The power of power laws and an interpretation of Lotkaian informetric systems as self-similar fractals

Abstract

Power laws as defined in 1926 by A. Lotka are increasing in importance because they have been found valid in varied social networks including the Internet. In this article some unique properties of power laws are proven. They are shown to characterize functions with the scale-free property (also called self-similarity property) as well as functions with the product property. Power laws have other desirable properties that are not shared by exponential laws, as we indicate in this paper. Specifically, Naranan (1970) proves the validity of Lotka's law based on the exponential growth of articles in journals and of the number of journals. His argument is reproduced here and a discrete-time argument is also given, yielding the same law as that of Lotka. This argument makes it possible to interpret the information production process as a self-similar fractal and show the relation between Lotka's exponent and the (self-similar) fractal dimension of the system. Lotkaian informetric systems are self-similar fractals, a fact revealed by Mandelbrot (1977) in relation to nature, but is also true for random texts, which exemplify a very special type of informetric system.