# The Mandelbrot Law, Bradford's Law, Vickery Law...

## Mandelbrot Law

Benoit Mandelbrot (B. Mandelbrot) in 1954 proposed a theoretical justification for the empirically discovered Zipf law. Representing words in the form of a sequence of letters separated by empty spaces and assigning the signs a certain "cost" (effort, time), Mandelbrot shows that words can be attributed to a priori probabilities, so that their overall "cost" on the average was minimal and the amount of information remained invariant. On the basis of these representations in a mathematical way, Mandelbrot showed that the resulting relationship between the frequency of a word and its rank corresponds to the empirical Zipf law with a small correction:

where p r i is the relative occurrence frequency of the word in the text; r i is the rank of the word; k is an empirical constant; γ is a value close to unity, but varies depending on the properties of the text.

The coefficient γ characterizes certain properties of the language - the degree of its formalization, while the degree of formalization of the language decreases with decreasing γ.

The law was opened in 1934 by an English chemist and bibliographer C. Bradford on the basis of the discovery of general principles for the distribution of publications for publications in different areas (for example, publications on geophysics and chemistry), became widely known after publication in 1948. The main sense of the law is as follows: if scientific journals in order of decreasing the number of articles on a specific issue, then the logs can be divided into three zones in such a way that the number of articles in each zone for a given topic is the same.

In this case, the first zone, called Bradford's zone core, includes profile magazines, directly devoted to the subject. The number of logs in the core zone is small. The second zone is made up of journals that are partially dedicated to a given area, and their number is significantly larger than the number of journals in the kernel. The third zone - the largest in number of publications - combines logs, the number of which is very far from the subject domain.

With an equal number of publications in each zone, the number of sources (in this example, journal titles) increases sharply when moving from one zone to another. Bradford found that the number of magazine titles in the third zone is about as many times as in the second zone, how many times the number of magazine titles in the second zone is greater than in the core:

where Р 1 2 3 - the number of journal titles in the 1st, 2nd and 3rd zones respectively.

## Vickery Law

B. Vickery1 clarified the model of S. Bradfod. He found that the journals, ranked in decreasing order of articles on a particular issue, can be broken down into any number of zones, not into three zones. The main sense of the Vikery's regularity (Figure 4.11): if the periodicals are arranged in decreasing order of the number of articles for a particular query, then in the list you can select a number of zones, each of which contains the same number of articles.

In this case, the number of logs in the first zone and the increasing number of them in subsequent zones are correlated as follows:

where x is the number of articles in each zone; T x - the number of logs containing x articles; T 2x , T 3x , T 4x , ... is the number of logs containing 2x, 3 x , 4 x , and so on, respectively.

Often this law is called Bradford's law in the interpretation of Vickery.

1 Vickery VS Bradford's law of scatering // J. Doc. 1948. Vol. 4. P. 198 -

Fig. 4.11. Vickery Law

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