# Lotka's law

Lotka's law,[1] 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. It states that the number of authors making ${\displaystyle x}$ contributions in a given period is a fraction of the number making a single contribution, following the formula ${\displaystyle 1/x^{a}}$ where ${\displaystyle a}$ nearly always equals two, i.e., an approximate inverse-square law, where the number of authors publishing a certain number of articles is a fixed ratio to the number of authors publishing a single article. 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. Though the law itself covers many disciplines, the actual ratios involved (as a function of 'a') are discipline-specific.

Graphical plot of the Lotka function described in the text, with C=1, n=2

The general formula says:

${\displaystyle X^{n}Y=C}$

or

${\displaystyle Y=C/X^{n},\,}$

where X is the number of publications, Y the relative frequency of authors with X publications, and n and ${\displaystyle C}$ are constants depending on the specific field (${\displaystyle n\approx 2}$).

## Example

Say 100 authors write at least one article each over a specific period, we assume for this table that C=100 and n=2. Then the number of authors writing portions of any particular articles in that time period is described as in the following table:

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 with 155 writers with an average of 1.9 articles for each writer.

This is an empirical observation rather than a necessary result. This form of the law is as originally published and is sometimes referred to as the "discrete Lotka power function".[2]

## 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.