Lotkaian informetrics and applications to social networks

Abstract

Two-dimensional informetrics is defined in the general context of sources that produce items and examples are given. These systems are called ``Information Production Processes'' (IPPs). They can be described by a size-frequency function $f$ or, equivalently, by a rank-frequency function $g$. If $f$ is a decreasing power law then we say that this function is the law of Lotka and it is equivalent with the power law $g$ which is called the law of Zipf. Examples in WWW are given. Next we discuss the scale-free property of $f$ also allowing for the interpretation of a Lotkaian IPP (i.e. for which $f$ is the law of Lotka) as a self-similar fractal. Then we discuss dynamical aspects of (Lotkaian) IPPs by introducing an item-transformation $\varphi$ and a source-transformation $\psi$. If these transformations are power functions we prove that the transformed IPP is Lotkaian and we present a formula for the exponent of the Lotka law. Applications are given on the evolution of WWW and on IPPs without low productive sources (e.g. sizes of countries, municipalities or databases). Lotka's law is then used to model the cumulative first citation distribution and examples of good fit are given. Finally, Lotka's law is applied to the study of performance indices such as the $h$-index (Hirsch) or the $g$-index (Egghe). Formulas are given for the $h$- and $g$-index in Lotkaian IPPs and applications are given.