The cumulative advantage function. A mathematical formulation based on conditional expectations and its application to scientometric distributions

Abstract

Cumulative advantage principle is a specific law underlying several social, particularly , bibliometric and scientometric processes. This phenomenon was described by single- and multiple-urn models (Price (1976). Tague (1981)). A theoretical model for cumulative advantage growth was developed by Schubert and Glaenzel (1984). This paper presents an exact measure of the cumulative advantage effect based on conditional expectations. For a given bibliometric random variable X (e.g. publication activity , citation rate) the cumulative advantage function i s defined as d k ) = E(iK-k)[(X-k) b O)/E(X). The 'extent of advantage' is studied on the basis of limit properties of this function. The behavior of ~ ( k ) is discussed for the urn-model distributions, particularly for its most prominent representants, the negative-binomial and the Waring distribution. The discussion is illustrated by several examples from bibliometric distributions.