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.

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.

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 n^th most common frequency will occur 1/n as often as the first.

Theoretical issues

Zipf's law may be stated mathematically as:

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

The law may also be written:

where H_N,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:

Empirical studies have found that in English, the frequencies of the approximately 1000 most-frequently-used words are approximately proportional to 1/n^s 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

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/(n^sζ(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

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

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 [1] for a proof.