# Zipf's law

Originally the term Zipf's law meant the observation of Harvard linguist George Kingsley Zipf (IPA: ) that the frequency of use of the nth-most-frequently-used word in any natural language is approximately inversely proportional to n.

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.

Zipf's law is an experimental law, not a theoretical one. Zipfian distributions are commonly observed in many kinds of phenomena. The causes of Zipfian distributions in real life are a matter of some controversy, however.

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.

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## Theoretical issues

Zipf's law may be stated mathematically as:

[itex]p_k(s,N)=\frac{1/k^s}{\sum_{n=1}^N 1/n^s}[itex]

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  will be equal to unity. pk  will then be the fraction of the time the k-th most common word occurs. It is easily seen that the distribution is normalized:

[itex]\sum_{k=1}^N p_k(s,N)=1.[itex]

The law may also be written:

[itex]p_k(s,N)=\frac{1/k^s}{H_{N,s}}[itex]

where HN,s is the Nth generalized harmonic number.

Mathematically, it is impossible for the classic version of Zipf's law to hold exactly if there are infinitely many words in a language, since the sum of all relative frequencies in the denominator above is equal to the harmonic series and therefore:

[itex]\sum_{n=1}^\infty \frac{1}{n}=\infty\![itex]

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

[itex]\sum_{n=1}^\infty \frac{1}{n^s}<\infty. \![itex]

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

[itex]f_n=[\mbox{constant}]/(q+n)^s. \![itex]

This is the Zipf-Mandelbrot law. The "constant" in this case is the reciprocal of the Hurwitz zeta function evaluated at s.

In the tail of the Yule-Simon distribution the frequencies are approximately

[itex]f_n \approx [\mbox{constant}]/n^{\rho+1}[itex]

for any choice of ρ > 0.

The log-normal distribution is the distribution of a random variable whose logarithm is normally distributed, useful when small fluctuations multiply a quantity rather than add to it.

In the parabolic fractal distribution, the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship (see external link below).

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.

It has been argued that Benford's law is a special case of Zipf's law. See  (http://home.zonnet.nl/galien8/factor/factor.html) for a proof.

## Examples of collections approximately obeying Zipf's law

• 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. "Zipf and Heaps Laws’ Coefficients Depend on Language" (http://www.cic.ipn.mx/~gelbukh/CV/Publications/2001/CICLing-2001-Zipf.htm). 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)

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