Zipf's law
Originally the term Zipf's law meant the observation of Harvard linguist George Kingsley Zipf (SAMPA: [zIf]) that the frequency of use of the nth-most-frequently-used word in any natural language is approximately inversely proportional to n.
Zipf's law is an experimental law, not a theoretical one. The causes of Zipfian distributions in real life are a matter of some controversy. However, Zipfian distributions are commonly observed in many kinds of phenomena.
Zipf's law is often demonstrated by scatterplotting the data, with the axes being log(rank order) and log(frequency). If the points are close to a single straight line, the distribution follows Zipf's law.
The classic case of Zipf's law is a "1/f function". Given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur 1/2 as often as the first. The third most common frequency will occur 1/3 as often as the first. The nth most common frequency will occur 1/n as often as the first.
Theoretical issues
Mathematically, it is impossible for Zipf's law to hold exactly if there are infinitely many words in a language, since for any constant of proportionality c > 0, the sum of all relative frequencies is proportional to the harmonic series and must be
- <math>\sum_{n=1}^\infty c/n=\infty\neq 1.<math>
Empirical studies have found that in English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to 1/ns where s is just slightly more than one.
As long as the exponent s exceeds 1, it is possible for such a law to hold with infinitely many words, since if s > 1 then
- <math>\sum_{n=1}^\infty 1/n^s<\infty.<math>
The value of this sum is ζ(s), where ζ is Riemann's zeta function.
Related laws
The term Zipf's law has consequently come to be used to refer to frequency distributions of "rank data" in which the relative frequency of the nth-ranked item is given by the Zeta distribution, 1/(nsζ(s)), where s > 1 is a parameter indexing this family of probability distributions. Indeed, the term Zipf's law sometimes simply means the zeta distribution, since probability distributions are sometimes called "laws". This distribution is sometimes called the Zipfian distribution or Yule distribution.
A more general law proposed by Benoit Mandelbrot has frequencies
- <math>f_n=[\mbox{constant}]/(q+n)^s.<math>
This is the Zipf-Mandelbrot law. The "constant" in this case is the reciprocal of the Hurwitz zeta function evaluated at s.
Examples of collections approximately obeying Zipf's law:
- frequency of accesses to web pages
- in particular the access counts on the BambooWeb most popular page, with s approximately equal to 0.3
- page access counts on Polish BambooWeb (data for late July 2003) approximately obey Zipf's law with s about 0.5
- words in the English language
- sizes of settlements
- income distribution amongst individuals
- size of earthquakes
- notes in musical performances
It has been pointed out (see external link below) that Zipfian distributions can also be regarded as being Pareto distributions with an exchange of variables.
See also
Further reading
- George K. Zipf, Human Behaviour and the Principle of Least-Effort, Addison-Wesley, Cambridge MA, 1949
- W. Li, "Random texts exhibit Zipf's-law-like word frequency distribution", IEEE Transactions on Information Theory, 38(6), pp.1842-1845, 1992.
- Alexander Gelbukh, Grigori Sidorov. . 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.
- Damian H. Zanette. Zipf's law and the creation of musical context. Online preprint at http://xxx.arxiv.org/abs/cs.CL/0406015
- Kali R. The city as a giant component: a random graph approach to Zipf's law. Applied Economics Letters, 15 September 2003, vol. 10, iss. 11, pp. 717-720(4)
