On the correction of the h-index for career length
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On the correction of the h-index for career length L. Egghe
Received: 3 October 2012 / Published online: 8 December 2012 Akade´miai Kiado´, Budapest, Hungary 2012
Abstract We describe mathematically the age-independent version of the h-index, defined by Abt (Scientometrics 91(3):863–868, 2012) and explain when this indicator is constant with age. We compare this index with the one where not the h-index is divided by career length but where all citation numbers are divided by career length and where we then calculate the new h-index. Both mathematical models are compared. A variant of this second method is by calculating the h-index of the citation data, divided by article age. Examples are given. Keywords
Age-independent h-index Career length
Introduction Let us have a researcher with T publications and let ci (i = 1, …, T) be the number of received citations of paper i. We suppose that the papers are arranged in decreasing order of number of received citations (i.e. ci C cj if and only if i B j). Then the Hirsch-index (Hirsch (2005)) (or h-index) is the largest rank r = h such that all papers on ranks i = 1, …, h have at least h citations (i.e. the largest rank r = h such that ch C h and hence ci C h for all i = 1, …, h). It is clear that the h-index is age-dependent (i.e. is dependent of career length). Long careers usually have higher values of T (total number of publications) and of ci (i = 1, …, T) (number of citations received by paper i), when compared to shorter careers (e.g. younger researchers). This fact was already noted in the defining paper Hirsch (2005). So, with the h-index, one should not compare researchers with different career length (as we should also not compare researchers from different fields, but that is the case for all citation-based indicators—in this paper we do not deal with this problem). L. Egghe (&) Universiteit Hasselt (UHasselt), Campus Diepenbeek, Agoralaan, 3590 Diepenbeek, Belgium e-mail: [email protected] L. Egghe Universiteit Antwerpen (UA), IBW, Stadscampus, Venusstraat 35, 2000 Antwerpen, Belgium
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Scientometrics (2013) 96:563–571
This has lead Abt (2012) to the following ‘‘age-independent’’ h-index (also implicit in Hirsch (2005)). Denote by h(t) the h-index of a researcher at career length t (starting at the time of the researcher’s first published paper). Then define aðtÞ ¼
hðtÞ ; t=10
ð1Þ
i.e. the h-index (at time t) divided by the (fractional) number of decades since the first published paper. The factor 10 is only useful in practical examples; in theoretical models we might use a ðtÞ ¼
hðtÞ t
ð2Þ
as well. Abt (2012) claims that a(t) is constant in t and gives practical evidence for it. Incidentally, a constant a(t) implies that h(t) increases linearly, as is trivial from (1) or (2). In this paper, based on the models developed in Egghe and Rousseau (2006) and Egghe (2009), we give a mathematical model for a(t) and a*(t) and present necessary and sufficient conditions for a(t) and a*(t) to be constant. Practical data on h(t) and a(t) for
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