Lotkaian Informetrics Research Articles

Relations among the h-, g-, ψ-, and p-index and offset-ability

• By their definitions, the h -, g -, and ψ- index are related through h ≤ g ≤ ψ. • If the Lotkaian exponent α ∈ ] 2 , 3 [ , then h ≤ p ≤ g ≤ ψ , where p is the mock h -index. • These relations are investigated in the fields of electrochemistry and gerontology. • These relations are discussed within a theoretical framework of offset-ability. We show that the h -index, g -index, ψ- index, and p -index, are related through the inequalities: h ≤ p ≤ g ≤ ψ . Moreover, this relation is proved theoretically in the mathematical framework of Lotkaian informetrics and is verified empirically by using two datasets from the Web of Science in the fields of electrochemistry and gerontology. For quantifying their relations, we estimate the g -index, ψ- index, and their cores and ratios of cores via a second-order Taylor series when the e -index, h -index, and C 1 (the maximum number of citations received by a paper) are known. Then we find for the two empirical cases, that ratios of cores and average citations are approximately stable. Compared with the g -index, the offset-ability of the h -index decreases by 20% but the average citations increase by 20%. A similar observation holds for the comparison of the g -index and ψ- index. To explore the possible applications of cores of different indices, we apply them to extract the core structure of a network. The h -core is the most efficient, while the ψ -core includes more nodes with high betweenness.

Solution by step functions of a minimum problem in L2[0,T], using generalized h- and g-indices

• Generalized h- and g-indices are studied. • It is shown that the h-index is a special case of the g-index. • These indices are applied to solve a minimum problem in L 2 . • Using step functions, this minimum problem is solved for decreasing functions. • Results are applied to the case of Lotkaian informetrics (Zipf, Lotka). In this article we solve a minimum problem involving step functions. The solution of this problem leads to an investigation into generalized h- and g-indices. This minimum problem and the related generalized h- and g-indices are studied in a general context of decreasing differentiable functions as well as in the specific case of Lotkaian informetrics. The study illustrates the use of h-and g-indices and their generalizations in a context which bears no relation to the research evaluation context in which these indices were originally introduced.

The source-effort coverage of an exponential informetric process

Lotkaian informetrics is the framework most often used to study statistical distributions in the production and usage of information. Although Lotkaian distributions are traditionally used to characterize the Information Production Process (IPP) , we have shown in a previous article that the IPP can successfully be studied using the effort function – the latter having been initially introduced to define the Exponential Informetric Process (EIP). These themes continue to be developed in this article , in which we present a necessary and sufficient condition for the existence of the EIP. Our current approach is similar to the one used to study IPPs. Inverse power and exponential distributions serve to illustrate the results obtained in the context of an EIP. Numerical examples are discussed .

The zynergy‐index and the formula for the h‐index

The h‐index, as originally proposed (Hirsch, 2005), is a purely heuristic construction. Burrell (2013) showed that efforts to derive formulae from the mathematical framework of Lotkaian informetrics could lead to misleading results. On this note, we argue that a simple heuristic “thermodynamical” model can enable a better three‐dimensional (3D) evaluation of the information production process leading to what we call the zynergy‐index.

Formulae for the h‐index: A lack of robustness in Lotkaian informetrics?

In one of the first attempts at providing a mathematical framework for the Hirsch index, Egghe and Rousseau (2006) assumed the standard Lotka model for an author's citation distribution to derive a delightfully simple closed formula for his/her h‐index. More recently, the same authors (Egghe & Rousseau, 2012b) have presented a new (implicit) formula based on the so‐called shifted Lotka function to allow for the objection that the original model makes no allowance for papers receiving zero citations. Here it is shown, through a small empirical study, that the formulae actually give very similar results whether or not the uncited papers are included. However, and more important, it is found that they both seriously underestimate the true h‐index, and we suggest that the reason for this is that this is a context—the citation distribution of an author—in which straightforward Lotkaian informetrics is inappropriate. Indeed, the analysis suggests that even if we restrict attention to the upper tail of the citation distribution, a simple Lotka/Pareto‐like model can give misleading results.

Lotkaian informetrics and applications to social networks

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.

Modelling successive h-indices

From a list of papers of an author, ranked in decreasing order of the number of citations to these papers one can calculate this author’s Hirsch index (or h-index). If this is done for a group of authors (e.g. from the same institute) then we can again list these authors in decreasing order of their h-indices and from this, one can calculate the h-index of (part of) this institute. One can go even further by listing institutes in a country in decreasing order of their h-indices and calculate again the h-index as described above. Such h-indices are called by Schubert [2007] “successive” h-indices. In this paper we present a model for such successive h-indices based on our existing theory on the distribution of the h-index in Lotkaian informetrics. We show that, each step, involves the multiplication of the exponent of the previous h-index by 1/α where α > 1 is a Lotka exponent. We explain why, in general, successive h-indices are decreasing. We also introduce a global h-index for which tables of individuals (authors, institutes,...) are merged. We calculate successive and global h-indices for the (still active) D. De Solla Price awardees.

