Lotka's law


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. It states that the number of authors making contributions in a given period is a fraction of the number making a single contribution, following the formula where 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 are discipline-specific.
The general formula says:
or
where X is the number of publications, Y the relative frequency of authors with X publications, and n and are constants depending on the specific field.

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 writtenNumber of authors writing that number of articles
10100/102 = 1
9100/92 ≈ 1
8100/82 ≈ 2
7100/72 ≈ 2
6100/62 ≈ 3
5100/52 = 4
4100/42 ≈ 6
3100/32 ≈ 11
2100/22 = 25
1100

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".

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