Zipf's law

G. K. Zipf proposed in his book Human Behavior and the Principle of the Least Effort from 1949 an empirical law on word frequencies in natural language speech and texts. Zipf's law states that while only a few words are used very often, many or most are used rarely.

Zipf's law states that in a tabulation of the occurrence of all words in a sufficiently comprehensive text, ranged by their frequency, will the product of rang number and frequency make up a constant. In addition will the number of different words in the vocabulary be equal to the frequency of the most common word (rank number 1).

Zipf's law may be stated mathematically as:
$f(k;s,N)=\frac{1/k^s}{\sum_{n=1}^N 1/n^s}$

where N  is the number of elements, k  is their rank, and s  is the exponent characterizing the distribution. In the example of the frequency of words in the English language, N  is the number of words in the English language and, if we use the classic version of Zipf's law, the exponent s is 1.

Zipf also provided a theoretical explanation for his law. He found that the law was an expression of a competition between two economic principles: "The economics of the speaker" that tend towards a reduction of the number of words in language and  "the economy of the listener" that tend to use a new word in each new linguistic act that the speaker wish to do. In all persons speaking a language fluently there is a balance and Zipf's law is an indication that this balance is reached. This balance is not present, however, by, for example, immigrants, who are in the process of learning a new language. In other words: Zipf's basic idea was that there are two opposing forces that guide the evolution of language: unification and diversification. From the speaker’s point of view, it is desirable in terms of effort minimization to communicate all meaning via a single word or sound. For the listener, it is desirable to have a different word associated with each separate meaning. Language evolves, Zipf suggests, in a way that optimizes the cost of communicative transactions between speakers and listeners.

Zipf's law is also an expression of more universal regularities. Zipf himself found that his law was valid in relation to populations in cities as plotted as a function of the rank (the most popular city is ranked number one, etc). Fedorowicz (1982) expresses the view that  Zipf's law can be applied on phenomena as different as distributions of income, the size of companies and biological arts and species. Zipf's law is believed to be equivalent with distributions in laws formulated by Yule, Lotka, Pareto, Bradford and Price. Zipf's law is often assumed to be related to other bibliometric laws (cf., for example, Chen & Leimkuhler, 1986; Kunz, 1988).

Zipf's law has been influential in Library and Information Science (LIS) in, for example, examinations of whether information retrieval languages are in accordance with it (cf., for example, Blair, 1990; Egghe, 1991; Fedorowicz, 1982; Ohly, 1982; Wyllys, 1981).

Literature:

Blair; D. C. (1990). Language and Representation in Information Retrieval. Amsterdam: Elsevier.

Brookes, B. C. (1969). The complete Bradford-Zipf 'Bibliograph'. Journal of Documentation, 25(1), 58-60.

Brookes, B. C. (1968). The derivation and application of the Bradford-Zipf distribution. Journal of Documentation, 24(4), 247-265.

Buckland, M. K. & Hindle, A. (1969). Library Zipf. Journal of Documentation, 25(1), 52-56.

Chen, Y.-S.; Leimkuhler, F. F. (1986). A relationship between Lotka's law, Bradford's law, and Zipf's law. Journal of the American Society for Information Science, 37(5), 307-314.

Egghe, L.: The exact place of Zipf's and Pareto's law amongst the classical information laws. Scientometrics, 20(1), 1991, 93-106.

Fairthorne, R. A. (1969). Empirical hyperbolic distributions (Bradford-Zipf-Mandelbrot) for bibliometric description and prediction. Journal of Documentation, 25(4), 319-343.

Fedorowicz, J. (1982). The theoretical foundation of Zipf's law and its application to the bibliographic database environment. Journal of the American Society for Information Science, 33(5), 285-293.

Gabaix, Xavier (1999). Zips's law for cities: An explanation. Quarterly Journal of Economics, 114(3), 739-67. Available at: http://econ-www.mit.edu/faculty/download_pdf.php?id=530  (Retrieved 2007-08-16).

Gelbukh, Alexander & Sidorov, Grigori (2004). Zipf and Heaps Laws’ Coefficients Depend on Language. Proc. CICLing-2001, Conference on Intelligent Text Processing and Computational Linguistics, February 18–24, 2001, Mexico City. Lecture Notes in Computer Science N 2004, ISSN 0302-9743, ISBN 3-540-41687-0, Springer-Verlag, pp. 332–335. Retrieved 2007-08-16 from: http://www.gelbukh.com/CV/Publications/2001/CICLing-2001-Zipf.htm

Kali R. (2003). The city as a giant component: a random graph approach to Zipf's law. Applied Economics Letters, 10(11), 717-720.

Kunz, M.: Lotka and Zipf: paper dragons with fuzzy tails. Scientometrics, 13(5-6), 1988, 289-­297.

Nicholls, P. T. (1987). Estimation of Zipf parameters. Journal of the American Society for Information Science, 38(6), 443-445.

Ohly, H. Peter: A procedure for comparing documentation language applications: the transformed Zipf curve. International Classification, 9(3), 1982, 125-128.

Wyllys, R. E. (1981). Empirical and theoretical bases of Zipf's Law. Library Trends, 30(1), 53-64.

Zipf, G. K. (1932). Selected Studies of the Principle of Relative Frequencies of Language. Cambridge, Massachusetts: Harvard University Press.

Zipf, G. K. (1949). Human behaviour and the Principle of the Least Effort. Reading, MA: Addison-Wesley.