View source for Zipf's law

<!-- EDITORS! Please see [[Wikipedia:WikiProject Probability123#Standards]] for a discussion of standards used for probability distribution articles such as this one.
-->{{Probability distribution|
  name       =Zipf's law|
  type       =mass|
  pdf_image  =[[Image:Zipf distribution PMF.png|325px|Plot of the Zipf PMF for ''N'' = 10]]<br /><small>Zipf PMF for ''N'' = 10 on a log–log scale. The horizontal axis is the index ''k''&nbsp;. (Note that the function is only defined at integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  cdf_image  =[[Image:Zipf distribution CMF.png|325px|Plot of the Zipf CDF for N=10]]<br /><small>Zipf CDF for ''N'' = 10. The horizontal axis is the index ''k''&nbsp;. (Note that the function is only defined at integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  parameters =<math>s \geq 0\,</math> ([[real number|real]])<br /><math>N \in \{1,2,3\ldots\}</math> ([[integer]])|
  support    =<math>k \in \{1,2,\ldots,N\}</math>|
  pdf        =<math>\frac{1/k^s}{H_{N,s}}</math> where ''H<sub>N,s</sub>'' is the ''N''th generalized [[harmonic number]]|
  cdf        =<math>\frac{H_{k,s}}{H_{N,s}}</math>|
  mean       =<math>\frac{H_{N,s-1}}{H_{N,s}}</math>|
  median     =|
  mode       =<math>1\,</math>|
  variance   =<math>\frac{H_{N,s-2}}{H_{N,s}}-\frac{H^2_{N,s-1}}{H^2_{N,s}}</math>|
  skewness   =|
  kurtosis   =|
  entropy    =<math>\frac{s}{H_{N,s}}\sum\limits_{k=1}^N\frac{\ln(k)}{k^s}
+\ln(H_{N,s})</math>|
  mgf        =<math>\frac{1}{H_{N,s}}\sum\limits_{n=1}^N \frac{e^{nt}}{n^s}</math>|
  char       =<math>\frac{1}{H_{N,s}}\sum\limits_{n=1}^N \frac{e^{int}}{n^s}</math>|
}}
'''Zipf's law''' ({{IPAc-en|z|ɪ|f}}) is an [[empirical law]] formulated using [[mathematical statistics]] that refers to the fact that many types of data studied in the [[physical science|physical]] and [[social science|social]] sciences can be approximated with a Zipfian distribution, one of a family of related discrete [[power law]] [[probability distribution]]s.  ''Zipf distribution'' is related to the [[zeta distribution]], but is not identical.

Zipf's law was originally formulated in terms of [[quantitative linguistics]], stating that given some [[Text corpus|corpus]] of [[natural language]] utterances, the frequency of any word is [[inversely proportional]] to its rank in the [[frequency table]]. Thus the most frequent word will occur approximately twice as often as the second most frequent word, three times as often as the third most frequent word, etc.: the [[rank-frequency distribution]] is an inverse relation. For example, in the [[Brown Corpus]] of American English text, the word ''[[English articles#Definite article|the]]'' is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's Law, the second-place word ''of'' accounts for slightly over 3.5% of words (36,411 occurrences), followed by ''and'' (28,852). Only 135 vocabulary items are needed to account for half the [[Brown Corpus]].<ref>{{citation|contribution=An introduction to textual econometrics|first1=Stephen|last1=Fagan|first2=Ramazan|last2=Gençay|pages=133–153|title=Handbook of Empirical Economics and Finance|editor1-first=Aman|editor1-last=Ullah|editor2-first=David E. A.|editor2-last=Giles|publisher=CRC Press|year=2010|isbn=9781420070361}}. [https://books.google.com/books?hl=en&lr=&id=QAUv9R6bJzwC&oi=fnd&pg=PA139 P.&nbsp;139]: "For example, in the Brown Corpus, consisting of over one million words, half of the word volume consists of repeated uses of only 135 words."</ref>

