2840:
2759:
2731:
39:
2874:
3044:
2817:
2886:
2906:
2794:
2930:
27:
519:
547:
2683:
6783:
2953:
6793:
2318:
is a special bounded case of Zipf's law, with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena. The ratios of probabilities in
Benford's law are not constant. The leading digits of
2709:
The actual rank-frequency plot of a natural language text deviates in some extent from the ideal Zipf distribution, especially at the two ends of the range. The deviations may depend on the language, on the topic of the text, on the author, on whether the text was translated from another language,
434:
The same relation for frequencies of words in natural language texts was observed by George Zipf in 1932, but he never claimed to have originated it. In fact, Zipf did not like mathematics. In his 1932 publication, the author speaks with disdain about mathematical involvement in linguistics, a. o.
3023:
In these cases, the observed frequency-rank relation can be modeled more accurately as by separate ZipfâMandelbrot laws distributions for different subsets or subtypes of words. This is the case for the frequency-rank plot of the first 10 million words of the
English Knowledge. In particular, the
3019:
Even in
English, the deviations from the ideal Zipf's law become more apparent as one examines large collections of texts. Analysis of a corpus of 30,000 English texts showed that only about 15% of the texts in it have a good fit to Zipf's law. Slight changes in the definition of Zipf's law can
480:
The same relationship was found to occur in many other contexts, and for other variables besides frequency. For example, when corporations are ranked by decreasing size, their sizes are found to be inversely proportional to the rank. The same relation is found for personal incomes (where it is
1989:
However, it may be partly explained by 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
2017:
is another possible explanation: Zipf himself proposed that neither speakers nor hearers using a given language wants 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.
2311:, 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. Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.
2839:
437:(âŠ) let me say here for the sake of any mathematician who may plan to formulate the ensuing data more exactly, the ability of the highly intense positive to become the highly intense negative, in my opinion, introduces the devil into the formula in the form of
2025:. If language is generated by a single monkey typing randomly, with fixed and nonzero probability of hitting each letter key or white space, then the words (letter strings separated by white spaces) produced by the monkey follows Zipf's law.
5253:
in our arms and legs are the scarcest. Being able to use a cell's size to estimate its frequency in the body could help doctors better understand certain body systems and hard-to-count cell types... The study suggests, for instance, that
1204:
2048:, which has been shown to fit word frequency versus rank in language and population versus city rank better than Zipf's law. It was originally derived to explain population versus rank in species by Yule, and applied to cities by Simon.
1399:
1300:
1073:
1881:
2956:
A log-log plot of word frequency in
Knowledge (November 27, 2006). '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
4399:
2758:
2730:
180:
2673:
Following
Auerbach's 1913 observation, there has been substantial examination of Zipf's law for city sizes. However, more recent empirical and theoretical studies have challenged the relevance of Zipf's law for cities.
260:
2173:
1966:= 1, one can also plot the reciprocal of the frequency (mean interword interval) against rank, or the reciprocal of rank against frequency, and compare the result with the line through the origin with slope 1.
4183:
1666:
2059:
with drift and variance parameters that depend only on the rank of the process. It has been shown mathematically that Zipf's law holds for Atlas models that satisfy certain natural regularity conditions.
1768:
2558:
2295:
1481:
3862:
3016:; a word of English being often translated to a compound of two such syllables. The rank-frequency table for those "words" deviates significantly from the ideal Zipf law, at both ends of the range.
1571:
500:
published a short paper showing that Zipf's law emerges even in randomly generated texts. It included proof that the power law form of Zipf's law was a byproduct of ordering words by rank.
2450:
922:
3074:
ciphers), the frequency-rank distribution is not affected. On the other hand, if separate occurrences of the same word may be mapped to two or more different words (as happens with the
192:" 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 "
5078:
3101:
The frequency-rank word distribution is often characteristic of the author and changes little over time. This feature has been used in the analysis of texts for authorship attribution.
855:
779:
182:
It is usually found that the most common word occurs approximately twice as often as the next common one, three times as often as the third most common, and so on. For example, in the
714:
651:
2905:
4371:
4451:
2885:
463:
338:
3649:, 2013 IEEE International Conference on Computer and Information Technology (BMSB'13), London, UK, June 2013. Suggests a heterogeneous Zipf-law TV channel-selection model
312:
2873:
599:
2985:, the frequencies of the dozen or so most frequent words deviate significantly from the ideal Zipf distribution, because of those words include articles inflected for
2929:
2793:
1931:, and then comparing the (log) likelihood ratio of the power law distribution to alternative distributions like an exponential distribution or lognormal distribution.
5369:
Zipf list for
English, French, Spanish, Italian, Swedish, Icelandic, Latin, Portuguese and Finnish from Gutenberg Project and online calculator to rank words in texts
1086:
2816:
2380:
3387:. Joint conference on new methods in language processing and computational natural language learning. Association for Computational Linguistics. pp. 151â160.
3062:
When a text is encrypted in such a way that every occurrence of each distinct plaintext word is always mapped to the same encrypted word (as in the case of simple
147:
951:
286:
207:
2346:
4563:
Pietronero, L.; Tosatti, E.; Tosatti, V.; Vespignani, A. (2001). "Explaining the uneven distribution of numbers in nature: The laws of
Benford and Zipf".
5441:
4962:
1313:
3619:
1217:
964:
4705:
1804:
3642:
3814:
4296:
Lin, Ruokuang; Ma, Qianli D. Y.; Bian, Chunhua (2014). "Scaling laws in human speech, decreasing emergence of new words and a generalized model".
5238:
4503:"Factorial randomness: the Laws of Benford and Zipf with respect to the first digit distribution of the factor sequence from the natural numbers"
4755:
4053:
3366:: "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."
4317:
Vitanov, Nikolay K.; Ausloos, Marcel; Bian, Chunhua (2015). "Test of two hypotheses explaining the size of populations in a system of cities".
34:. The lower plot shows the remainder when the Zipf law is divided away. It shows that there remains significant pattern not fitted by Zipf law.
5070:
412:
observed an inverse proportionality between the population sizes of cities, and their ranks when sorted by decreasing order of that variable.
3789:
2052:
5306:
5225:
3108:
have been found to satisfy Zipf's law, suggesting that text is most likely not a hoax but rather written in an obscure language or cipher.
1306:
2082:
6847:
5570:
3852:
3729:
3705:
6796:
6053:
1586:
2006:, the first-order truncation of the series results in Zipf's law. Further, a second-order truncation of the Taylor series resulted in
5961:
3977:
3577:
5234:, vol. 330, no. 1 (January 2024), pp. 94â95. "'As you double the volume of a cell, the frequency of cells of that size is halved,'"
3834:
Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-Law
Distributions in Empirical Data. SIAM Review, 51(4), 661â703.
6748:
3095:
4421:
1693:
6614:
5826:
5585:
5434:
4941:
4076:
47:
4129:
2456:
6509:
6273:
5395:
4997:
4839:
2229:
1420:
5342:
5334:
4502:
3347:
Fagan, Stephen; Gençay, Ramazan (2010), "An introduction to textual econometrics", in Ullah, Aman; Giles, David E. A. (eds.),
5947:
5007:
4481:
3906:
3613:
3464:
1509:
6268:
6212:
6110:
5872:
5510:
3363:
5359:
4275:
3885:
396:
Some sets of time-dependent empirical data deviate somewhat from Zipf's law. Such empirical distributions are said to be
6554:
6288:
6141:
5816:
5560:
6018:
5371:
3680:
Lazzardi, Silvia; Valle, Filippo; Mazzolini, Andrea; Scialdone, Antonio; Caselle, Michele; Osella, Matteo (2021-06-17).
6822:
6786:
6458:
6434:
6013:
5427:
5208:
Simple
Explanation of Zipf's Mystery via New Rank-Share Distribution, Derived from Combinatorics of the Ranking Process
1986:, "A Brief History of Generative Models for Power Law and Lognormal Distributions", and Simkin, "Re-inventing Willis".
5051:
An essential rephrasing of the Zipf-Mandelbrot law to solve authorship attribution applications by
Gaussian statistics
6852:
6655:
6532:
6493:
6465:
6439:
6357:
6283:
5706:
5454:
5193:
3356:
2213:
482:
6643:
6609:
6475:
6470:
6315:
6123:
5821:
5575:
4164:
2973:)) line. These lines correspond to three distinct parameterizations of the ZipfâMandelbrot distribution, overall a
2385:
800:
4647:
3854:
Investigating Esperanto's statistical proportions relative to other languages using neural networks and Zipf's law
3553:
3388:
868:
6817:
6393:
6306:
6278:
6187:
6136:
6008:
5791:
5756:
5272:
2209:
5291:
807:
727:
518:
6407:
6324:
6161:
6085:
5908:
5786:
5761:
5625:
5620:
5615:
5053:
5034:
2308:
4972:
3597:
Chapter 1 - The Principal Problem in Political Economy: Income Distribution in the History of Economic Thought
546:
38:
6723:
6589:
6297:
6146:
6078:
6063:
5956:
5930:
5862:
5701:
5595:
5590:
5532:
5517:
5412:
3446:
Auerbach F. (1913) Das Gesetz der Bevölkerungskonzentration. Petermannâs Geographische Mitteilungen 59, 74â76
3037:
70:
20:
669:
6559:
6549:
6240:
6166:
5867:
5726:
3254:
609:
6619:
3198: â An application of Zipf's law describing the frequency of publication by authors in any given field
2694:
In many texts in human languages, word frequencies approximately follow a Zipf distribution with exponent
1990:
macro-trend of Zipf's law (the more probable words are the shortest and have equal probability). In 1959,
6867:
6862:
6604:
6599:
6544:
6480:
6424:
6245:
6232:
6023:
5968:
5920:
5711:
5640:
5505:
5407:
3639:
3595:
3243:
3117:
2220:
2045:
1982:, the reason is still not well understood. Recent reviews of generative processes for Zipf's law include
1928:
1685:
378:
42:
A plot of the frequency of each word as a function of its frequency rank for two English language texts:
4528:
Eftekhari, Ali (2006). "Fractal geometry of texts: An initial application to the works of Shakespeare".
3802:
3043:
6738:
6514:
6333:
6115:
6068:
5937:
5913:
5893:
5736:
5610:
5490:
1210:
5228:, "Cells by Count and Size: The larger a cell type is, the rarer it is in the body â and vice versa",
4727:
4669:
3999:
6832:
6743:
6527:
6488:
6362:
6199:
6043:
5988:
5886:
5850:
5721:
5686:
3228:
2014:
720:
5390:
5169:
4090:
188:
6429:
6217:
5983:
5942:
5857:
5811:
5751:
5716:
5605:
5500:
5450:
1995:
374:
351:
109:
5101:"Keywords and Co-Occurrence Patterns in the Voynich Manuscript: An Information-Theoretic Analysis"
3781:
3086:
Zipf's law has been used for extraction of parallel fragments of texts out of comparable corpora.
