domain. Interestingly, the estimated models indicate higher degree of preferentially attached citations and smaller share of randomness than previous studies.