Lotka's law

Lotka's law, named after Alfred J. Lotka, is one of a variety of special applications of Zipf's law. It describes the frequency of publication by authors in any given field. Let X be the number of publications, $$Y$$ be the number of authors with $$X$$ publications, and $$k$$ be a constants depending on the specific field. Lotka's law states that $$Y \propto X^{-k}$$.

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 $$Y' \propto X^{-(k-1)}$$, where $$Y'$$ is the number of authors with at least $$X$$ publications. Their equivalence can be proved by taking the derivative.



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 by National Institute of Statistics.
 * - Software to fit a Lotka power law distribution to observed frequency data.