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The Informetric Laws

(originally, a given text). It has been shown (Egghe, 2005) that Lotka’s law
and Zipf’s law are equivalent: mathematically one follows from the other and
vice versa, using formula (9.6) or its continuous version. The corresponding
relation between the exponents α and β is given by formula (9.10):
1
β 5 (9.10)
α 2 1
Note that by formula (9.10) α 5 2 corresponds with β 5 1. The so-
called law of Zipf in linguistics (sources are word types, and items are
occurrences of these words in a text) was promoted and made popular by
him (Zipf, 1949), but he was not the first to note this regularity: in lin-
guistics Estoup (1916) and Condon (1928) were precursors. Moreover,
this same relation had already been observed earlier by Auerbach (1913)
in the context of cities and inhabitants. More information about Zipf’s
life can be found in Rousseau (2002a).
The same regularity was also observed by Pareto (1895) in the context
of income inequalities. These facts show that for many years different
fields have found the same regularities without realizing this.
Mandelbrot (1967) formulated a generalization of Zipf’s law, conse-
quently known as Mandelbrot’s law, in the context of fractal theory.
Mandelbrot’s law has the form:

D
grðÞ 5 (9.11)
ð 11ErÞ β
This function has one parameter more than Zipf’s and hence fits
observed data easier. We recall that fractals are mathematical objects that
represent the limit of infinitely many iterations. An interesting aspect of
fractals is that they are often self-similar. This means that a part of the
fractal is identical to the entire fractal itself. Fractal dimensions, related to
the exponent β in Mandelbrot’s law, are used to measure the complexity
of objects. For more on fractals, see Mandelbrot (1977), Feder (1988),
and Egghe (2005).
Besides the functions f and g and the related laws of Lotka and Zipf,
also the laws of Bradford (1934) and Leimkuhler (1967) belong to the
group of informetric laws. It can be shown that they correspond to the
special case that α 5 2, or β 5 1.
Bradford’s law was observed in the context of journals and the number
of articles they published on a given topic (historically Bradford studied
the topics Lubrication and Applied Geophysics). It is formulated as follows.