Page 315 - Becoming Metric Wise

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The Informetric Laws

g(r)

h

0 h r
Figure 9.7 Illustration of the h-index in a continuous model.

Proof. In this framework the total number of sources with n or more
items equals
1N
P
f ðkÞ. Now, as already mentioned before, we use a continuous
k5n
framework to facilitate calculations. This leads to:
1N ð N ð N
X C C 12α
f ðkÞ f ð jÞ dj 5 α dj 5 n (9.16)
k5n n n j α 2 1
where α . 1 (otherwise this integral makes no sense). The total number
of sources, T, is equal to
N C
ð
T 5 f ð jÞdj 5 (9.17)
1 α 2 1
Combining Eqs. (9.16) and (9.17) yields that the number of sources
with n or more items is equal to Tn 12α . We conclude that the h-index
12α
h is equal to that number n such that Tn 5 n. Consequently:
α
T: h 12α 5 h or T 5 h , leading to the expression h 5 T 1=α .
This is the expression of the h-index in a Lotkaian framework. Note
that in its original context of publications and citations formula (9.15)
includes the number of publications and the citation curve, through its
parameter α. If this parameter is known also the number of citations is
known, but clearly knowing that the citation curve has a Lotka distribu-
tion with exponent α contains more information. Of course, if data do
not satisfy Lotka’s distribution then this expression for the h-index cannot
be applied. Schubert and Gla ¨nzel (2007) proposed the following relation