# 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 n contributions is about $1/n^{a}$ of those making one contribution, where a nearly always equals two. More plainly, 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 very discipline-specific.

The general formula says:

$X^n Y = C$

or

$Y = C / X^n, \,$

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

This law is believed to have applications in other fields, for example in the military for fighter pilot kills.

## Example

Say 100 authors write one article each over a specific period, we assume for this table that C=1 and n=2:

Number 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
Graphical plot of the Lotka function described above, with C=1, n=2

That would be a total of 293 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]

## References

1. ^ Lotka, Alfred J. (1926). "The frequency distribution of scientific productivity". Journal of the Washington Academy of Sciences 16 (12): 317–324.
2. ^ Egghe, Leo (2005). "Relations between the continuous and the discrete Lotka power function". Journal of the American Society for Information Science and Technology 56 (7): 664–668. doi:10.1002/asi.20157.