441:
317:
6837:
6728:
6670:
6341:
6128:
6038:
5993:
5978:
5796:
5746:
5741:
5542:
5522:
3144:
2721:, the plot takes a staircase shape, because each word can occur only an integer number of times.
2069:
2029:
2022:
2007:
366:
201:
5898:
5314:
5185:
291:
6857:
6594:
6582:
6571:
6453:
6349:
6156:
5600:
5580:
5485:
4085:
3234:
1199:{\displaystyle {\frac {s}{H_{N,s}}}\sum \limits _{k=1}^{N}{\frac {\ln(k)}{k^{s}}}+\ln(H_{N,s})}
577:
122:
5402:
4670:"Zipf's law and city size distribution: A survey of the literature and future research agenda"
3898:
3867:
3726:
3681:
6718:
6675:
6519:
6194:
6048:
6028:
5925:
5495:
4999:
The Zoologist's Guide to the Galaxy: What Animals on Earth Reveal About Aliens--and Ourselves
2854:
2183:
1795:
662:
570:
58:
5351:
3129: â Observation that in many real-life datasets, the leading digit is likely to be small
3032:
lower than 1, while open-ended vocabulary growth with document size and corpus size require
117:
stating that when a list of measured values is sorted in decreasing order, the value of the
6768:
6763:
6758:
6753:
6690:
6660:
6539:
6182:
6073:
5676:
5635:
5630:
5527:
5286:
5112:
4883:
4799:
4681:
4584:
4336:
4192:
4021:
2353:
1983:
416:
347:
5973:
3574:
3120: â Hypothesis that more people will lurk in a virtual community than will participate
2686:
Zipf's law plot for the first 10 million words in 30 Wikipedias (as of October 2015) in a
1974:
Although Zipf's Law holds for most natural languages, and even some non-natural ones like
8:
6827:
6702:
6227:
6207:
6177:
6151:
6105:
6033:
5845:
5781:
5340:
Distributions de type "fractal parabolique" dans la Nature (French, with English summary)
5230:
3939:
3207:
3063:
3009:
2865:
2830:
2784:
2205:
2194:
1999:
1935:
1079:
935:
390:
265:
5116:
4887:
4803:
4685:
4588:
4340:
4196:
4025:
3850:
3300:"Zipf's word frequency law in natural language: A critical review and future directions"
1946:
of rank order, and logarithm of frequency. The data conform to Zipf's law with exponent
1923:
Empirically, a data set can be tested to see whether Zipf's law applies by checking the
6733:
6222:
6003:
5998:
5903:
5840:
5835:
5691:
5681:
5565:
5329:
5199:
Kali R. (2003) "The city as a giant component: a random graph approach to Zipf's law,"
5143:
5100:
4967:
4906:
4873:
4861:
4831:
4789:
4639:
4574:
4545:
4443:
4391:
4352:
4326:
4297:
4267:
4074:
Wentian Li (1992). "Random Texts Exhibit Zipf's-Law-Like Word Frequency Distribution".
4045:
4011:
3969:
3697:
3660:
3605:
3545:
3324:
3299:
3216: â Statistical principle about ratio of effects to causes, a.k.a. the "80â20 rule"
3201:
3105:
2990:
2986:
2331:
4930:
4596:
4215:
4178:
3431:
3075:
3056:
6631:
6058:
5801:
5731:
5696:
5645:
5242:
5189:
5181:
5148:
5130:
5013:
5003:
4911:
4835:
4823:
4815:
4747:
4697:
4631:
4477:
4395:
4259:
4220:
4114:
4037:
3961:
3902:
3810:
3701:
3659:
Zanette, DamiĂĄn H. (June 7, 2004). "Zipf's law and the creation of musical context".
3609:
3537:
3460:
3352:
3329:
3132:
2982:
2799:
2073:
2056:
1887:
928:
43:
4777:
4549:
4447:
4356:
4049:
3921:
3727:
Evaluation of the Suitability of a Zipfian Gap Model for Pairwise Sequence Alignment
3078:), the Zipf distribution will typically have a flat part at the high-frequency end.
6842:
5806:
5480:
5419:
5339:
5249:
are by far the most common cells in our bodies, whereas the comparatively gigantic
5221:
5138:
5120:
4993:
4901:
4891:
4807:
4739:
4689:
4623:
4592:
4537:
4506:
4433:
4383:
4344:
4271:
4251:
4210:
4200:
4153:
4095:
4029:
3973:
3956:
3951:
3894:
3851:
Bill Manaris; Luca Pellicoro; George Pothering; Harland Hodges (13 February 2006).
3835:
3759:
3689:
3601:
3529:
3319:
3311:
3249:
3213:
3126:
3001:
2974:
2826:
2776:
2772:
2749:
2741:
2315:
1991:
1394:{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{int}}{n^{s}}}}
386:
355:
141:
87:
4743:
4348:
4033:
1295:{\displaystyle {\frac {1}{H_{N,s}}}\sum \limits _{n=1}^{N}{\frac {e^{nt}}{n^{s}}}}
5375:
5363:
5346:
5302:
5217:
5125:
5057:
5038:
4896:
3785:
3733:
3646:
3581:
3165:
3159:
2862:
2807:
2803:
2745:
2737:
1955:
1951:
1924:
1500:
1068:{\displaystyle {\frac {H_{N,s-2}}{H_{N,s}}}-{\frac {H_{N,s-1}^{2}}{H_{N,s}^{2}}}}
793:
474:
133:
5245:
and his fellow Zipf's-law researchers found, says Hatton. "Teensy, nonnucleated
4693:
3940:"A Brief History of Generative Models for Power Law and Lognormal Distributions"
3594:
Sandmo, Agnar (2015-01-01), Atkinson, Anthony B.; Bourguignon, François (eds.),
3584:, Science, 293, 5536, 1818, American Association for the Advancement of Science.
1576:
The distribution is sometimes generalized to an inverse power law with exponent
26:
5879:
5356:
5246:
5180:, February 18â24, 2001, Mexico City. Lecture Notes in Computer Science N 2004,
4958:
4239:
4184:
Proceedings of the National Academy of Sciences of the United States of America
3219:
3195:
3150:
3091:
3087:
2711:
2041:
1927:
of an empirical distribution to the hypothesized power law distribution with a
1891:
1876:{\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}<\infty \ .}
861:
490:
409:
359:
5017:
4811:
4541:
3747:
3693:
3640:
Efficient Interactive Multicast over DVB-T2 - Utilizing Dynamic SFNs and PARPS
3315:
3246: â Observation that no scientific discovery is named after its discoverer
3180: â Phenomenon in statistics where highest-ranked data points are outliers
196:" accounts for slightly over 3.5% of words (36,411 occurrences), followed by "
6811:
6502:
6250:
5368:
5134:
4819:
4751:
4701:
4635:
4263:
4041:
3965:
3541:
3171:
3067:
3025:
3005:
2936:
2912:
2892:
2822:
2003:
428:
424:
175:{\displaystyle {\text{word frequency}}\propto {\frac {1}{\text{word rank}}}.}
114:
31:
4627:
4255:
4205:
4156:
1773:
The generalized Zipf distribution can be extended to infinitely many items (
255:{\displaystyle {\text{frequency}}\propto {\frac {1}{({\text{rank}}+b)^{a}}}}
5385:
5250:
5152:
4915:
4827:
4240:"Power laws for monkeys typing randomly: the case of unequal probabilities"
4224:
3333:
3192: â Graphical representation of the distribution of income or of wealth
3189:
2780:
486:
183:
4938:
Proceedings of the Ninth Workshop on Building and Using Comparable Corpora
5255:
4579:
4438:
3736:, International Conference on Bioinformatics Computational Biology: 2011.
3600:, Handbook of Income Distribution, vol. 2, Elsevier, pp. 3â65,
3177:
3138:
3090:
and others have suggested the application of Zipf's law for detection of
2768:
2710:
and on the spelling rules used. Some deviation is inevitable because of
602:
344:
137:
53:
5380:
4963:"Why Alien Language Would Stand Out Among All the Noise of the Universe"
4387:
4016:
2682:
354:. It has been found to apply to many other types of data studied in the
5259:
5178:
Conference on Intelligent Text Processing and Computational Linguistics
4643:
4611:
3549:
3517:
3382:
2916:
2846:
497:
415:
Zipf's law has been discovered before Zipf, by the French stenographer
4099:
3839:
3763:
2044:" or "success breeds success"). Such a growth process results in the
5235:
5032:
Handling the Zipf distribution in computerized authorship attribution
5031:
4565:
3183:
3052:
2997:
2764:
1979:
1975:
1943:
382:
5211:
5207:
5050:
3533:
3504:
Selected Studies on the Principle of Relative Frequency in Language.
2168:{\displaystyle f(k;N,q,s)={\frac {1}{C}}\,{\frac {1}{(k+q)^{s}}}.\,}
4878:
4794:
4331:
4176:
3665:
3071:
3013:
2952:
957:
4562:
4369:
4302:
3748:"Random texts exhibit Zipf's-law-like word frequency distribution"
5173:
4728:"Is the Zipf law spurious in explaining city-size distributions?"
2944:
2940:
2920:
2850:
2687:
1939:
1661:{\displaystyle f(k;N,s)={\frac {1}{H_{N,s}}}\,{\frac {1}{k^{s}}}}
654:
63:
5071:"Mystery text's language-like patterns may be an elaborate hoax"
4668:
Arshad, Sidra; Hu, Shougeng; Ashraf, Badar Nadeem (2018-02-15).
5330:
Seeing Around Corners (Artificial societies turn up Zipf's law)
3679:
3518:"The Unity of Nature, Least-Action, and Natural Social Science"
2961:) line, while the early part is closer to the magenta (1/
2858:
4471:
3682:"Emergent Statistical Laws in Single-Cell Transcriptomic Data"
4859:
3928:, 1959, réédité en 1988, traduction anglaise rééditée en 2004
3168: â Word that only appears once in a given text or record
3153: â Bare list of a language's words in corpus linguistics
2896:
1763:{\displaystyle H_{N,s}=\sum _{k=1}^{N}{\frac {1}{k^{s}}}\ .}
4500:
4179:"Least effort and the origins of scaling in human language"
3638:
M. Eriksson, S.M. Hasibur Rahman, F. Fraille, M. Sjöström,
560: . (The function is only defined at integer values of
200:" (28,852). It is often used in the following form, called
2553:{\displaystyle {\frac {\log(P(n)/P(n-1))}{\log(n/(n-1))}}}
2021:
A minimal explanation assumes that words are generated by
528:= 10 on a logâlog scale. The horizontal axis is the index
5099:
Montemurro, Marcelo A.; Zanette, DamiĂĄn H. (2013-06-21).