The law is named after the American [[linguistics|linguist]] [[George Kingsley Zipf]] (1902–1950), who popularized it and sought to explain it (Zipf 1935, 1949), though he did not claim to have originated it.<ref name=Powers1998>{{cite journal| last = Powers | first = David M W| url=http://aclweb.org/anthology/W98-1218 |title=Applications and explanations of Zipf's law| year = 1998| conference = Joint conference on new methods in language processing and computational natural language learning| pages = 151–160| publisher = Association for Computational Linguistics }}</ref> The French stenographer [[Jean-Baptiste Estoup]] (1868–1950) appears to have noticed the regularity before Zipf.<ref>Christopher D. Manning, Hinrich Schütze ''Foundations of Statistical Natural Language Processing'', MIT Press (1999), {{isbn|978-0-262-13360-9}}, p. 24</ref>{{not in body|reason=This inline citation may well be adequate to verify the claim, but something like this should not be in the lead if it is not also in the body.|date=May 2016}} It was also noted in 1913 by German physicist [[Felix Auerbach]]<ref name="Auerbach1913">Auerbach F. (1913) Das Gesetz der Bevölkerungskonzentration. Petermann’s Geographische Mitteilungen 59, 74–76</ref> (1856–1933).

==Other data sets==
The same relationship occurs in many other rankings unrelated to language, such as the population ranks of cities in various countries, corporation sizes, income rankings, ranks of number of people watching the same TV channel,<ref>M. Eriksson, S.M. Hasibur Rahman, F. Fraille, M. Sjöström, [http://apachepersonal.miun.se/~mageri/myresearch/bmsb2013-Eriksson.pdf Efficient Interactive Multicast over DVB-T2 - Utilizing Dynamic SFNs and PARPS] {{webarchive|url=https://web.archive.org/web/20140502183246/http://apachepersonal.miun.se/~mageri/myresearch/bmsb2013-Eriksson.pdf |date=2014-05-02}}, 2013 IEEE International Conference on Computer and Information Technology (BMSB'13), London, UK, June 2013. Suggests a heterogeneous Zipf-law TV channel-selection model</ref> and so on. The appearance of the distribution in rankings of cities by population was first noticed by Felix Auerbach in 1913.<ref name="Auerbach1913"/> Empirically, a data set can be tested to see whether Zipf's law applies by checking the goodness of fit of an empirical distribution to the hypothesized power law distribution with a [[Kolmogorov–Smirnov test]], and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.<ref name="Clausetetal2009">Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review, 51(4), 661–703. doi:10.1137/070710111</ref> When Zipf's law is checked for cities, a better fit has been found with exponent ''s'' = 1.07; i.e. the <math>n^{th}</math> largest settlement is <math>\frac{1}{n^{1.07}}</math> the size of the largest settlement.

==Theoretical review==
Zipf's law is most easily observed by [[graph of a function|plotting]] the data on a [[log-log]] graph, with the axes being [[logarithm|log]] (rank order) and log (frequency). For example, the word ''"the"'' (as described above) would appear at ''x'' = log(1), ''y'' = log(69971). It is also possible to plot reciprocal rank against frequency or reciprocal frequency or interword interval against rank.<ref name=Powers1998/> The data conform to Zipf's law to the extent that the plot is [[linear equation|linear]].

