Page 309 - Becoming Metric Wise

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The Informetric Laws

elitist situation. We feel it is important to point out that these regularities
are not laws in the sense of the natural sciences, but just explainable prob-
abilistic regularities that are in practice more or less valid. Yet we will fol-
low the tradition and use the term “informetric laws” (actually, the
traditional terminology is “bibliometric laws”). These remarks are valid
whenever these “laws” are applicable, whatever the field of application
(economy, demography, biology, etc.)
The first function we will discuss is the size-frequency function f.For
each natural number n 5 1,2,3,.. . f(n) denotes the number of sources
with n items, e.g., the number of authors who wrote n articles. The next
function is the rank-frequency function g. A rank-frequency function is
obtained as follows: rank sources decreasingly according to the number of
items. The resulting ranks are r 5 1,2,3,.. .. Then g(r) denotes the number
of items of the source at rank r. Functions f and g of a given IPP are said
to be dual functions. Their relation is given by formula (9.6):
N
X
21
r 5 g ðÞ 5 fkðÞ (9.6)
n
k5n
The function g 21 is the inverse function of the function g (assumed to
21
be injective): If g maps the point x to the point y, then g maps y to x.
Formula (9.6) expresses the relation between the variables r and n, through
the functions f and g. It further shows that if the function f is known, also
g 21 is known and hence g itself. Theoretical work using discrete sums is dif-
ficult and often not possible. For this reason one makes models in which f
and g are continuous functions: they can take any positive real value (some-
times restricted to a certain, possibly infinitely long, interval). In this con-
tinuous model the function f is obtained from the function g by taking the
derivative of g 21 (Egghe, 2005). The use of continuous models to describe
real life phenomena are standard techniques in probability theory, the most
classical example being the Gauss curve, also known as the bell curve.
The term Lotkaian informetrics (Egghe, 2005) refers to those models
and their application in which one assumes that the size-frequency func-
tion f is a power function, which in the field of informetrics is often
referred to as Lotka’s law. Concretely we have
C
fnðÞ 5 (9.7)
n α
with C . 0and α . 1. When referring to observed data the variable n is a
strictly positive natural number, while in models n is a real number usually