Lotka's law

For the law of energetics, see Lotka's principle.

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.^[2][3]

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.

Definition

Let ${\displaystyle X}$ be the number of publications, ${\displaystyle Y}$ be the number of authors with ${\displaystyle X}$ publications, and ${\displaystyle k}$ be a constant depending on the specific field. Lotka's law states that ${\displaystyle Y\propto X^{-k}}$.

In Lotka's original publication, he claimed ${\displaystyle k=2}$. Subsequent research showed that ${\displaystyle k}$ varies depending on the discipline.

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

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

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.

Other applicaitons

A generalized version of Lotka's Law has been used to model the number of gold disks certified by the Recording Industry Association of America from 1958 to 1989, and was found to be an almost perfect fit to the data.^[4]

Relationship to Riemann Zeta

Lotka's law may be described using the Zeta distribution:

${\displaystyle f(x)={\frac {1}{\zeta (s)}}\cdot {\frac {1}{x^{s}}}}$

for ${\displaystyle x=1,2,3,4,\dots }$ and where

${\displaystyle \zeta (s)=\sum _{x=1}^{\infty }{\frac {1}{x^{s}}}}$

is the Riemann zeta function. It is the limiting case of Zipf's law where an individual's maximum number of publications is infinite.

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.

See also

• Price's law
• Riemann zeta function
• Zeta distribution