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. Let be the number of publications, be the number of authors with publications, and be a constants depending on the specific field. Lotka's law states that .

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.

In Lotka's original publication, he claimed . Subsequent research showed that varies depending on the discipline.

Equivalently, Lotka's law can be stated as , where is the number of authors with at least 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

edit

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

edit
  • 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.

Relationship to Riemann Zeta

edit

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

 

for   and where

 

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

See also

edit

References

edit
  1. ^ Lotka, Alfred J. (1926). "The frequency distribution of scientific productivity". Journal of the Washington Academy of Sciences. 16 (12): 317–324.

Further reading

edit
  • Kee H. Chung and Raymond A. K. Cox (March 1990). "Patterns of Productivity in the Finance Literature: A Study of the Bibliometric Distributions". Journal of Finance. 45 (1): 301–309. doi:10.2307/2328824. JSTOR 2328824. — 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".
edit