4476:(second ed.). New York: John Wiley & Sons, Inc.
2879:
Cervantes's Don Quixote, Part I (1605) and Part II (1615)
2290:{\displaystyle f(k;\rho )\approx {\frac {}{k^{\rho +1}}}}
1476:{\displaystyle f(k;N)={\frac {1}{H_{N}}}\,{\frac {1}{k}}}
93:
4157:"Statistical metalinguistics and Zipf/Pareto/Mandelbrot"
2677:
465:. The only mathematical expression Zipf used looks like
5262:
are far more common than biologists realized." (p. 94.)
3224:
Pages displaying short descriptions of redirect targets
3155:
Pages displaying short descriptions of redirect targets
3122:
Pages displaying short descriptions of redirect targets
2698:
close to 1: that is, the most common word occurs about
5396:
Benford's law, Zipf's law, and the Pareto distribution
5170:"Zipf and Heaps Lawsâ Coefficients Depend on Language"
4860:
Moreno-SĂĄnchez, I.; Font-Clos, F.; Corral, A. (2016).
4776:
Verbavatz, Vincent; Barthelemy, Marc (November 2020).
4370:
Ricardo T. Fernholz; Robert Fernholz (December 2020).
3866:. Innsbruck, Austria. pp. 102â108. Archived from
3457:
Foundations of Statistical Natural Language Processing
3436:. Cambridge, Massachusetts: Addison-Wesley. p. 1.
2965:) line while the later part is closer to the cyan (1/(
2028:
Another possible cause for the Zipf distribution is a
1994:
observed that if any of a large class of well-behaved
1566:{\displaystyle H_{N}=\sum _{k=1}^{N}{\frac {1}{k}}\ .}
5326:âAn article on Zipf's law applied to city populations
4862:"Large-scale analysis of Zipf's Law in English texts"
4674:
Physica A: Statistical Mechanics and Its Applications
4115:"On the statistical laws of linguistic distributions"
2459:
2388:
2356:
2334:
2232:
2085:
1807:
1781:
exceeds 1. In that case, the normalization constant
1696:
1589:
1512:
1423:
1316:
1220:
1089:
967:
938:
871:
810:
730:
672:
612:
580:
532:. (The function is only defined at integer values of
444:
320:
294:
268:
210:
150:
132:
The best known instance of Zipf's law applies to the
96:
90:
5449:
3239:
Pages displaying wikidata descriptions as a fallback
3231: â Idea that agents prefer to do what's easiest
3104:
The word-like sign groups of the 15th-century codex
2717:
At the low-frequency end, where the rank approaches
2002:) is expressed in terms of rank and expanded into a
4931:"Parallel Document Identification using Zipf's Law"
4775:
4177:Ramon Ferrer i Cancho & Ricard V. Sole (2003).
3222: â Physicist and science historian (1922â1983)
2036:of an item tends to grow at a rate proportional to
564:. The connecting lines do not indicate continuity.)
536:. The connecting lines do not indicate continuity.)
4316:
4237:
3997:
3174: â Heuristic for distinct words in a document
2552:
2444:
2374:
2340:
2289:
2167:
1875:
1762:
1660:
1565:
1475:
1393:
1294:
1198:
1067:
945:
916:
849:
773:
708:
645:
593:
485:), number of people watching the same TV channel,
457:
332:
306:
280:
254:
174:
5098:
4940:. LREC 2016. PortoroĆŸ, Slovenia. pp. 21â25.
3998:Simkin, M. V.; Roychowdhury, V. P. (2011-05-01).
3782:Zipf, power-laws, and Pareto - a ranking tutorial
3186: â Feature of some statistical distributions
3135: â Pattern of references in science journals
6809:
4726:Gan, Li; Li, Dong; Song, Shunfeng (2006-08-01).
4472:N. L. Johnson; S. Kotz & A. W. Kemp (1992).
4422:"E Pluribus Unum: From Complexity, Universality"
3433:Human Behavior and the Principle of Least Effort
4122:Annales de la Société Scientifique de Bruxelles
3828:
3779:
1886:The infinite item case is characterized by the
4667:
3489:Condon, EDWARD U. "Statistics of vocabulary."
3429:
2977:with three segments: a head, middle, and tail.
2200:The Zipf distribution is sometimes called the
5435:
5168:Alexander Gelbukh and Grigori Sidorov (2001)
4161:SRI International Computer Science Laboratory
3020:increase this percentage up to close to 50%.
2445:{\displaystyle \log _{10}(n+1)-\log _{10}(n)}
5391:Complex systems: Unzipping Zipf's law (2011)
3937:
3376:
3374:
3372:
917:{\displaystyle {\frac {H_{N,s-1}}{H_{N,s}}}}
703:
679:
640:
619:
19:For the linguistics law on word length, see
4992:
4419:
4112:
3775:
3773:
3440:
3384:Applications and explanations of Zipf's law
3349:Handbook of Empirical Economics and Finance
3346:
3293:
3291:
3289:
2193:Zipfian distributions can be obtained from
1969:
1950:to the extent that the plot approximates a
5442:
5428:
4986:
4951:
4501:Johan Gerard van der Galien (2003-11-08).
4238:Conrad, B.; Mitzenmacher, M. (July 2004).
4073:
3478:Relativ frequency of English speech sounds
3425:
3423:
3421:
3407:
3405:
3297:
2204:because it is analogous to the continuous
2055:, systems of exchangeable positive-valued
850:{\displaystyle {\frac {H_{k,s}}{H_{N,s}}}}
774:{\displaystyle {\frac {1/k^{s}}{H_{N,s}}}}
5142:
5124:
5062:
4928:
4905:
4895:
4877:
4793:
4725:
4578:
4527:
4496:
4494:
4437:
4330:
4301:
4295:
4214:
4204:
4089:
4015:
3955:
3926:La science et la théorie de l'information
3664:
3369:
3323:
2164:
2129:
1898:is 1 or less, the normalization constant
1640:
1462:
942:
590:
369:, the concept has been formalized as the
66:scale. The dotted line is the ideal law
5386:Zipf's Law examples and modelling (1985)
5301:
5092:
5043:
5024:
4922:
4853:
4769:
4719:
4661:
4603:
4556:
4521:
4465:
4413:
4363:
4310:
4289:
4170:
4147:
4106:
4067:
3915:
3883:
3877:
3863:Artificial Intelligence and Applications
3844:
3770:
3719:
3673:
3652:
3632:
3567:
3509:
3496:
3455:Christopher D. Manning, Hinrich SchĂŒtze
3340:
3286:
3096:search for extraterrestrial intelligence
3042:
2951:
2681:
1410:elements assigns to the element of rank
37:
25:
5381:Citations and the ZipfâMandelbrot's law
4612:"Zipf's Law for Cities: An Explanation"
4244:IEEE Transactions on Information Theory
4077:IEEE Transactions on Information Theory
3752:IEEE Transactions on Information Theory
3658:
3449:
3418:
3402:
556:= 10. The horizontal axis is the index
350:, and is still an important concept in
6810:
4609:
4491:
3938:Mitzenmacher, Michael (January 2004).
3593:
3506:Harvard, MA: Harvard University Press.
3380:
3036:greater than 1 for convergence of the
2725:Zipf's law plots for several languages
2068:A generalization of Zipf's law is the
709:{\displaystyle k\in \{1,2,\ldots ,N\}}
5423:
4957:
4135:from the original on 15 December 2020
3298:Piantadosi, Steven (March 25, 2014).
2678:Word frequencies in natural languages
646:{\displaystyle N\in \{1,2,3\ldots \}}
473:= constant, which he "borrowed" from
343:This law is named after the American
108:
6792:
3515:
3237: â distribution of size by rank
3028:in English is better described with
1918:
503:
186:of American English text, the word "
5049:Frans J. Van Droogenbroeck (2019):
5030:Frans J. Van Droogenbroeck (2016):
3899:10.36505/ExLing-2020/11/0047/000462
3024:frequencies of the closed class of
3012:, each "word" consists of a single
2752:(1840), and Medieval English (1460)
1406:Formally, the Zipf distribution on
1341:
1245:
1114:
13:
6848:Tails of probability distributions
5352:An analysis of income distribution
5162:
4616:The Quarterly Journal of Economics
4530:Journal of Quantitative Linguistic
3745:
3606:10.1016/B978-0-444-59428-0.00002-3
3575:Zipf distribution of US firm sizes
2051:A similar explanation is based on
1864:
1839:
1414:(counting from 1) the probability
14:
6879:
5307:"Guest Column: Math and the City"
5266:
5212:https://ssrn.com/abstract=2918642
5068:
4474:Univariate Discrete Distributions
3480:. Harvard University Press, 1923.
1934:Zipf's law can be visuallized by
1495:is a normalization constant, the
6791:
6782:
6781:
5335:PlanetMath article on Zipf's law
4947:from the original on 2018-03-23.
4113:Belevitch V (18 December 1959).
3210: â Probability distribution
2928:
2904:
2884:
2872:
2838:
2815:
2792:
2757:
2729:
2319:data satisfying Zipf's law with
545:
543:Cumulative distribution function
517:
121:th entry is often approximately
86:
5081:from the original on 2022-05-18
4842:from the original on 2021-10-29
4778:"The growth equation of cities"
4758:from the original on 2019-04-13
4708:from the original on 2023-10-29
4650:from the original on 2021-10-26
4454:from the original on 2021-08-05
4402:from the original on 2021-01-29
4278:from the original on 2022-10-17
4231:
4056:from the original on 2012-01-29
3991:
3980:from the original on 2023-07-22
3931:
3817:from the original on 2001-11-07
3792:from the original on 2023-04-01
3739:
3708:from the original on 2021-06-17
3622:from the original on 2023-10-29
3587:
3556:from the original on 2022-11-20
3483:
3391:from the original on 2015-09-10
3351:, CRC Press, pp. 133â153,
3081:
2214:continuous uniform distribution
2063:
1942:graph, with the axes being the
373:: a family of related discrete
4376:Journal of Applied Probability
3957:10.1080/15427951.2004.10129088
3516:Zipf, George Kingsley (1942).