Formally, let:
* ''N'' be the number of elements;
* ''k'' be their rank;
* ''s'' be the value of the exponent characterizing the distribution.
Zipf's law then predicts that out of a population of ''N'' elements, the normalized frequency of elements of rank ''k'', ''f''(''k'';''s'',''N''), is:

:<math>f(k;s,N)=\frac{1/k^s}{\sum\limits_{n=1}^N (1/n^s)}</math>

Zipf's law holds if the number of elements with a given frequency is a random variable with power law distribution <math>p(f) = \alpha f^{-1-1/s}.</math><ref name=Adamic2000>Adamic, Lada A. (2000) [http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html "Zipf, Power-laws, and Pareto - a ranking tutorial", originally published at http://www.parc.xerox.com/istl/groups/iea/papers/ranking/ranking.html] {{webarchive|url=https://web.archive.org/web/20071026062626/http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html |date=2007-10-26 }}</ref>

It has been claimed that this representation of Zipf's law is more suitable for statistical testing, and in this way it has been analyzed in more than 30,000 English texts. The goodness-of-fit tests yield that only about 15% of the texts are statistically compatible with this form of Zipf's law. Slight variations in the definition of Zipf's law can increase this percentage up to close to 50%.<ref>{{Cite journal|last = Moreno-Sánchez|first = I|last2 = Font-Clos|first2 = F|last3 = Corral|first3 = A|date = 2016|title = Large-Scale Analysis of Zipf's Law in English Texts|url =|journal = PLoS ONE|volume = 11|pages = e0147073|doi = 10.1371/journal.pone.0147073|pmid =|access-date =|arxiv = 1509.04486}}</ref>

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. ''f''(''k'';&nbsp;''s'',''N'') will then be the fraction of the time the ''k''th most common word occurs.

The law may also be written:

:<math>f(k;s,N)=\frac{1}{k^s H_{N,s}}</math>

where ''H<sub>N,s</sub>'' is the ''N''th generalized [[harmonic number]].

The simplest case of Zipf's law is a "{{sfrac|1|''f''}} function." Given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur half as often as the first, the third most common frequency will occur {{sfrac|1|3}} as often as the first, and the ''n''th most common frequency will occur {{sfrac|1|''n''}} as often as the first. However, this cannot hold exactly, because items must occur an integer number of times; there cannot be 2.5 occurrences of a word. Nevertheless, over fairly wide ranges, and to a fairly good approximation, many natural phenomena obey Zipf's law.

In human languages, word frequencies have a very heavy-tailed distribution, and can therefore be modeled reasonably well by a Zipf distribution with an ''s'' close to 1.

As long as the exponent ''s'' exceeds 1, it is possible for such a law to hold with infinitely many words, since if ''s''&nbsp;>&nbsp;1 then
:<math>\zeta (s) = \sum_{n=1}^\infty \frac{1}{n^s}<\infty. \!</math>
where ''ζ'' is [[Riemann zeta function|Riemann's zeta function]].

==Statistical explanation==

[[File:Zipf 30wiki en labels.png|thumbnail|A plot of the rank versus frequency for the first 10 million words in 30 Wikipedias (dumps from October 2015) in a [[log-log]] scale.]]

