# Lotka's law Lotka law for the 15 most populated categories on arXiv (2023-07). It is a log-log plot. The x-axis is the number of publications, and the y-axis is the number of authors with at least that many publications.

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:

Portion of articles written Number of authors writing that number of articles
10 100/102 = 1
9 100/92 ≈ 1 (1.23)
8 100/82 ≈ 2 (1.56)
7 100/72 ≈ 2 (2.04)
6 100/62 ≈ 3 (2.77)
5 100/52 = 4
4 100/42 ≈ 6 (6.25)
3 100/32 ≈ 11 (11.111...)
2 100/22 = 25
1 100

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. ISSN 1018-046X
• B Rousseau and R Rousseau (2000). "LOTKA: A program to fit a power law distribution to observed frequency data". Cybermetrics. 4. ISSN 1137-5019. - Software to fit a Lotka power law distribution to observed frequency data.