3470:
3267:
2833:, all with separated syllables
2663:
2544:
2541:
2529:
2518:
2507:
2504:
2492:
2481:
2475:
2469:
2439:
2433:
2414:
2402:
2366:
2360:
2309:parabolic fractal distribution
2265:
2257:
2248:
2236:
2149:
2136:
2113:
2089:
1817:
1811:
1611:
1593:
1439:
1427:
1193:
1174:
1149:
1143:
408:In 1913, the German physicist
385:relation. They are related to
240:
225:
1:
5002:. Penguin. pp. 251â256.
4744:10.1016/j.econlet.2006.03.004
4597:10.1016/S0378-4371(00)00633-6
4372:"Zipf's law for atlas models"
4349:10.1080/02664763.2015.1047744
4319:Journal of Applied Statistics
4034:10.1016/j.physrep.2010.12.004
3413:The Psychobiology of Language
3280:
2668:
2210:discrete uniform distribution
2197:by an exchange of variables.
1938:the item frequency data on a
5292:Resources in other libraries
5201:Applied Economics Letters 10
5126:10.1371/journal.pone.0066344
4897:10.1371/journal.pone.0147073
3255:Most common words in English
2219:The tail frequencies of the
2202:discrete Pareto distribution
2032:process, in which the value
458:{\displaystyle {\sqrt {-i}}}
333:{\displaystyle b\approx 2.7}
288:are fitted parameters, with
7:
5408:Encyclopedia of Mathematics
5196:, Springer-Verlag: 332â335.
4694:10.1016/j.physa.2017.10.005
3111:
3038:Generalized Harmonic Series
2323:= 1 satisfy Benford's law.
1686:generalized harmonic number
379:rank-frequency distribution
10:
6884:
6615:Wrapped asymmetric Laplace
5586:Extended negative binomial
3884:Skotarek, Dariusz (2020).
3725:Ramu Chenna, Toby Gibson;
3381:Powers, David M W (1998).
3162: â Economic principle
3118:1% rule (Internet culture)
2810:(500-800), all with vowels
1777:= â) only if the exponent
403:
307:{\displaystyle a\approx 1}
21:Zipf's law of abbreviation
18:
6823:Computational linguistics
6777:
6711:
6669:
6570:
6406:
6384:
6375:
6274:Generalized extreme value
6259:
6094:
6054:Relativistic BreitâWigner
5770:
5667:
5658:
5551:
5471:
5462:
5451:Probability distributions
5357:Zipf List of French words
5287:Resources in your library
4929:Mohammadi, Mehdi (2016).
4812:10.1038/s41586-020-2900-x
4542:10.1080/09296170600850106
3694:10.1101/2021.06.16.448706
3573:Axtell, Robert L (2001):
3316:10.3758/s13423-014-0585-6
3229:Principle of least effort
2208:in the same way that the
2076:, whose frequencies are:
2015:principle of least effort
1996:statistical distributions
1310:
1305:
1214:
1209:
1083:
1078:
961:
956:
932:
927:
865:
860:
804:
799:
724:
719:
666:
661:
594:{\displaystyle s\geq 0\,}
574:
569:
541:
515:Probability mass function
513:
496:In 1992 bioinformatician
375:probability distributions
6853:Quantitative linguistics
3780:Adamic, Lada A. (2000).
3766:– via IEEE Xplore.
3493:67.1733 (1928): 300-300.
3260:
3244:Stigler's law of eponymy
2935:First four books of the
2911:First five books of the
2899:) in Hebrew, with vowels
2891:First five books of the
2314:It has been argued that
1970:Statistical explanations
352:quantitative linguistics
16:Probability distribution
6269:Generalized chi-squared
6213:Normal-inverse Gaussian
4628:10.1162/003355399556133
4610:Gabaix, Xavier (1999).
4256:10.1109/TIT.2004.830752
4206:10.1073/pnas.0335980100
3887:Zipf's law in Toki Pona
3502:George K. Zipf (1932):
3430:George K. Zipf (1949).
3411:George K. Zipf (1935):
3145:Demographic gravitation
3141: â Linguistics law
2221:YuleâSimon distribution
2046:YuleâSimon distribution
2030:preferential attachment
2023:monkeys typing randomly
1958:) function with slope â
1929:KolmogorovâSmirnov test
1796:Riemann's zeta function
1580:instead of 1. Namely,
423:(4th ed) in 1916, with
367:mathematical statistics
6818:Discrete distributions
6581:Univariate (circular)
6142:Generalized hyperbolic
5571:ConwayâMaxwellâPoisson
5561:Beta negative binomial
3803:"originally published"
3235:Rank-size distribution
3204: â Linguistic law
3059:
2978:
2691:
2554:
2446:
2376:
2342:
2291:
2169:
1877:
1843:
1764:
1736:
1662:
1567:
1546:
1477:
1395:
1360:
1296:
1264:
1200:
1133:
1069:
947:
918:
851:
775:
710:
647:
595:
459:
421:Gammes Stenographiques
334:
308:
282:
256:
176:
136:of words in a text or
123:inversely proportional
78:
35:
6626:Bivariate (spherical)
6124:Kaniadakis Îș-Gaussian
5210:, Available at SSRN:
4000:"Re-inventing Willis"
3688:: 2021â06.16.448706.
3147: â Social effect
3046:
2955:
2855:Russian Synodal Bible
2685:
2555:
2447:
2377:
2375:{\displaystyle P(n)=}
2343:
2292:
2184:Hurwitz zeta function
2170:
1878:
1823:
1765:
1716:
1663:
1568:
1526:
1478:
1396:
1340:
1297:
1244:
1201:
1113:
1070:
948:
919:
852:
776:
711:
648:
596:
477:'s 1926 publication.
460:
335:
309:
283:
257:
177:
59:The War of the Worlds
41:
29:
6691:Dirac delta function
6638:Bivariate (toroidal)
6595:Univariate von Mises
6466:Multivariate Laplace
6358:Shifted log-logistic
5707:Continuous Bernoulli
4439:10.1162/DAED_a_00158
4420:Terence Tao (2012).
3944:Internet Mathematics
3746:Li, Wentian (1992).
3459:, MIT Press (1999),
3273:as Zipf acknowledged
3064:substitution ciphers
3051:in plain text, in a
2706:th most common one.
2457:
2386:
2354:
2332:
2230:
2212:is analogous to the
2195:Pareto distributions
2083:
1805:
1694:
1587:
1510:
1421:
1314:
1218:
1087:
965:
936:
869:
808:
728:
670:
610:
578:
442:
435:ibidem, p. 21:
417:Jean-Baptiste Estoup
371:Zipfian distribution
348:George Kingsley Zipf
318:
292:
266:
208:
148:
6739:Natural exponential
6644:Bivariate von Mises
6610:Wrapped exponential
6476:Multivariate stable
6471:Multivariate normal
5792:Benktander 2nd kind
5787:Benktander 1st kind
5576:Discrete phase-type
5231:Scientific American
5117:2013PLoSO...866344M
4888:2016PLoSO..1147073M
4804:2020Natur.587..397V
4686:2018PhyA..492...75A
4589:2001PhyA..293..297P
4388:10.1017/jpr.2020.64
4341:2015JApSt..42.2686V
4197:2003PNAS..100..788C
4026:2011PhR...502....1S
3415:. Houghton-Mifflin.
3208:Pareto distribution
3000:languages, such as
2206:Pareto distribution
2070:ZipfâMandelbrot law
2057:diffusion processes
2042:the rich get richer
2000:normal distribution
1915:tends to infinity.
1062:
1041:
946:{\displaystyle 1\,}
510:
391:Pareto distribution
281:{\displaystyle a,b}
202:Zipf-Mandelbrot law
6868:1949 introductions
6863:Corpus linguistics
6394:Rectified Gaussian
6279:Generalized Pareto
6137:Generalized normal
6009:Matrix-exponential
5374:2011-04-08 at the
5362:2007-06-23 at the
5345:2004-10-24 at the
5311:The New York Times
5206:Shyklo A. (2017);
5056:2023-09-30 at the
5037:2023-10-04 at the
4968:Nautilus Quarterly
4959:Doyle, Laurance R.
3893:. ExLing Society.
3807:www.parc.xerox.com
3732:2014-03-06 at the
3645:2014-05-02 at the
3580:2023-10-17 at the
3106:Voynich Manuscript
3060:
2987:grammatical gender
2979:
2692:
2550:
2442:
2372:
2338:
2300:for any choice of
2287:
2223:are approximately
2165:
1894:. If the exponent
1873:
1760:
1658:
1563:
1473:
1391:
1292:
1196:
1065:
1042:
1015:
943:
914:
847:
771:
706:
643:
591:
508:
455:
427:in 1923, and with
330:
304:
278:
252:
172:
79:
36:
6805:
6804:
6402:
6401:
6371:
6370:
6262:whose type varies
6208:Normal (Gaussian)
6162:Hyperbolic secant
6111:Exponential power
6014:MaxwellâBoltzmann
5762:Wigner semicircle
5654:
5653:
5626:Parabolic fractal
5616:Negative binomial
5273:Library resources
5243:McGill University
5009:978-1-9848-8197-7
4994:Kershenbaum, Arik
4788:(7834): 397â401.
4732:Economics Letters
4483:978-0-471-54897-3
4325:(12): 2686â2693.
4154:Neumann, Peter G.
4100:10.1109/18.165464
3908:978-618-84585-1-2
3840:10.1137/070710111
3811:Xerox Corporation
3764:10.1109/18.165464
3615:978-0-444-59430-3
3465:978-0-262-13360-9
3049:War of the Worlds
2983:Romance languages
2661:
2660:
2548:
2341:{\displaystyle n}
2285:
2263:
2159:
2127:
2074:Benoit Mandelbrot
1954:(more precisely,
1919:Empirical testing
1888:Zeta distribution
1869:
1859:
1756:
1752:
1656:
1638:
1559:
1555:
1471:
1460:
1404:
1403:
1389:
1338:
1290:
1242:
1163:
1111:
1063:
1008:
912:
845:
769:
565:
537:
504:Formal definition
453:
250:
231:
214:
167:
166:
154:
6875:
6833:Statistical laws
6795:
6794:
6785:
6784:
6724:Compound Poisson
6699:
6687:
6656:von MisesâFisher
6652:
6640:
6628:
6590:Circular uniform
6586:
6506:
6450:
6421:
6382:
6381:
6284:MarchenkoâPastur
6147:Geometric stable
6064:Truncated normal
5957:Inverse Gaussian
5863:Hyperexponential
5702:Beta rectangular
5670:bounded interval
5665:
5664:
5533:Discrete uniform
5518:Poisson binomial
5469:
5468:
5444:
5437:
5430:
5421:
5420:
5416:
5325:
5323:
5322:
5313:. Archived from
5303:Strogatz, Steven
5222:Jen Christiansen
5157:
5156:
5146:
5128:
5096:
5090:
5089:
5087:
5086:
5069:Boyle, Rebecca.