Although Zipf's Law holds for all languages, even non-natural ones like [[Esperanto]],<ref>{{cite conference |author1=Bill Manaris |author2=Luca Pellicoro |author3=George Pothering |author4=Harland Hodges |title=INVESTIGATING ESPERANTO’S STATISTICAL PROPORTIONS RELATIVE TO OTHER LANGUAGES USING NEURAL NETWORKS AND ZIPF’S LAW |url=http://www.cs.cofc.edu/~manaris/uploads/Main/IASTED2006.pdf |journal=[[Artificial Intelligence and Applications]] |date=13 February 2006 |location=Innsbruck, Austria |pages=102–108 |url-status=live |archiveurl=https://web.archive.org/web/20160305040450/http://www.cs.cofc.edu/~manaris/uploads/Main/IASTED2006.pdf |archivedate=5 March 2016 }}</ref> the reason is still not well understood.<ref>[[Léon Brillouin]], ''La science et la théorie de l'information'', 1959, réédité en 1988, traduction anglaise rééditée en 2004</ref> However, it may be partially explained by the statistical analysis of randomly generated texts. Wentian Li has shown that in a document in which each character has been chosen randomly from a uniform distribution of all letters (plus a space character), the "words" with different lengths follow the macro-trend of the Zipf's law (the more probable words are the shortest with equal probability).<ref>{{cite journal |author=Wentian Li |title=Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution |journal=[[IEEE Transactions on Information Theory]] |volume=38 |issue=6 |year=1992 |pages=1842–1845 |doi=10.1109/18.165464 |df= |citeseerx=10.1.1.164.8422 }}</ref>  [[Vitold Belevitch]] in a paper, ''On the Statistical Laws of Linguistic Distribution'' offered a mathematical derivation.  He took a large class of well-behaved [[statistical distribution]]s (not only the [[normal distribution]]) and expressed them in terms of rank. He then expanded each expression into a [[Taylor series]]. In every case Belevitch obtained the remarkable result that a first-order truncation of the series resulted in Zipf's law. Further, a second-order truncation of the Taylor series resulted in [[Zipf–Mandelbrot law|Mandelbrot's law]].<ref>[[Peter G. Neumann|Neumann, Peter G.]] [http://www.csl.sri.com/users/neumann/#12a "Statistical metalinguistics and Zipf/Pareto/Mandelbrot"], ''SRI International Computer Science Laboratory'', accessed and [https://web.archive.org/web/20110605012951/http://www.csl.sri.com/users/neumann/ archived] 29 May 2011.</ref><ref>{{cite journal
| author = Belevitch V
| title = On the statistical laws of linguistic distributions
| journal = Annales de la Société Scientifique de Bruxelles
| volume = 73
| series = I
| date = 18 December 1959
| pages = 310–326
}}</ref>

The [[principle of least effort]] is another possible explanation:
Zipf himself proposed that neither speakers nor hearers using a given language want to work any harder than necessary to reach understanding, and the process that results in approximately equal distribution of effort leads to the observed Zipf distribution.<ref>{{cite book
| author = Zipf GK
| title = Human Behavior and the Principle of Least Effort
| location = Cambridge, Massachusetts
| publisher = Addison-Wesley
| year =1949
| page = 1
}}</ref><ref>{{cite journal
 |author1       = Ramon Ferrer i Cancho
 |author2       = Ricard V. Sole
 |lastauthoramp = yes
 |year          = 2003
 |title         = Least effort and the origins of scaling in human language
 |url           = http://www.pnas.org/content/100/3/788.abstract?sid=cc7fae18-87c9-4b67-863a-4195bb47c1d1
 |journal       = [[Proceedings of the National Academy of Sciences of the United States of America]]
 |volume        = 100
 |pages         = 788–791
 |issue         = 3
 |doi           = 10.1073/pnas.0335980100
 |pmid          = 12540826
 |pmc           = 298679
 |url-status       = live
 |archiveurl    = https://web.archive.org/web/20111201040141/http://www.pnas.org/content/100/3/788.abstract?sid=cc7fae18-87c9-4b67-863a-4195bb47c1d1
 |archivedate   = 2011-12-01
}}</ref>

Similarly, [[preferential attachment]] (intuitively, "the rich get richer" or "success breeds success") that results in the [[Yule–Simon distribution]] has been shown to fit word frequency versus rank in language<ref>{{cite arXiv |title=Scaling laws in human speech, decreasing emergence of new words and a generalized model |df= |eprint = 1412.4846|last1 = Lin|first1 = Ruokuang|last2 = Ma|first2 = Qianli D. Y.|last3 = Bian|first3 = Chunhua|class = cs.CL|year = 2014}}</ref> and population versus city rank<ref>{{cite journal |title=Test of two hypotheses explaining the size of populations in a system of cities |journal = Journal of Applied Statistics|volume = 42|issue = 12|pages = 2686–2693|df= |arxiv = 1506.08535|doi = 10.1080/02664763.2015.1047744|year = 2015|last1 = Vitanov|first1 = Nikolay K.|last2 = Ausloos|first2 = Marcel|last3 = Bian|first3 = Chunhua}}</ref> better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.