Time-dependent Lotkaian informetrics incorporating growth of sources and items

In a previous article, static Lotkaian theory was extended by introducing a growth function for the items. In this article, a second general growth function–this time for the sources–is introduced. Hence this theory now comprises real growth situations, where items and sources grow, starting from zero, and at possibly different paces. The time-dependent size- and rank-frequency functions are determined and, based on this, we calculate the general, time-dependent, expressions for the h - and g -index. As in the previous article we can prove that both indices increase concavely with a horizontal asymptote, but the proof is more complicated: we need the result that the generalized geometric average of concavely increasing functions is concavely increasing.

Study of different h‐indices for groups of authors

AbstractIn this article, for any group of authors, we define three different h‐indices. First, there is the successive h‐index h2 based on the ranked list of authors and their h‐indices h1 as defined by Schubert (2007). Next, there is the h‐index hP based on the ranked list of authors and their number of publications. Finally, there is the h‐index hC based on the ranked list of authors and their number of citations. We present formulae for these three indices in Lotkaian informetrics from which it also follows that h2 < hp < hc. We give a concrete example of a group of 167 authors on the topic “optical flow estimation.” Besides these three h‐indices, we also calculate the two‐by‐two Spearman rank correlation coefficient and prove that these rankings are significantly related.

In this issue

In this issue

Untangling Herdan's law and Heaps' law: Mathematical and informetric arguments

AbstractHerdan's law in linguistics and Heaps' law in information retrieval are different formulations of the same phenomenon. Stated briefly and in linguistic terms they state that vocabularies' sizes are concave increasing power laws of texts' sizes. This study investigates these laws from a purely mathematical and informetric point of view. A general informetric argument shows that the problem of proving these laws is, in fact, ill‐posed. Using the more general terminology of sources and items, the author shows by presenting exact formulas from Lotkaian informetrics that the total number T of sources is not only a function of the total number A of items, but is also a function of several parameters (e.g., the parameters occurring in Lotka's law). Consequently, it is shown that a fixed T (or A) value can lead to different possible A (respectively, T) values. Limiting the T(A)‐variability to increasing samples (e.g., in a text as done in linguistics) the author then shows, in a purely mathematical way, that for large sample sizes T≈LAθ, where θ is a constant, θ6<1 but close to 1, hence roughly, Heaps' or Herdan's law can be proved without using any linguistic or informetric argument. The author also shows that for smaller samples, u is not a constant but essentially decreases as confirmed by practical examples. Finally, an exact informetric argument on random sampling in the items shows that, in most cases, T = T(A) is a concavely increasing function, in accordance with practical examples.

The source-item coverage of the exponential function

Statistical distributions in the production of information are most often studied in the framework of Lotkaian informetrics. In this paper, we recall some results of basic theory of Lotkaian informetrics, then we transpose methods (Theorem 1) applied to Lotkaian distributions by Leo Egghe (Theorem 2) to the exponential distributions (Theorem 3, Theorem 4). We give examples and compare the results (Theorem 5). Finally, we propose to widen the problem using the concept of exponential informetric process (Theorem 6).

Item-time-dependent Lotkaian informetrics and applications to the calculation of the time-dependent [formula omitted]-index and [formula omitted]-index

The model for the cumulative n th citation distribution, as developed in [L. Egghe, I.K. Ravichandra Rao, Theory of first-citation distributions and applications, Mathematical and Computer Modelling 34 (2001) 81–90] is extended to the general source–item situation. This yields a time-dependent Lotka function based on a given (static) Lotka function (considered to be valid for time t = ∞ ). Based on this function, a time-dependent Lotkaian informetrics theory is then further developed by e.g. deriving the corresponding time-dependent rank–frequency function. These tools are then used to calculate the dynamical (i.e. time-dependent) g -index (of Egghe) while also an earlier proved result on the time-dependent h -index (of Hirsch) is refound. It is proved that both indexes are concavely increasing to their steady state values for t = ∞ .

An interpretation of the effort function through the mathematical formalism of Exponential Informetric Process

Statistical distributions in the production or utilization of information are most often studied in the framework of Lotkaian informetrics. In this article, we show that an Information Production Process (IPP), traditionally characterized by Lotkaian distributions, can be fruitfully studied using the effort function, a concept introduced in an earlier article to define an Exponential Informetric Process. We thus propose replacing the concept of Lotkaian distribution by the logarithmic effort function. In particular, we show that an effort function defines an Exponential Informetric Process if its asymptotic behavior is equivalent to the logarithmic function β · Log( x) with β > 1, which is the effort function of a Lotkaian distribution.

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

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.

Relations between the continuous and the discrete Lotka power function

AbstractThe discrete Lotka power function describes the number of sources (e.g., authors) with n = 1, 2, 3, … items (e.g., publications). As in econometrics, informetrics theory requires functions of a continuous variable j, replacing the discrete variable n. Now j represents item densities instead of number of items. The continuous Lotka power function describes the density of sources with item density j. The discrete Lotka function one obtains from data, obtained empirically; the continuous Lotka function is the one needed when one wants to apply Lotkaian informetrics, i.e., to determine properties that can be derived from the (continuous) model. It is, hence, important to know the relations between the two models. We show that the exponents of the discrete Lotka function (if not too high, i.e., within limits encountered in practice) and of the continuous Lotka function are approximately the same. This is important to know in applying theoretical results (from the continuous model), derived from practical data.