5066:
5060:
5047:
5041:
5028:
5022:
5021:
4990:
4984:
4983:
4981:
4980:
4971:. Archived from
4955:
4949:
4948:
4946:
4935:
4926:
4920:
4919:
4909:
4899:
4881:
4857:
4851:
4850:
4848:
4847:
4797:
4773:
4767:
4766:
4764:
4763:
4723:
4717:
4716:
4714:
4713:
4665:
4659:
4658:
4656:
4655:
4607:
4601:
4600:
4582:
4580:cond-mat/9808305
4573:(1â2): 297â304.
4560:
4554:
4553:
4536:(2â3): 177â193.
4525:
4519:
4518:
4516:
4514:
4505:. Archived from
4498:
4489:
4487:
4469:
4463:
4462:
4460:
4459:
4441:
4417:
4411:
4410:
4408:
4407:
4382:(4): 1276â1297.
4367:
4361:
4360:
4334:
4314:
4308:
4307:
4305:
4293:
4287:
4286:
4284:
4283:
4250:(7): 1403â1414.
4235:
4229:
4228:
4218:
4208:
4174:
4168:
4151:
4145:
4144:
4142:
4140:
4134:
4119:
4110:
4104:
4103:
4093:
4084:(6): 1842â1845.
4071:
4065:
4064:
4062:
4061:
4019:
3995:
3989:
3988:
3986:
3985:
3959:
3935:
3929:
3919:
3913:
3912:
3892:
3881:
3875:
3874:
3873:on 5 March 2016.
3872:
3859:
3848:
3842:
3832:
3826:
3825:
3823:
3822:
3800:
3798:
3797:
3777:
3768:
3767:
3758:(6): 1842â1845.
3743:
3737:
3723:
3717:
3716:
3714:
3713:
3677:
3671:
3670:
3668:
3656:
3650:
3636:
3630:
3629:
3628:
3627:
3591:
3585:
3571:
3565:
3564:
3562:
3561:
3513:
3507:
3500:
3494:
3487:
3481:
3476:Dewey, Godfrey.
3474:
3468:
3453:
3447:
3444:
3438:
3437:
3427:
3416:
3409:
3400:
3399:
3397:
3396:
3378:
3367:
3361:
3344:
3338:
3337:
3327:
3310:(5): 1112â1130.
3304:Psychon Bull Rev
3295:
3274:
3271:
3250:Letter frequency
3240:
3225:
3214:Pareto principle
3156:
3123:
2975:broken power law
2932:
2908:
2888:
2876:
2866:Majority version
2845:Biblical texts:
2842:
2819:
2802:(14th century),
2796:
2761:
2733:
2720:
2697:
2559:
2557:
2556:
2551:
2549:
2547:
2528:
2510:
2488:
2461:
2451:
2449:
2448:
2443:
2429:
2428:
2398:
2397:
2381:
2379:
2378:
2373:
2347:
2345:
2344:
2339:
2326:
2325:
2322:
2296:
2294:
2293:
2288:
2286:
2284:
2283:
2268:
2264:
2261:
2255:
2181:
2174:
2172:
2171:
2166:
2160:
2158:
2157:
2156:
2131:
2128:
2120:
2039:
2035:
2008:Mandelbrot's law
1992:Vitold Belevitch
1965:
1962:. For exponent
1961:
1949:
1914:
1909:
1905:
1901:
1897:
1882:
1880:
1879:
1874:
1867:
1860:
1858:
1857:
1845:
1842:
1837:
1792:
1788:
1784:
1780:
1776:
1769:
1767:
1766:
1761:
1754:
1753:
1751:
1750:
1738:
1735:
1730:
1712:
1711:
1682:
1678:
1674:
1667:
1665:
1664:
1659:
1657:
1655:
1654:
1642:
1639:
1637:
1636:
1618:
1579:
1572:
1570:
1569:
1564:
1557:
1556:
1548:
1545:
1540:
1522:
1521:
1498:
1493:
1489:
1482:
1480:
1479:
1474:
1472:
1464:
1461:
1459:
1458:
1446:
1413:
1409:
1400:
1398:
1397:
1392:
1390:
1388:
1387:
1378:
1377:
1362:
1359:
1354:
1339:
1337:
1336:
1318:
1301:
1299:
1298:
1293:
1291:
1289:
1288:
1279:
1278:
1266:
1263:
1258:
1243:
1241:
1240:
1222:
1205:
1203:
1202:
1197:
1192:
1191:
1164:
1162:
1161:
1152:
1135:
1132:
1127:
1112:
1110:
1109:
1091:
1074:
1072:
1071:
1066:
1064:
1061:
1056:
1040:
1035:
1014:
1009:
1007:
1006:
991:
990:
969:
952:
950:
949:
944:
923:
921:
920:
915:
913:
911:
910:
895:
894:
873:
856:
854:
853:
848:
846:
844:
843:
828:
827:
812:
780:
778:
777:
772:
770:
768:
767:
752:
751:
750:
741:
732:
715:
713:
712:
707:
652:
650:
649:
644:
600:
598:
597:
592:
551:
549:
523:
521:
511:
507:
489:in music, cells
483:Pareto principle
464:
462:
461:
456:
454:
446:
339:
337:
336:
331:
313:
311:
310:
305:
287:
285:
284:
279:
261:
259:
258:
253:
251:
249:
248:
247:
232:
229:
220:
215:
212:
181:
179:
178:
173:
168:
164:
160:
155:
152:
142:natural language
112:
110:[tÍĄsÉȘpf]
107:
103:
102:
99:
98:
95:
92:
6883:
6882:
6878:
6877:
6876:
6874:
6873:
6872:
6808:
6807:
6806:
6801:
6773:
6749:Maximum entropy
6707:
6695:
6683:
6673:
6665:
6648:
6636:
6624:
6579:
6566:
6503:Matrix-valued:
6500:
6446:
6417:
6409:
6398:
6386:
6377:
6367:
6261:
6255:
6172:
6098:
6096:
6090:
6019:MaxwellâJĂŒttner
5868:Hypoexponential
5774:
5772:
5771:supported on a
5766:
5727:Noncentral beta
5687:BaldingâNichols
5669:
5668:supported on a
5660:
5650:
5553:
5547:
5543:ZipfâMandelbrot
5473:
5464:
5458:
5448:
5401:
5398:by Terence Tao.
5376:Wayback Machine
5364:Wayback Machine
5347:Wayback Machine
5320:
5318:
5298:
5297:
5296:
5281:
5280:
5276:
5269:
5247:red blood cells
5218:Clara Moskowitz
5165:
5163:Further reading
5160:
5097:
5093:
5084:
5082:
5067:
5063:
5058:Wayback Machine
5048:
5044:
5039:Wayback Machine
5029:
5025:
5010:
4991:
4987:
4978:
4976:
4956:
4952:
4944:
4933:
4927:
4923:
4872:(1): e0147073.
4858:
4854:
4845:
4843:
4774:
4770:
4761:
4759:
4724:
4720:
4711:
4709:
4666:
4662:
4653:
4651:
4608:
4604:
4561:
4557:
4526:
4522:
4512:
4510:
4499:
4492:
4484:
4470:
4466:
4457:
4455:
4418:
4414:
4405:
4403:
4368:
4364:
4315:
4311:
4294:
4290:
4281:
4279:
4236:
4232:
4175:
4171:
4163:, accessed and
4152:
4148:
4138:
4136:
4132:
4117:
4111:
4107:
4091:10.1.1.164.8422
4072:
4068:
4059:
4057:
4017:physics/0601192
4004:Physics Reports
3996:
3992:
3983:
3981:
3936:
3932:
3920:
3916:
3909:
3890:
3882:
3878:
3870:
3857:
3849:
3845:
3833:
3829:
3820:
3818:
3801:
3795:
3793:
3786:Hewlett-Packard
3778:
3771:
3744:
3740:
3734:Wayback Machine
3724:
3720:
3711:
3709:
3678:
3674:
3657:
3653:
3647:Wayback Machine
3637:
3633:
3625:
3623:
3616:
3592:
3588:
3582:Wayback Machine
3572:
3568:
3559:
3557:
3534:10.2307/2784953
3514:
3510:
3501:
3497:
3488:
3484:
3475:
3471:
3454:
3450:
3445:
3441:
3428:
3419:
3410:
3403:
3394:
3392:
3379:
3370:
3359:
3345:
3341:
3296:
3287:
3283:
3278:
3277:
3272:
3268:
3263:
3238:
3223:
3202:Menzerath's law
3166:Hapax legomenon
3154:
3121:
3114:
3084:
3076:VigenĂšre cipher
3057:VigenĂšre cipher
2948:
2943:) in the Latin
2933:
2924:
2919:) in the Latin
2909:
2900:
2889:
2880:
2877:
2868:
2863:Byzantine Greek
2849:from the Latin
2843:
2834:
2820:
2811:
2797:
2788:
2762:
2753:
2734:
2718:
2695:
2680:
2671:
2666:
2524:
2511:
2484:
2462:
2460:
2458:
2455:
2454:
2424:
2420:
2393:
2389:
2387:
2384:
2383:
2382:
2355:
2352:
2351:
2350:Benford's law:
2333:
2330:
2329:
2320:
2273:
2269:
2260:
2256:
2254:
2231:
2228:
2227:
2179:
2152:
2148:
2135:
2130:
2119:
2084:
2081:
2080:
2066:
2040:(intuitively, "
2037:
2033:
1972:
1963:
1959:
1947:
1925:goodness of fit
1921:
1912:
1910:
1907:
1903:
1899:
1895:
1853:
1849:
1844:
1838:
1827:
1806:
1803:
1802:
1793:
1790:
1786:
1782:
1778:
1774:
1746:
1742:
1737:
1731:
1720:
1701:
1697:
1695:
1692:
1691:
1683:
1680:
1676:
1672:
1650:
1646:
1641:
1626:
1622:
1617:
1588:
1585:
1584:
1577:
1547:
1541:
1530:
1517:
1513:
1511:
1508:
1507:
1501:harmonic number
1496:
1494:
1491:
1487:
1463:
1454:
1450:
1445:
1422:
1419:
1418:
1411:
1407:
1383:
1379:
1367:
1363:
1361:
1355:
1344:
1326:
1322:
1317:
1315:
1312:
1311:
1284:
1280:
1271:
1267:
1265:
1259:
1248:
1230:
1226:
1221:
1219:
1216:
1215:
1181:
1177:
1157:
1153:
1136:
1134:
1128:
1117:
1099:
1095:
1090:
1088:
1085:
1084:
1057:
1046:
1036:
1019:
1013:
996:
992:
974:
970:
968:
966:
963:
962:
937:
934:
933:
900:
896:
878:
874:
872:
870:
867:
866:
833:
829:
817:
813:
811:
809:
806:
805:
794:harmonic number
792:th generalized
786:
757:
753:
746:
742:
737:
733:
731:
729:
726:
725:
671:
668:
667:
611:
608:
607:
606:
579:
576:
575:
550:
544:
522:
516:
506:
475:Alfred J. Lotka
445:
443:
440:
439:
406:
319:
316:
315:
293:
290:
289:
267:
264:
263:
243:
239:
228:
224:
219:
211:
209:
206:
205:
159:
151:
149:
146:
145:
134:frequency table
105:
89:
85:
49:Complete Herbal
24:
17:
12:
11:
5:
6881:
6871:
6870:
6865:
6860:
6855:
6850:
6845:
6840:
6838:Empirical laws
6835:
6830:
6825:
6820:
6803:
6802:
6800:
6799:
6789:
6778:
6775:
6774:
6772:
6771:
6766:
6761:
6756:
6751:
6746:
6744:Locationâscale
6741:
6736:
6731:
6726:
6721:
6715:
6713:
6709:
6708:
6706:
6705:
6700:
6693:
6688:
6680:
6678:
6667:
6666:
6664:
6663:
6658:
6653:
6646:
6641:
6634:
6629:
6622:
6617:
6612:
6607:
6605:Wrapped Cauchy
6602:
6600:Wrapped normal
6597:
6592:
6587:
6576:
6574:
6568:
6567:
6565:
6564:
6563:
6562:
6557:
6555:Normal-inverse
6552:
6547:
6537:
6536:
6535:
6525:
6517:
6512:
6507:
6498:
6497:
6496:
6486:
6478:
6473:
6468:
6463:
6462:
6461:
6451:
6444:
6443:
6442:
6437:
6427:
6422:
6414:
6412:
6404:
6403:
6400:
6399:
6397:
6396:
6390:
6388:
6379:
6373:
6372:
6369:
6368:
6366:
6365:
6360:
6355:
6347:
6339:
6331:
6322:
6313:
6304:
6295:
6286:
6281:
6276:
6271:
6265:
6263:
6257:
6256:
6254:
6253:
6248:
6246:Variance-gamma
6243:
6238:
6230:
6225:
6220:
6215:
6210:
6205:
6197:
6192:
6191:
6190:
6180:
6175:
6170:
6164:
6159:
6154:
6149:
6144:
6139:
6134:
6126:
6121:
6113:
6108:
6102:
6100:
6092:
6091:
6089:
6088:
6086:Wilks's lambda
6083:
6082:
6081:
6071:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
6026:
6024:Mittag-Leffler
6021:
6016:
6011:
6006:
6001:
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5965:
5964:
5954:
5945:
5940:
5935:
5934:
5933:
5923:
5921:gamma/Gompertz
5918:
5917:
5916:
5911:
5901:
5896:
5891:
5890:
5889:
5877:
5876:
5875:
5870:
5865:
5855:
5854:
5853:
5843:
5838:
5833:
5832:
5831:
5830:
5829:
5819:
5809:
5804:
5799:
5794:
5789:
5784:
5778:
5776:
5773:semi-infinite
5768:
5767:
5765:
5764:
5759:
5754:
5749:
5744:
5739:
5734:
5729:
5724:
5719:
5714:
5709:
5704:
5699:
5694:
5689:
5684:
5679:
5673:
5671:
5662:
5656:
5655:
5652:
5651:
5649:
5648:
5643:
5638:
5633:
5628:
5623:
5618:
5613:
5608:
5603:
5598:
5593:
5588:
5583:
5578:
5573:
5568:
5563:
5557:
5555:
5552:with infinite
5549:
5548:
5546:
5545:
5540:
5535:
5530:
5525:
5520:
5515:
5514:
5513:
5506:Hypergeometric
5503:
5498:
5493:
5488:
5483:
5477:
5475:
5466:
5460:
5459:
5447:
5446:
5439:
5432:
5424:
5418:
5417:
5399:
5393:
5388:
5383:
5378:
5366:
5354:
5349:
5337:
5332:
5327:
5305:(2009-05-29).
5295:
5294:
5289:
5283:
5282:
5278:Zipf's law
5271:
5270:
5268:
5267:External links
5265:
5264:
5263:
5215:
5204:
5197:
5164:
5161:
5159:
5158:
5091:
5061:
5042:
5023:
5008:
4996:(2021-03-16).
4985:
4961:(2016-11-18).
4950:
4921:
4852:
4768:
4738:(2): 256â262.
4718:
4660:
4622:(3): 739â767.
4602:
4555:
4520:
4490:
4482:
4464:
4412:
4362:
4309:
4288:
4230:
4191:(3): 788â791.
4169:
4146:
4105:
4066:
3990:
3950:(2): 226â251.
3930:
3922:LĂ©on Brillouin
3914:
3907:
3876:
3843:
3827:
3769:
3738:
3718:
3672:
3651:
3631:
3614:
3586:
3566:
3508:
3495:
3482:
3469:
3448:
3439:
3417:
3401:
3368:
3357:
3339:
3284:
3282:
3279:
3276:
3275:
3265:
3264:
3262:
3259:
3258:
3257:
3252:
3247:
3241:
3232:
3226:
3217:
3211:
3205:
3199:
3193:
3187:
3181:
3175:
3169:
3163:
3157:
3151:Frequency list
3148:
3142:
3136:
3133:Bradford's law
3130:
3124:
3113:
3110:
3092:alien language
3088:Laurance Doyle
3083:
3080:
3068:Caesar ciphers
3026:function words
2950:
2949:
2934:
2927:
2925:
2910:
2903:
2901:
2890:
2883:
2881:
2878:
2871:
2869:
2844:
2837:
2835:
2821:
2814:
2812:
2798:
2791:
2789:
2763:
2756:
2754:
2735:
2728:
2726:
2712:sampling error
2679:
2676:
2670:
2667:
2665:
2662:
2659:
2658:
2655:
2652:
2648:
2647:
2644:
2641:
2637:
2636:
2633:
2630:
2626:
2625:
2622:
2619:
2615:
2614:
2611:
2608:
2604:
2603:
2600:
2597:
2593:
2592:
2589:
2586:
2582:
2581:
2578:
2575:
2571:
2570:
2568:
2565:
2561:
2560:
2546:
2543:
2540:
2537:
2534:
2531:
2527:
2523:
2520:
2517:
2514:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2487:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2452:
2441:
2438:
2435:
2432:
2427:
2423:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2396:
2392:
2371:
2368:
2365:
2362:
2359:
2348:
2337:
2298:
2297:
2282:
2279:
2276:
2272:
2267:
2259:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2176:
2175:
2163:
2155:
2151:
2147:
2144:
2141:
2138:
2134:
2126:
2123:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2072:, proposed by
2065:
2062:
1998:(not only the
1971:
1968:
1920:
1917:
1902:
1890:and is called
1884:
1883:
1872:
1866:
1863:
1856:
1852:
1848:
1841:
1836:
1833:
1830:
1826:
1822:
1819:
1816:
1813:
1810:
1785:
1771:
1770:
1759:
1749:
1745:
1741:
1734:
1729:
1726:
1723:
1719:
1715:
1710:
1707:
1704:
1700:
1675:
1669:
1668:
1653:
1649:
1645:
1635:
1632:
1629:
1625:
1621:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1574:
1573:
1562:
1554:
1551:
1544:
1539:
1536:
1533:
1529:
1525:
1520:
1516:
1490:
1484:
1483:
1470:
1467:
1457:
1453:
1449:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1402:
1401:
1386:
1382:
1376:
1373:
1370:
1366:
1358:
1353:
1350:
1347:
1343:
1335:
1332:
1329:
1325:
1321:
1309:
1303:
1302:
1287:
1283:
1277:
1274:
1270:
1262:
1257:
1254:
1251:
1247:
1239:
1236:
1233:
1229:
1225:
1213:
1207:
1206:
1195:
1190:
1187:
1184:
1180:
1176:
1173:
1170:
1167:
1160:
1156:
1151:
1148:
1145:
1142:
1139:
1131:
1126:
1123:
1120:
1116:
1108:
1105:
1102:
1098:
1094:
1082:
1076:
1075:
1060:
1055:
1052:
1049:
1045:
1039:
1034:
1031:
1028:
1025:
1022:
1018:
1012:
1005:
1002:
999:
995:
989:
986:
983:
980:
977:
973:
960:
954:
953:
941:
931:
925:
924:
909:
906:
903:
899:
893:
890:
887:
884:
881:
877:
864:
858:
857:
842:
839:
836:
832:
826:
823:
820:
816:
803:
797:
796:
784:
766:
763:
760:
756:
749:
745:
740:
736:
723:
717:
716:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
665:
659:
658:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
589:
586:
583:
573:
567:
566:
542:
539:
538:
514:
505:
502:
491:transcriptomes
452:
449:
410:Felix Auerbach
405:
402:
381:is an inverse
329:
326:
323:
303:
300:
297:
277:
274:
271:
246:
242:
238:
235:
227:
223:
218:
171:
163:
158:
153:word frequency
30:Zipf's Law on
15:
9:
6:
4:
3:
2:
6880:
6869:
6866:
6864:
6861:
6859:
6858:Bibliometrics
6856:
6854:
6851:
6849:
6846:
6844:
6841:
6839:
6836:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6815:
6813:
6798:
6790:
6788:
6780:
6779:
6776:
6770:
6767:
6765:
6762:
6760:
6757:
6755:
6752:
6750:
6747:
6745:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6716:
6714:
6710:
6704:
6701:
6698:
6694:
6692:
6689:
6686:
6682:
6681:
6679:
6677:
6672:
6668:
6662:
6659:
6657:
6654:
6651:
6647:
6645:
6642:
6639:
6635:
6633:
6630:
6627:
6623:
6621:
6618:
6616:
6613:
6611:
6608:
6606:
6603:
6601:
6598:
6596:
6593:
6591:
6588:
6585:
6584:
6578:
6577:
6575:
6573:
6569:
6561:
6558:
6556:
6553:
6551:
6548:
6546:
6543:
6542:
6541:
6538:
6534:
6531:
6530:
6529:
6526:
6524:
6523:
6518:
6516:
6515:Matrix normal
6513:
6511:
6508:
6505:
6504:
6499:
6495:
6492:
6491:
6490:
6487:
6485:
6484:
6481:Multivariate
6479:
6477:
6474:
6472:
6469:
6467:
6464:
6460:
6457:
6456:
6455:
6452:
6449:
6445:
6441:
6438:
6436:
6433:
6432:
6431:
6428:
6426:
6423:
6420:
6416:
6415:
6413:
6411:
6408:Multivariate
6405:
6395:
6392:
6391:
6389:
6383:
6380:
6374:
6364:
6361:
6359:
6356:
6354:
6352:
6348:
6346:
6344:
6340:
6338:
6336:
6332:
6330:
6328:
6323:
6321:
6319:
6314:
6312:
6310:
6305:
6303:
6301:
6296:
6294:
6292:
6287:
6285:
6282:
6280:
6277:
6275:
6272:
6270:
6267:
6266:
6264:
6260:with support
6258:
6252:
6249:
6247:
6244:
6242:
6239:
6237:
6236:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6203:
6198:
6196:
6193:
6189:
6186:
6185:
6184:
6181:
6179:
6176:
6174:
6173:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6133:
6132:
6127:
6125:
6122:
6120:
6119:
6114:
6112:
6109:
6107:
6104:
6103:
6101:
6097:on the whole
6093:
6087:
6084:
6080:
6077:
6076:
6075:
6072:
6070:
6069:type-2 Gumbel
6067:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
6000:
5997:
5995:
5992:
5990:
5987:
5985:
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5963:
5960:
5959:
5958:
5955:
5953:
5951:
5946:
5944:
5941:
5939:
5938:Half-logistic
5936:
5932:
5929:
5928:
5927:
5924:
5922:
5919:
5915:
5912:
5910:
5907:
5906:
5905:
5902:
5900:
5897:
5895:
5894:Folded normal
5892:
5888:
5885:
5884:
5883:
5882:
5878:
5874:
5871:
5869:
5866:
5864:
5861:
5860:
5859:
5856:
5852:
5849:
5848:
5847:
5844:
5842:
5839:
5837:
5834:
5828:
5825:
5824:
5823:
5820:
5818:
5815:
5814:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5779:
5777:
5769:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5740:
5738:
5737:Raised cosine
5735:
5733:
5730:
5728:
5725:
5723:
5720:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5675:
5674:
5672:
5666:
5663:
5657:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5612:
5611:Mixed Poisson
5609:
5607:
5604:
5602:
5599:
5597:
5594:
5592:
5589:
5587:
5584:
5582:
5579:
5577:
5574:
5572:
5569:
5567:
5564:
5562:
5559:
5558:
5556:
5550:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5526:
5524:
5521:
5519:
5516:
5512:
5509:
5508:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5491:Beta-binomial
5489:
5487:
5484:
5482:
5479:
5478:
5476:
5470:
5467:
5461:
5456:
5452:
5445:
5440:
5438:
5433:
5431:
5426:
5425:
5422:
5414:
5410:
5409:
5404:
5400:
5397:
5394:
5392:
5389:
5387:
5384:
5382:
5379:
5377:
5373:
5370:
5367:
5365:
5361:
5358:
5355:
5353:
5350:
5348:
5344:
5341:
5338:
5336:
5333:
5331:
5328:
5317:on 2015-09-27
5316:
5312:
5308:
5304:
5300:
5299:
5293:
5290:
5288:
5285:
5284:
5279:
5274:
5261:
5257:
5252:
5248:
5244:
5240:
5239:Ian A. Hatton
5237:
5233:
5232:
5227:
5223:
5219:
5216:
5213:
5209:
5205:
5202:
5198:
5195:
5194:3-540-41687-0
5191:
5187:
5183:
5179:
5175:
5171:
5167:
5166:
5154:
5150:
5145:
5140:
5136:
5132:
5127:
5122:
5118:
5114:
5111:(6): e66344.
5110:
5106:
5102:
5095:
5080:
5076:
5075:New Scientist
5072:
5065:
5059:
5055:
5052:
5046:
5040:
5036:
5033:
5027:
5019:
5015:
5011:
5005:
5001:
5000:
4995:
4989:
4975:on 2020-07-29
4974:
4970:
4969:
4964:
4960:
4954:
4943:
4939:
4932:
4925:
4917:
4913:
4908:
4903:
4898:
4893:
4889:
4885:
4880:
4875:
4871:
4867:
4863:
4856:
4841:
4837:
4833:
4829:
4825:
4821:
4817:
4813:
4809:
4805:
4801:
4796:
4791:
4787:
4783:
4779:
4772:
4757:
4753:
4749:
4745:
4741:
4737:
4733:
4729:
4722:
4707:
4703:
4699:
4695:
4691:
4687:
4683:
4679:
4675:
4671:
4664:
4649:
4645:
4641:
4637:
4633:
4629:
4625:
4621:
4617:
4613:
4606:
4598:
4594:
4590:
4586:
4581:
4576:
4572:
4568:
4567:
4559:
4551:
4547:
4543:
4539:
4535:
4531:
4524:
4509:on 2007-03-05
4508:
4504:
4497:
4495:
4485:
4479:
4475:
4468:
4453:
4449:
4445:
4440:
4435:
4431:
4427:
4423:
4416:
4401:
4397:
4393:
4389:
4385:
4381:
4377:
4373:
4366:
4358:
4354:
4350:
4346:
4342:
4338:
4333:
4328:
4324:
4320:
4313:
4304:
4299:
4292:
4277:
4273:
4269:
4265:
4261:
4257:
4253:
4249:
4245:
4241:
4234:
4226:
4222:
4217:
4212:
4207:
4202:
4198:
4194:
4190:
4186:
4185:
4180:
4173:
4166:
4162:
4158:
4155:
4150:
4131:
4127:
4123:
4116:
4109:
4101:
4097:
4092:
4087:
4083:
4079:
4078:
4070:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4018:
4013:
4009:
4005:
4001:
3994:
3979:
3975:
3971:
3967:
3963:
3958:
3953:
3949:
3945:
3941:
3934:
3927:
3923:
3918:
3910:
3904:
3900:
3896:
3889:
3888:
3880:
3869:
3865:
3864:
3856:
3855:
3847:
3841:
3837:
3831:
3816:
3812:
3808:
3804:
3791:
3787:
3783:
3776:
3774:
3765:
3761:
3757:
3753:
3749:
3742:
3735:
3731:
3728:
3722:
3707:
3703:
3699:
3695:
3691:
3687:
3683:
3676:
3667:
3662:
3655:
3648:
3644:
3641:
3635:
3621:
3617:
3611:
3607:
3603:
3599:
3598:
3590:
3583:
3579:
3576:
3570:
3555:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3512:
3505:
3499:
3492:
3486:
3479:
3473:
3466:
3462:
3458:
3452:
3443:
3435:
3434:
3426:
3424:
3422:
3414:
3408:
3406:
3390:
3386:
3385:
3377:
3375:
3373:
3365:
3360:
3358:9781420070361
3354:
3350:
3343:
3335:
3331:
3326:
3321:
3317:
3313:
3309:
3305:
3301:
3294:
3292:
3290:
3285:
3270:
3266:
3256:
3253:
3251:
3248:
3245:
3242:
3236:
3233:
3230:
3227:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3127:Benford's law
3125:
3119:
3116:
3115:
3109:
3107:
3102:
3099:
3097:
3093:
3089:
3079:
3077:
3073:
3069:
3065:
3058:
3054:
3050:
3045:
3041:
3039:
3035:
3031:
3027:
3021:
3017:
3015:
3011:
3007:
3006:Lhasa Tibetan
3003:
2999:
2994:
2992:
2988:
2984:
2976:
2972:
2969: +
2968:
2964:
2960:
2954:
2946:
2942:
2938:
2937:New Testament
2931:
2926:
2922:
2918:
2914:
2913:Old Testament
2907:
2902:
2898:
2894:
2893:Old Testament
2887:
2882:
2875:
2870:
2867:
2864:
2860:
2856:
2852:
2848:
2841:
2836:
2832:
2828:
2824:
2823:Lhasa Tibetan
2818:
2813:
2809:
2805:
2801:
2795:
2790:
2786:
2782:
2778:
2774:
2770:
2766:
2760:
2755:
2751:
2747:
2743:
2739:
2732:
2727:
2724:
2723:
2722:
2715:
2713:
2707:
2705:
2701:
2689:
2684:
2675:
2656:
2653:
2650:
2649:
2645:
2642:
2639:
2638:
2634:
2631:
2628:
2627:
2623:
2620:
2617:
2616:
2612:
2609:
2606:
2605:
2601:
2598:
2595:
2594:
2590:
2587:
2584:
2583:
2579:
2576:
2573:
2572:
2569:
2566:
2563:
2562:
2538:
2535:
2532:
2525:
2521:
2515:
2512:
2501:
2498:
2495:
2489:
2485:
2478:
2472:
2466:
2463:
2453:
2436:
2430:
2425:
2421:
2417:
2411:
2408:
2405:
2399:
2394:
2390:
2369:
2363:
2357:
2349:
2335:
2328:
2327:
2324:
2317:
2316:Benford's law
2312:
2310:
2305:
2303:
2280:
2277:
2274:
2270:
2251:
2245:
2242:
2239:
2233:
2226:
2225:
2224:
2222:
2217:
2215:
2211:
2207:
2203:
2198:
2196:
2191:
2189:
2186:evaluated at
2185:
2178:The constant
2161:
2153:
2145:
2142:
2139:
2132:
2124:
2121:
2116:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2086:
2079:
2078:
2077:
2075:
2071:
2061:
2058:
2054:
2049:
2047:
2043:
2031:
2026:
2024:
2019:
2016:
2011:
2009:
2005:
2004:Taylor series
2001:
1997:
1993:
1987:
1985:
1981:
1977:
1967:
1957:
1953:
1945:
1941:
1937:
1932:
1930:
1926:
1916:
1893:
1889:
1870:
1861:
1854:
1850:
1846:
1834:
1831:
1828:
1824:
1820:
1814:
1808:
1801:
1800:
1799:
1797:
1757:
1747:
1743:
1739:
1732:
1727:
1724:
1721:
1717:
1713:
1708:
1705:
1702:
1698:
1690:
1689:
1688:
1687:
1651:
1647:
1643:
1633:
1630:
1627:
1623:
1619:
1614:
1608:
1605:
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398:quasi-Zipfian
394:
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387:Benford's law
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6650:Multivariate
6649:
6637:
6625:
6620:Wrapped LĂ©vy
6580:
6528:Matrix gamma
6521:
6501:
6489:Normal-gamma
6482:
6448:Continuous:
6447:
6418:
6363:Tukey lambda
6350:
6342:
6337:-exponential
6334:
6326:
6317:
6308:
6299:
6293:-exponential
6290:
6234:
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6117:
6044:Poly-Weibull
5989:Log-logistic
5949:
5948:Hotelling's
5880:
5722:Logit-normal
5596:GaussâKuzmin
5591:FloryâSchulz
5537:
5472:with finite
5406:
5319:. Retrieved
5315:the original
5310:
5277:
5256:immune cells
5251:muscle cells
5229:
5203:: 717â720(4)
5200:
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4973:the original
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4507:the original
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4167:29 May 2011.