==Related laws==

[[Image:Wikipedia-n-zipf.png|thumb|A plot of word frequency in Wikipedia (November 27, 2006). The plot is in [[log-log]] coordinates. ''x''&nbsp; is rank of a word in the frequency table; ''y''&nbsp; is the total number of the word's occurrences. Most popular words are "the", "of" and "and", as expected. Zipf's law corresponds to the middle linear portion of the curve, roughly following the green (1/''x'')&nbsp; line, while the early part is closer to the magenta (1/''x''<sup>0.5</sup>) line while the later part is closer to the cyan (1/(''k''&nbsp;+&nbsp;''x'')<sup>2.0</sup>) line. These lines correspond to three distinct parameterizations of the Zipf–Mandelbrot distribution, overall a [[broken power law]] with three segments: a head, middle, and tail.]]

''Zipf's law'' in fact refers more generally to frequency distributions of "rank data," in which the relative frequency of the ''n''th-ranked item is given by the [[Zeta distribution]], 1/(''n''<sup>''s''</sup>''ζ''(''s'')), where the parameter ''s''&nbsp;>&nbsp;1 indexes the members of this family of [[probability distribution]]s.  Indeed, ''Zipf's law'' is sometimes synonymous with "zeta distribution," since probability distributions are sometimes called "laws". This distribution is sometimes called the '''Zipfian''' distribution.

A generalization of Zipf's law is the [[Zipf–Mandelbrot law]], proposed by [[Benoit Mandelbrot]], whose frequencies are:

:<math>f(k;N,q,s)=\frac{[\text{constant}]}{(k+q)^s}.\,</math>

The "constant" is the reciprocal of the [[Hurwitz zeta function]] evaluated at ''s''. In practice, as easily observable in distribution plots for large corpora, the observed distribution can be modelled more accurately as a sum of separate distributions for different subsets or subtypes of words that follow different parameterizations of the Zipf–Mandelbrot distribution, in particular the closed class of functional words exhibit ''s'' lower than 1, while open-ended vocabulary growth with document size and corpus size require ''s'' greater than 1 for convergence of the [[harmonic series (mathematics)|Generalized Harmonic Series]].<ref name=Powers1998/>

Zipfian distributions can be obtained from [[Pareto distribution]]s by an exchange of variables.<ref name="Adamic2000"/>

The Zipf distribution is sometimes called the '''discrete Pareto distribution'''<ref>{{cite book|title=Univariate Discrete Distributions|edition=second|year=1992|author1=N. L. Johnson |author2=S. Kotz |author3=A. W. Kemp  |last-author-amp=yes |publisher=John Wiley & Sons, Inc.|location=New York|isbn=978-0-471-54897-3|ref=harv}}, p. 466.</ref> because it is analogous to the continuous [[Pareto distribution]] in the same way that the [[Uniform distribution (discrete)|discrete uniform distribution]] is analogous to the [[Uniform distribution (continuous)|continuous uniform distribution]].

The tail frequencies of the [[Yule–Simon distribution]] are approximately

:<math>f(k;\rho) \approx \frac{[\text{constant}]}{k^{\rho+1}}</math>

for any choice of ''ρ'' > 0.

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.<ref name="Galien">{{cite web |url=http://home.zonnet.nl/galien8/factor/factor.html |title=Factorial randomness: the Laws of Benford and Zipf with respect to the first digit distribution of the factor sequence from the natural numbers |author=Johan Gerard van der Galien |date=2003-11-08 |accessdate=8 July 2016 |archiveurl=https://web.archive.org/web/20070305150334/http://home.zonnet.nl/galien8/factor/factor.html |archivedate=2007-03-05}}</ref> Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.<ref>Ali Eftekhari (2006) Fractal geometry of texts. ''Journal of Quantitative Linguistic'' 13(2-3): 177–193.</ref>

