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306 Becoming Metric-Wise

members of the dataset. The King effect has been observed for Chinese
character use with respect to Zipf’s law, the possessive 的 5 de (in
pinyin) 5 of (in English) being the Dragon King (Rousseau & Zhang,
1992) and for French cities (Paris as King) and country population sizes
(China and India being the Dragon Kings) in the case of a so-called
stretched exponential, another distribution with a fat tail (Laherre `re &
Sornette, 1998). In the case of a Zipf rank-frequency relation and a unique
Dragon King the Leimkuhler form can be written as R(r) 5 k 1 a ln
(1 1 br).

9.4 TWO APPLICATIONS OF LOTKA’S LAW

In this section, we provide two applications of Lotka’s law. The first gives a
model for the h-index. The second deals with the first-citation function.

9.4.1 The h-index in a Lotkaian Framework (Egghe &
Rousseau, 2006a)
In this framework we consider the continuous Lotka size-frequency func-
tion f:1; N -Š0; CŠ of the form
½
½
C
f ð jÞ 5 (9.14)
j α
where C . 0 and α . 1 (Egghe, 2005) to describe the citation function.
In a discrete setting f( j) denotes the number of sources, here articles,
with production j, here with j citations. We will describe the h-index,
h, as a function of the exponent α and the total number of sources T.
First we note that, with g(r) the number of citations of the source at
rank, the h-index is characterized as the rank r such that g(r) 5 r, see
Fig. 9.7. Indeed, in its discrete interpretation, this equality states that
there are r sources with r or more items while the other have no more
than r items.

Theorem
Suppose that a Lotkaian system with T sources and parameter α is given.
Then the h-index is:
1=α
h 5 T (9.15)
Readers who are not familiar with integrals can safely skip this proof.