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3269:
3190:Lorenz curve
3160:Gibrat's law
3103:
3100:
3085:
3082:Applications
3070:, or simple
3061:
3048:
3033:
3029:
3022:
3018:
2995:
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2781:Dom Casmurro
2775:, 1605) and
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2192:
2187:
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2067:
2064:Related laws
2053:atlas models
2050:
2027:
2020:
2012:
1988:
1984:Mitzenmacher
1973:
1933:
1922:
1911:diverges as
1885:
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6734:Exponential
6583:directional
6572:Directional
6459:Generalized
6430:Multinomial
6385:continuous-
6325:Kaniadakis
6316:Kaniadakis
6307:Kaniadakis
6298:Kaniadakis
6289:Kaniadakis
6241:TracyâWidom
6218:Skew normal
6200:Noncentral
5984:Log-Laplace
5962:Generalized
5943:Half-normal
5909:Generalized
5873:Logarithmic
5858:Exponential
5812:Chi-squared
5752:U-quadratic
5717:Kumaraswamy
5659:Continuous
5606:Logarithmic
5501:Categorical
5260:lymphocytes
4128:: 310â326.
4010:(1): 1â35.
3364:P. 139
3220:Price's law
3196:Lotka's law
3178:King effect
3139:Brevity law
3066:, like the
3055:, and in a
2857:, the four
2769:Don Quixote
2664:Occurrences
2657:â0.9462848
2654:0.04575749
2646:â0.9397966
2643:0.05115252
2635:â0.9315169
2632:0.05799195
2624:â0.9205788
2621:0.06694679
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2602:â0.8830605
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2591:â0.8463832
2588:0.12493874
2580:â0.7735840
2577:0.17609126
2567:0.30103000
1892:Lotka's law
54:H. G. Wells
52:(1652) and
6828:Power laws
6812:Categories
6729:Elliptical
6685:Degenerate
6671:Degenerate
6419:Discrete:
6378:univariate
6233:Student's
6188:Asymmetric
6167:Johnson's
6095:supported
6039:Phase-type
5994:Log-normal
5979:Log-Cauchy
5969:Kolmogorov
5887:Noncentral
5817:Noncentral
5797:Beta prime
5747:Triangular
5742:Reciprocal
5712:IrwinâHall
5661:univariate
5641:YuleâSimon
5523:Rademacher
5465:univariate
5403:"Zipf law"
5321:2009-05-29
5226:Ni-Ka Ford
5085:2022-02-25
5018:1242873084
4979:2020-08-30
4879:1509.04486
4846:2021-10-26
4795:2011.09403
4762:2021-10-26
4712:2021-10-26
4654:2021-10-26
4458:2021-03-26
4406:2021-03-26
4332:1506.08535
4282:2023-08-20
4060:2023-07-25
3984:2023-07-25
3821:2016-02-23
3796:2023-10-12
3784:(Report).
3712:2021-06-18
3666:cs/0406015
3626:2023-07-11
3560:2022-11-20
3522:Sociometry
3395:2015-02-02
3281:References
3172:Heaps' law
3010:Vietnamese
2998:East Asian
2917:Pentateuch
2847:Pentateuch
2831:Vietnamese
2785:Portuguese
2702:times the
2669:City sizes
571:Parameters
509:Zipf's law
498:Wentian Li
362:sciences.
82:Zipf's law
6454:Dirichlet
6435:Dirichlet
6345:-Gaussian
6320:-Logistic
6157:Holtsmark
6129:Gaussian
6116:Fisher's
6099:real line
5601:Geometric
5581:Delaporte
5486:Bernoulli
5463:Discrete
5413:EMS Press
5236:ecologist
5186:0302-9743
5135:1932-6203
4836:227012701
4820:1476-4687
4752:0165-1765
4702:0378-4371
4680:: 75â92.
4636:0033-5533
4566:Physica A
4488:, p. 466.
4396:146808080
4303:1412.4846
4264:1557-9654
4086:CiteSeerX
4042:0370-1573
3966:1542-7951
3788:Company.
3702:235482777
3542:0038-0431
3184:Long tail
3053:book code
2861:from the
2765:Cervantes
2736:Texts in
2536:−
2516:
2499:−
2467:
2431:
2418:−
2400:
2275:ρ
2252:≈
2246:ρ
1980:Toki Pona
1976:Esperanto
1944:logarithm
1865:∞
1840:∞
1825:∑
1809:ζ
1718:∑
1528:∑
1342:∑
1246:∑
1172:
1141:
1115:∑
1030:−
1011:−
985:−
889:−
695:…
677:∈
638:…
617:∈
585:≥
448:−
431:in 1928.
429:E. Condon
383:power law
325:≈
299:≈
217:∝
213:frequency
165:word rank
157:∝
6787:Category
6719:Circular
6712:Families
6697:Singular
6676:singular
6440:Negative
6387:discrete
6353:-Weibull
6311:-Weibull
6195:Logistic
6079:Discrete
6049:Rayleigh
6029:Nakagami
5952:-squared
5926:Gompertz
5775:interval
5511:Negative
5496:Binomial
5372:Archived
5360:Archived
5343:Archived
5172:. Proc.
5153:23805215
5105:PLOS ONE
5079:Archived
5054:Archived
5035:Archived
4942:Archived
4916:26800025
4866:PLOS ONE
4840:Archived
4828:33208958
4756:Archived
4706:Archived
4648:Archived
4550:17657731
4452:Archived
4448:14535989
4426:Daedalus
4400:Archived
4357:10599428
4276:Archived
4225:12540826
4165:archived
4139:24 April
4130:Archived
4054:Archived
4050:88517297
3978:Archived
3815:Archived
3790:Archived
3730:Archived
3706:Archived
3643:Archived
3620:archived
3578:Archived
3554:Archived
3389:Archived
3334:24664880
3112:See also
3072:codebook
3014:syllable
2996:In many
2981:In some
2806:(~650),
2771:Part I (
2748:(1865),
2744:(1972),
2740:(1669),
2304:> 0.
2262:constant
1936:plotting
1794:becomes
958:Variance
425:G. Dewey
389:and the
356:physical
345:linguist
113:) is an
44:Culpeper
6843:Eponyms
6797:Commons
6769:Wrapped
6764:Tweedie
6759:Pearson
6754:Mixture
6661:Bingham
6560:Complex
6550:Inverse
6540:Wishart
6533:Inverse
6520:Matrix
6494:Inverse
6410:(joint)
6329:-Erlang
6183:Laplace
6074:Weibull
5931:Shifted
5914:Inverse
5899:Fréchet
5822:Inverse
5757:Uniform
5677:Arcsine
5636:Skellam
5631:Poisson
5554:support
5528:Soliton
5481:Benford
5474:support
5415:, 2001
5258:called
5176:-2001,
5174:CICLing
5144:3689824
5113:Bibcode
4907:4723055
4884:Bibcode
4800:Bibcode
4682:Bibcode
4644:2586883
4585:Bibcode
4337:Bibcode
4272:8913575
4193:Bibcode
4022:Bibcode
3974:1671059
3686:bioRxiv
3550:2784953
3491:Science
3467:, p. 24
3325:4176592
3094:in the
3047:Well's
3002:Chinese
2947:version
2945:Vulgate
2941:Gospels
2923:version
2921:Vulgate
2859:Gospels
2851:Vulgate
2827:Chinese
2787:, 1899)
2773:Spanish
2750:Italian
2742:Russian
2688:log-log
2307:In the
2182:is the
1940:log-log
1080:Entropy
788:is the
663:Support
655:integer
481:called
404:History
106:German:
64:log-log
6703:Cantor
6545:Normal
6376:Mixed
6302:-Gamma
6228:Stable
6178:Landau
6152:Gumbel
6106:Cauchy
6034:Pareto
5846:Erlang
5827:Scaled
5782:Benini
5621:Panjer
5275:about
5224:, and
5192:
5184:
5151:
5141:
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5016:
5006:
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4782:Nature
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4513:8 July
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3008:, and
2991:number
2808:Hebrew
2804:Arabic
2746:French
2738:German
1956:affine
1952:linear
1868:
1755:
1671:where
1558:
1486:where
781:where
377:whose
360:social
314:, and
262:where
138:corpus
6425:Ewens
6251:Voigt
6223:Slash
6004:Lomax
5999:Log-t
5904:Gamma
5851:Hyper
5841:Davis
5836:Dagum
5692:Bates
5682:ARGUS
5566:Borel
4945:(PDF)
4934:(PDF)
4874:arXiv
4832:S2CID
4790:arXiv
4640:JSTOR
4575:arXiv
4546:S2CID
4444:S2CID
4392:S2CID
4353:S2CID
4327:arXiv
4298:arXiv
4268:S2CID
4133:(PDF)
4118:(PDF)
4046:S2CID
4012:arXiv
3970:S2CID
3891:(PDF)
3871:(PDF)
3858:(PDF)
3698:S2CID
3661:arXiv
3546:JSTOR
3261:Notes
2939:(the
2915:(the
2897:Torah
2895:(the
2800:Ge'ez
2777:Assis
2690:scale
1684:is a
487:notes
6674:and
6632:Kent
6059:Rice
5974:LĂ©vy
5802:Burr
5732:PERT
5697:Beta
5646:Zeta
5538:Zipf
5455:list
5190:ISBN
5182:ISSN
5149:PMID
5131:ISSN
5014:OCLC
5004:ISBN
4912:PMID
4824:PMID
4816:ISSN
4748:ISSN
4698:ISSN
4632:ISSN
4515:2016
4478:ISBN
4260:ISSN
4221:PMID
4141:2020
4038:ISSN
3962:ISSN
3903:ISBN
3610:ISBN
3538:ISSN
3461:ISBN
3353:ISBN
3330:PMID
2989:and
2853:and
2013:The
1978:and
1862:<
929:Mode
862:Mean
603:real
358:and
230:rank
6510:LKJ
5807:Chi
5241:of
5139:PMC
5121:doi
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4892:doi
4808:doi
4786:587
4740:doi
4690:doi
4678:492
4624:doi
4620:114
4593:doi
4571:293
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4434:doi
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4384:doi
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4211:PMC
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3952:doi
3895:doi
3836:doi
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3690:doi
3602:doi
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3320:PMC
3312:doi
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801:CDF
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