Lotkaian Informetrics and applications to social networks

Leo Egghe
Universiteit Hasselt

We start by describing growth in one-dimensional informetrics and give examples in WWW. Then two-dimensional informetrics is defined in terms of Information Production Processes (IPPs) which contain sources that produce items. Many examples are given. Two-dimensional informetrics is, mathematically, described by a size-frequency function (number of sources with n items) and by a rank-frequency function (number of items in the source on rank r). We have Lotkaian informetrics if the size-frequency function is a power law: the law of Lotka. Lotka's law is equivalent with Zipf's law: a power law for the rank-frequency function. Most IPPs are Lotkaian: this is illustrated by some examples in WWW.

Then the scale-free property of Lotkaian systems is discussed. Lotka's law can be explained based on exponential growth of sources and items (Naranan (1970)) and, from this, Lotkaian IPPs can be interpreted as self-similar fractals (Egghe (2005)). A formula for the fractal dimension is given in function of the Lotka exponent, generalizing results of Mandelbrot.

We then go into some applications of Lotkaian informetrics. We describe dynamical aspects of Lotkaian IPPs via transformations on the sources and on the items. A formula for the size-frequency function of the transformed IPP is given (Egghe (2004, 2007)). The link with three-dimensional informetrics is given. In the case of power law transformations in Lotkaian systems we can show that the transformed size-frequency function is also Lotkaian and a formula is given for the new Lotka exponent. This has been applied twice. V. Cothey (2007) applies this in order to study the evolution of an IPP in the case of parts of WWW. Egghe and Rousseau (2006) use these formulae to describe IPPs without low productive sources. There one shows that these IPPs have low Lotka exponents which is illustrated by giving examples of country and municipality sizes or the sizes of databases. Finally we show how Lotka's law can be used in the modelling of the cumulative first-citation distribution, i.e. the distribution over time at which an article receives its first citation. Both the concavely increasing distributions and the S-shaped distributions are explained in one formula in which the Lotka exponent is a determining parameter (Egghe (2000)).

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