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308 Becoming Metric-Wise

between the h-index, the number of publications (T) and the total num-
ber of received citations, denoted as CIT:

2 1=3

ð CITÞ
h 5 c (9.18)
T
where c is a constant. We note that the fact that the variable T appears in
the denominator of Eq. (9.18) does not contradict Eq. (9.15); Eq. (9.18)
also increases in T since the variable CIT is increasingly dependent on T.

9.4.2 A Combination of One and Two-Dimensional
Informetrics: the Cumulative First-Citation Distribution
Consider a bibliography, i.e., a set of related articles. For each article one
determines the time when it received its first citation and the time
between publication and the moment of first citation (in days, months or
years, in practice depending on the availability of data). Rousseau (1994)
noted that one may distinguish two types of cumulative first-citation
curves: a concave increasing curve and an increasing S-shaped curve.
Occurrence of these two types of curves was explained by Egghe
(2000). In this explanation he used the exponential obsolescence function
and Lotka’s law. Let Eq. (9.2) represent the distribution of all citations
received t years after publication and let Eq. (9.7) represent the number
of articles with n citations. If we denote the cumulative first-citation
curve by Φ t 1 ðÞ, i.e., the cumulative fraction of all articles which received
at least one citation at time t 1 , then Egghe (2000) showed that, with γ
the fraction of cited articles (ever, not at time t 1 ) that

α21
t 1
Φ t 1 ðÞ 5 γ 12að Þ k (9.19)
He further proved that Φ is concave for 1 , α # 2 and that it is
S-shaped for α . 2. We note the special role played by α 5 2. Figs. 9.8
and 9.9 show examples of the two types of curves.

9.5 MEASURING INEQUALITY

Garfield (1972) showed that at that time the top 152 journals in the SCI
accounted for 50% of all references to journals. It is, moreover, clear that
the informetric laws describe situations in which a large inequality is pres-
ent. This leads to the problem of how to measure this inequality. The