The stochastic h-index

A variant of the h-index, named the stochastic h-index, is proposed. This new index is obtained by adding to the h-index the probability, under a specific stochastic model, that the h-index will increase by one or more within a given time interval. The stochastic h-index thus extends the h-index to the real line and has a direct interpretation as the distance to the next higher index value. We show how the stochastic h-index can be evaluated and compare it with other variants of the h-index which purportedly indicate the distance to a higher h-index.     

Definitions of time series in citation analysis with special attention to the h-index

The structure of different types of time series in citation analysis is revealed, using an adapted form of the Frandsen–Rousseau notation. Special cases where this approach can be used include time series of impact factors and time series of h-indices, or h-type indices. This leads to a tool describing dynamic aspects of citation analysis. Time series of h-indices are calculated in some specific models.     

An axiomatic analysis of Egghe’s g-index

The g-index is a well-known index for measuring and comparing the output of scientific researchers, which has been introduced by Leo Egghe in 2006 as an improvement of the Hirsch-index. This article gives an axiomatic characterization of the g-index in terms of three natural axioms.     

A modification of the h-index: The hm-index accounts for multi-authored manuscripts

In order to take multiple co-authorship appropriately into account, a straightforward modification of the Hirsch index was recently proposed. Fractionalised counting of the papers yields an appropriate measure which is called the h_m-index. The effect of this procedure is compared in the present work with other variants of the h-index and found to be superior to the fractionalised counting of citations and to the normalization of the h-index with the average number of authors in the h-core. Three fictitious examples for model cases and one empirical case are analysed.     

Ranking forestry journals using the h-index

An expert ranking of forestry journals was compared with Journal Impact Factors and h-indices computed from the ISI Web of Science and internet-based data. Citations reported by Google Scholar offer an efficient way to rank all journals objectively, in a manner consistent with other indicators. This h-index exhibited a high correlation with the Journal Impact Factor (r = 0.92), but is not confined to journals selected by any particular commercial provider. A ranking of 180 forestry journals is presented, on the basis of this index.     

A multilevel meta-analysis of studies reporting correlations between the h index and 37 different h index variants

This paper presents the first meta-analysis of studies that computed correlations between the h index and variants of the h index (such as the g index; in total 37 different variants) that have been proposed and discussed in the literature. A high correlation between the h index and its variants would indicate that the h index variants hardly provide added information to the h index. This meta-analysis included 135 correlation coefficients from 32 studies. The studies were based on a total sample size of N = 9005; on average, each study had a sample size of n = 257. The results of a three-level cross-classified mixed-effects meta-analysis show a high correlation between the h index and its variants: Depending on the model, the mean correlation coefficient varies between .8 and .9. This means that there is redundancy between most of the h index variants and the h index. There is a statistically significant study-to-study variation of the correlation coefficients in the information they yield. The lowest correlation coefficients with the h index are found for the h index variants MII and m index. Hence, these h index variants make a non-redundant contribution to the h index.     

Monotonicity and the Hirsch index

The Hirsch index is a number that synthesizes a researcher's output. It is the maximum number h such that the researcher has h papers with at least h citations each. Woeginger [Woeginger, G. J. (2008a). An axiomatic characterization of the Hirsch-index. Mathematical Social Sciences, 56(2), 224–232; Woeginger, G. J. (2008b). A symmetry axiom for scientific impact indices. Journal of Informetrics, 2(3), 298–303] characterizes the Hirsch index when indices are assumed to be integer-valued. In this note, the Hirsch index is characterized, when indices are allowed to be real-valued, by adding to Woeginger's monotonicity two axioms in a way related to the concept of monotonicity.     

Axiomatics for the Hirsch index and the Egghe index

The Hirsch index and the Egghe index are both numbers that synthesize a researcher's output. The h-index associated with researcher r is the maximum number h such that r has h papers with at least h citations each. The g-index is the maximum number g of papers by r such that the average number of citations of the g papers is at least g. Both indices are characterized in terms of four axioms. One identifies outputs deserving index at most one. A second one establishes a strong monotonicity condition. A third one requires the index to satisfy a property of subadditivity. The last one consists of a monotonicity condition, for the h-index, and an aggregate monotonicity condition, for the g-index.     

