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Lotka's law, named after Alfred J. Lotka, is one of a variety of special applications of
Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...
. It describes the frequency of publication by authors in any given field.


Definition

Let X be the number of publications, Y be the number of authors with X publications, and k be a constant depending on the specific field. Lotka's law states that Y \propto X^. In Lotka's original publication, he claimed k=2. Subsequent research showed that k varies depending on the discipline. Equivalently, Lotka's law can be stated as Y' \propto X^, where Y' is the number of authors with ''at least'' X publications. Their equivalence can be proved by taking the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
.


Example

Assume that n=2 in a discipline, then as the number of articles published increases, authors producing that many publications become less frequent. There are 1/4 as many authors publishing two articles within a specified time period as there are single-publication authors, 1/9 as many publishing three articles, 1/16 as many publishing four articles, etc. And if 100 authors wrote ''exactly'' one article each over a specific period in the discipline, then: That would be a total of 294 articles and 155 writers, with an average of 1.9 articles for each writer.


Software

* Friedman, A. 2015. "The Power of Lotka’s Law Through the Eyes of R" The Romanian Statistical Review. Published b
National Institute of Statistics
*

to fit a Lotka power law distribution to observed frequency data.


Relationship to Riemann Zeta

Lotka's law may be described using the Zeta distribution: :f(x) = \frac \cdot \frac for x = 1, 2, 3, 4, \dots and where : \zeta (s) = \sum_^\infty \frac is the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
. It is the limiting case of
Zipf's law Zipf's law (; ) is an empirical law stating that when a list of measured values is sorted in decreasing order, the value of the -th entry is often approximately inversely proportional to . The best known instance of Zipf's law applies to the ...
where an individual's maximum number of publications is infinite.


See also

* Price's law *
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
* Zeta distribution


References


Further reading

* — Chung and Cox analyze a bibliometric regularity in finance literature, relating Lotka's law to the maxim that " the rich get richer and the poor get poorer", and equating it to the maxim that "success breeds success".


External links

*
''The Journal of the Washington Academy of Sciences'', vol. 16
{{DEFAULTSORT:Lotka's Law Bibliometrics Statistical laws