It has been argued that [[Benford's law]] is a special bounded case of Zipf's law,<ref name="Galien"/> with the connection between these two laws being explained by  their both originating from scale invariant functional relations from statistical physics and critical phenomena.<ref>L. Pietronero, E. Tosatti, V. Tosatti, A. Vespignani (2001) Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf. ''Physica A'' 293: 297–304.</ref> The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with s = 1 satisfy Benford's law.
{| class="wikitable" style="text-align: center;"
|-
!<math>n</math>
!Benford's law: <math>P(n) = </math><br/><math>\log_{10}(n+1)-\log_{10}(n)</math>
!<math>\frac{\log(P(n)/P(n-1))}{\log(n/(n-1))}</math>
|-
| 1
| 0.30103000
|  
|-
| 2
| 0.17609126
| −0.7735840
|-
| 3
| 0.12493874
| −0.8463832
|-
| 4
| 0.09691001
| −0.8830605
|-
| 5
| 0.07918125
| −0.9054412
|-
| 6
| 0.06694679
| −0.9205788
|-
| 7
| 0.05799195
| −0.9315169
|-
| 8
| 0.05115252
| −0.9397966
|-
| 9
| 0.04575749
| −0.9462848
|}

== Applications ==
In [[information theory]], a symbol (event, signal) of probability <math>p</math> contains <math>\log_2(1/p)</math> [[bit]]s of information. Hence, Zipf law for natural numbers: <math>\Pr(x) \approx 1/x</math> is equivalent with number <math>x</math> containing <math>\log_2(x)</math> bits of information. To add information from a symbol of probability <math>p</math> into information already stored in a natural number <math>x</math>, we should go to <math>x'</math> such that <math>\log_2(x') \approx \log_2(x) + \log_2(1/p)</math>, or equivalently <math>x' \approx x/p</math>. For instance, in standard binary system we would have <math>x' = 2x + s</math>, what is optimal for <math>\Pr(s=0) = \Pr(s=1) = 1/2</math> probability distribution. Using <math>x' \approx x/p</math> rule for a general probability distribution is the base of [[Asymmetric Numeral Systems]] family of  [[entropy coding]] methods used in [[data compression]], which state distribution is also governed by Zipf law.

Zipf's law also has been used for extraction of parallel fragments of texts out of comparable corpora.<ref>{{cite conference |url=https://comparable.limsi.fr/bucc2016/pdf/BUCC04.pdf |title=Parallel Document Identification using Zipf's Law |last1=Mohammadi |first1=Mehdi |date=2016 |book-title=Proceedings of the Ninth Workshop on Building and Using Comparable Corpora |pages=21–25 |location=Portorož, Slovenia |conference=LREC 2016 |url-status=live |archiveurl=https://web.archive.org/web/20180323154706/https://comparable.limsi.fr/bucc2016/pdf/BUCC04.pdf |archivedate=2018-03-23 }}</ref>

==See also==
{{div col|colwidth=20em}}
* [[Brevity law]]
* [[Heaps' law]]
* [[Menzerath's law]]
* [[Bradford's law]]
* [[Benford's law]]
* [[Demographic gravitation]]
* [[Frequency list]]
* [[Gibrat's law]]
* [[Hapax legomenon]]
* [[Lorenz curve]]
* [[Lotka's law]]
* [[Pareto distribution]]
* [[Pareto principle]], a.k.a. the "80–20 rule"
* [[Principle of least effort]]
* [[Price's law]]
* [[Rank-size distribution]]
* [[King effect]]
* [[Stigler's law of eponymy]]
* [[1% rule (Internet culture)]]
{{Div col end}}