How to derive an advantage from the arbitrariness of the g-index

The definition of the g-index is as arbitrary as that of the h-index, because the threshold number g² of citations to the g most cited papers can be modified by a prefactor at one's discretion, thus taking into account more or less of the highly cited publications within a dataset. In a case study I investigate the citation records of 26 physicists and show that the prefactor influences the ranking in terms of the generalized g-index less than for the generalized h-index. I propose specifically a prefactor of 2 for the g-index, because then the resulting values are of the same order of magnitude as for the common h-index. In this way one can avoid the disadvantage of the original g-index, namely that the values are usually substantially larger than for the h-index and thus the precision problem is substantially larger; while the advantages of the g-index over the h-index are kept. Like for the generalized h-index, also for the generalized g-index different prefactors might be more useful for investigations which concentrate only on top scientists with high citation frequencies or on junior researchers with small numbers of citations.     

Characteristic scores and scales based on h-type indices

Based on the rank-order citation distribution of e.g. a researcher, one can define certain points on this distribution, hereby summarizing the citation performance of this researcher. Previous work of Glänzel and Schubert defined these so-called “characteristic scores and scales” (CSS), based on average citation data of samples of this ranked publication–citation list.In this paper we will define another version of CSS, based on diverse h-type indices such as the h-index, the g-index, the Kosmulski's h⁽²⁾-index and the g-variant of it, the g⁽²⁾-index.Mathematical properties of these new CSS are proved in a Lotkaian framework. These CSS also provide an improvement of the single h-type indices in the sense that they give h-type index values for different parts of the ranked publication–citation list.     

On the h-index,the size of the Hirsch core and Jin's A-index

Hirsch's h-index seeks to give a single number that in some sense summarizes an author's research output and its impact. Essentially, the h-index seeks to identify the most productive core of an author's output in terms of most received citations. This most productive set we refer to as the Hirsch core, or h-core. Jin's A-index relates to the average impact, as measured by the average number of citations, of this “most productive” core. In this paper, we investigate both the total productivity of the Hirsch core – what we term the size of the h-core – and the A-index using a previously proposed stochastic model for the publication/citation process, emphasising the importance of the dynamic, or time-dependent, nature of these measures. We also look at the inter-relationships between these measures. Numerical investigations suggest that the A-index is a linear function of time and of the h-index, while the size of the Hirsch core has an approximate square-law relationship with time, and hence also with the A-index and the h-index.     

A mathematical characterization of the Hirsch-index by means of minimal increments

The minimum configuration to have a h-index equal to h is h papers each having h citations, hence h² citations in total. To increase the h-index to h + 1 we minimally need (h + 1)² citations, an increment of I₁(h) = 2h + 1. The latter number increases with 2 per unit increase of h. This increment of the second order is denoted I₂(h) = 2.If we define I₁ and I₂ for a general Hirsch configuration (say n papers each having f(n) citations) we calculate I₁(f) and I₂(f) similarly as for the h-index. We characterize all functions f for which I₂(f) = 2 and show that this can be obtained for functions f(n) different from the h-index. We show that f(n) = n (i.e. the h-index) if and only if I₂(f) = 2, f(1) = 1 and f(2) = 2.We give a similar characterization for the threshold index (where n papers have a constant number C of citations). Here we deal with second order increments I₂(f) = 0.     

Modifying h-index by allocating credit of multi-authored papers whose author names rank based on contribution

In the present work we introduce a modification of the h-index for multi-authored papers with contribution based author name ranking. The modified h-index is denoted by h_mc-index. It employs the framework of the h_m-index, which in turn is a straightforward modification of the Hirsch index, proposed by Schreiber. To retain the merit of requiring no additional rearrangement of papers in the h_m-index and in order to overcome its shortage of benefiting secondary authors at the expense of primary authors, h_mc-index uses combined credit allocation (CCA) to replace fractionalized counting in the h_m-index. The h_m-index is a special form of h_mc-index and fits for papers with equally important authors or alphabetically ordered authorship. There is a possibility of an author of lower contribution to the whole scientific community obtaining a higher h_mc-index. Rational h_mc-index, denoted by h_mcr-index, can avoid it. A fictitious example as a model case and two empirical cases are analyzed. The correlations of the h_mcr-index with the h-index and its several variants considering multiple co-authorship are inspected with 30 researchers’ citation data. The results show that the h_mcr-index is more reasonable for authors with different contributions. A researcher playing more important roles in significant work will obtain higher h_mcr-index.     

