Positive reinforcement and 3-dimensional informetrics

We show that the composition of two information production processes (IPPs), where the items of the first IPP are the sources of the second, and where the ranks of the sources in the first IPP agree with the ranks of the sources in the second IPP, yields an IPP which is positively reinforced with respect to the first IPP. This means that the rank-frequency distribution of the composition is the composition of the rank-frequency distribution of the first IPP and an increasing function φ, which is explicitly calculable from the two IPPs' distributions. From the rank-frequency distribution of the composition, we derive its size-frequency distribution in terms of the size-frequency distribution of the first IPP and of the function φ. The paper also relates the concentration of the reinforced IPP to that of the original one. This theory solves part of the problem of the determination of a third IPP from two given ones (so-called three-dimensional informetrics). In this paper we solved the “linear” case, i.e., where the third IPP is the composition of the other two IPPs. This revised version was published online in June 2006 with corrections to the Cover Date.     

An explanation of disproportionate growth using linear 3-dimensional informetrics and its relation with the fractal dimension

Summary We study new and existing data sets which show that growth rates of sources usually are different from growth rates of items. Examples: references in publications grow with a rate that is different (usually higher) from the growth rate of the publications themselves; article growth rates are different from journal growth rates and so on. In this paper we interpret this phenomenon of “disproportionate growth' in terms of Naranan's growth model and in terms of the self-similar fractal dimension of such an information system, which follows from Naranan's growth model. The main part of the paper is devoted to explain disproportionate growth. We show that the “simple' 2-dimensional informetrics models of source-item relations are not able to explain this but we also show that linear 3-dimensional informetrics (i.e. adding a new source set) is capable to model disproportionate growth. Formulae of such different growth rates are presented using Lotkaian informetrics and new and existing data sets are presented and interpreted in terms of the used linear 3-dimensional model.     

On the 80/20 rule

In a recent paper¹ Burrell shows that libraries with lower average borrowings tend to require a larger proportion of their collections to account for 80% of the borrowings, than those with higher average borrowings. In that study, the underlying frequency distribution was a negative binomial. We are dealing with a case when the underlying distribution is of Lotka type. It is also shown that the 80/20-effect is much stronger in this case.     

The share of items of highly productive sources as a function of the size of the system

Summary The research in this paper is based on the paper of D.W. Aksnes & G. Sivertsen: The effect of highly cited papers on national citation indicators, Scientometrics 59 (2) (2004), 213-224, where one states that “the few highly cited papers account for the highest share of the citations in the smallest fields”. This, at first sight, evident property is examined in the theoretical models that exist in the literature. We first define exactly what we mean by “size of a field” (i.e. when is a field “smaller” or “larger” than another one). We show that there are two, non-equivalent possible definitions. Next we define exactly the possible property under study. This leads us again to two possible, non-equivalent formulations. Hence, in total, there are four different formulations to consider. We show, by giving counterexamples, that none of these four formulations are true in general. We also express conditions (in Lotkaian and Zipfian informetrics), under which the property of Aksnes and Sivertsen is true. All these results are not only valid in the papers-citations relationships but in any informetric source-item relationship. In this connection we present formulae describing the share of items of highly productive sources as a function of the parameters of the system (e.g. the size of the system).     

Theory and practice of the shifted Lotka function

One of the major drawbacks of the classical Lotka function is that arguments only start from the value 1. However, in many applications one may want to start from the value 0, e.g. when including zero received citations. In this article we consider the shifted Lotka function, which includes the case of zero items. Basic results for the total number of sources, the total number of items and the average number of items per source are given in this framework. Next we give the rank-frequency function (Zipf-type function) corresponding to the shifted Lotka function and prove their exact relation. The article ends with a practical example which can be fitted by a shifted Lotka function.     

The h-index formalism

Scientometrics - This article provides an overview of the development of the h-index formalism. We begin with the original formulation as provided by Hirsch and move on to the latest versions. In...     

A proposal to define a core of a scientific subject: A definition using concentration and fuzzy sets

Determining the core of a field"s literature, i.e. its "most important" sources, has been and still is an important problem in bibliometrics. In this article an exact definition of a core of a bibliography or a conglomerate is presented. The main ingredients for this definition are: fuzzy set theory, Lorenz curves and concentration measures. If one prefers a strict delineation, the fuzzy core can easily be defuzzified. The method we propose does not depend on the subjective notion of "importance". It is, moreover, completely reproducible. The method and the resulting core is also independent of the mathematical function (Lotka, Zipf, Bradford, etc.) that may be used to describe the relation between the set of sources and that of items.     

The Distribution of N-Grams

N-grams are generalized words consisting of N consecutive symbols, as they are used in a text. This paper determines the rank-frequency distribution for redundant N-grams. For entire texts this is known to be Zipf's law (i.e., an inverse power law). For N-grams, however, we show that the rank (r)-frequency distribution is