==References==
{{Reflist}}

==Further reading==
Primary:
* [[George K. Zipf]] (1949) ''Human Behavior and the Principle of Least Effort''. Addison-Wesley. "Online text [https://archive.org/details/in.ernet.dli.2015.90211]" 
* George K. Zipf (1935) ''The Psychobiology of Language''. Houghton-Mifflin.
Secondary:
* Alexander Gelbukh and Grigori Sidorov (2001) [http://www.gelbukh.com/CV/Publications/2001/CICLing-2001-Zipf.htm "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: 332–335.
* Damián H. Zanette (2006) "[http://xxx.arxiv.org/abs/cs.CL/0406015 Zipf's law and the creation of musical context,]" ''Musicae Scientiae 10'': 3–18.
* Frans J. Van Droogenbroeck (2016), [https://www.academia.edu/24147736/ Handling the Zipf distribution in computerized authorship attribution]
* Frans J. Van Droogenbroeck (2019), [https://www.academia.edu/40029629 An essential rephrasing of the Zipf-Mandelbrot law to solve authorship attribution applications by Gaussian statistics  ]
* Kali R. (2003) "The city as a giant component: a random graph approach to Zipf's law," ''Applied Economics Letters 10'': 717–720(4)
*{{cite journal |last= Gabaix|first= Xavier|authorlink= Xavier Gabaix|date=August 1999|title= Zipf's Law for Cities: An Explanation |journal= Quarterly Journal of Economics|volume= 114|issue= 3|pages= 739–67|issn= 0033-5533|url= http://pages.stern.nyu.edu/~xgabaix/papers/zipf.pdf|doi= 10.1162/003355399556133|citeseerx= 10.1.1.180.4097}}
* Axtell, Robert L; [http://www.sciencemag.org/content/293/5536/1818.short Zipf distribution of US firm sizes], Science, 293, 5536, 1818, 2001, American Association for the Advancement of Science
* Ramu Chenna, Toby Gibson; [http://www.worldcomp-proceedings.com/proc/p2011/BIC4329.pdf Evaluation of the Suitability of a Zipfian Gap Model for Pairwise Sequence Alignment],
International Conference on Bioinformatics Computational Biology: 2011.
* Shyklo A. (2017); [https://ssrn.com/abstract=2918642 Simple Explanation of Zipf's Mystery via New Rank-Share Distribution, Derived from Combinatorics of the Ranking Process], Available at SSRN: https://ssrn.com/abstract=2918642.

==External links==
{{commons category}}
*{{Cite news | last = Strogatz | first = Steven  | authorlink = Steven Strogatz | title = Guest Column: Math and the City  | date = 2009-05-29 | url = http://judson.blogs.nytimes.com/2009/05/19/math-and-the-city/ | accessdate = 2009-05-29 | postscript = <!--None--> | work=The New York Times}}—An article on Zipf's law applied to city populations
*[https://www.theatlantic.com/issues/2002/04/rauch.htm Seeing Around Corners (Artificial societies turn up Zipf's law)]
*[https://web.archive.org/web/20021018011011/http://planetmath.org/encyclopedia/ZipfsLaw.html PlanetMath article on Zipf's law]
*[http://www.hubbertpeak.com/laherrere/fractal.htm Distributions de type "fractal parabolique" dans la Nature (French, with English summary)]
*[https://www.newscientist.com/article.ns?id=mg18524904.300 An analysis of income distribution]
*[http://www.lexique.org/listes/liste_mots.txt Zipf List of French words]
*[http://1.1o1.in/en/webtools/semantic-depth Zipf list for English, French, Spanish, Italian, Swedish, Icelandic, Latin, Portuguese and Finnish from Gutenberg Project and online calculator to rank words in texts]
*[https://arxiv.org/abs/physics/9901035 Citations and the Zipf–Mandelbrot's law]
*[http://www.geoffkirby.co.uk/ZIPFSLAW.pdf Zipf's Law examples and modelling (1985)]
*[http://www.nature.com/nature/journal/v474/n7350/full/474164a.html Complex systems: Unzipping Zipf's law (2011)]
*[http://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-law-and-the-pareto-distribution/ Benford’s law, Zipf’s law, and the Pareto distribution] by Terence Tao.

{{ProbDistributions|discrete-finite}}

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