An axiomatic approach to bibliometric rankings and indices

This paper analyzes several well-known bibliometric indices using an axiomatic approach. We concentrate on indices aiming at capturing the global impact of a scientific output and do not investigate indices aiming at capturing an average impact. Hence, the indices that we study are designed to evaluate authors or groups of authors but not journals. The bibliometric indices that are studied include classic ones such as the number of highly cited papers as well as more recent ones such as the h-index and the g-index. We give conditions that characterize these indices, up to the multiplication by a positive constant. We also study the bibliometric rankings that are induced by these indices. Hence, we provide a general framework for the comparison of bibliometric rankings and indices.     

An index for evaluating journals in a small domestic citation index database whose citation rate is generally very low: A test based on the Korea Citation Index (KCI) database

This study describes the meaning of and the formula for S-index, which is a novel evaluation index based on the number of citations of each article in a particular journal and the rank of the article according to the number of citations. This study compares S-index with Impact Factor (IF), which is the most well-known evaluation index, using the Korea Citation Index data. It is shown that S-index is positively correlated with the number of articles published in a journal. Tapered h-index (h_T-index), which is based on all articles of a journal like S-index, is compared with S-index. It is shown that there is a very strong positive correlation between S-index and h_T-index. Although S-index is similar to h_T-index, S-index has a slightly better differentiating power and ranks the journal with evenly cited articles higher.     

Real and rational variants of the h-index and the g-index

The definitions of the rational and real-valued variants of the h-index and g-index are reviewed. It is shown how they can be obtained both graphically and by calculation. Formulae are derived expressing the exact relations between the h-variants and between the g-variants. Subsequently these relations are examined. In a citation context the real h-index is often, but not always, smaller than the rational h-index. It is also shown that the relation between the real and the rational g-index depends on the number of citations of the article ranked g + 1. Maximum differences between h, h_r and h_rat on the one hand and between g, g_r and g_rat on the other are determined.     

Regularity in the research output of individual scientists: An empirical analysis by recent bibliometric tools

This paper proposes an empirical analysis of several scientists based on their time regularity, defined as the ability of generating an active and stable research output over time, in terms of both quantity/publications and impact/citations. In particular, we empirically analyse three recent bibliometric tools to perform qualitative/quantitative evaluations under the new perspective of regularity. These tools are respectively (1) the PY/CY diagram, (2) the publication/citation Ferrers diagram and triad indicators, and (3) a year-by-year comparison of the scientists’ output (Borda's ranking). Results of the regularity analysis are then compared with those obtained under the classical perspective of overall production.The proposed evaluation tools can be applied to competitive examinations for research position/promotion, as complementary instruments to the commonly adopted bibliometric techniques.     

Peer review and the h-index: Two studies

Research was undertaken that examined what, if any, correlation there was between the h-index and rankings by peer assessment, and what correlation there was between the 2008 UK RAE rankings and the collective h-index of submitting departments. About 100 international scholars in Library and Information Science were ranked by their peers on the quality of their work. These rankings were correlated with the h and g scores the scholars had achieved. The results showed that there was a correlation between their median rankings and the indexes. The 2008 RAE grade point averages (GPA) achieved by departments from three UoAs – Anthropology, Library and Information Management and Pharmacy were compared with each of their collective h and g index scores. Results were mixed, with a strong correlation between pharmacy departments and index scores, followed by library and information management to anthropology where negative and non-significant results were found. Taken together, the findings from the research indicate that individual ranking by peer assessment and their h-index or variants was generally good. Results for the RAE 2008 gave correlations between GPA and successive versions of the h-index which varied in strength, except for anthropology where, it is suggested detailed cited reference searches must be undertaken to maximise citation counts.