{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T10:15:17Z","timestamp":1785233717826,"version":"3.55.0"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,17]],"date-time":"2020-02-17T00:00:00Z","timestamp":1581897600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"mineco","doi-asserted-by":"publisher","award":["PGC-FIS2018-099629-B-I00"],"award-info":[{"award-number":["PGC-FIS2018-099629-B-I00"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>An important body of quantitative linguistics is constituted by a series of statistical laws about language usage. Despite the importance of these linguistic laws, some of them are poorly formulated, and, more importantly, there is no unified framework that encompasses all them. This paper presents a new perspective to establish a connection between different statistical linguistic laws. Characterizing each word type by two random variables\u2014length (in number of characters) and absolute frequency\u2014we show that the corresponding bivariate joint probability distribution shows a rich and precise phenomenology, with the type-length and the type-frequency distributions as its two marginals, and the conditional distribution of frequency at fixed length providing a clear formulation for the brevity-frequency phenomenon. The type-length distribution turns out to be well fitted by a gamma distribution (much better than with the previously proposed lognormal), and the conditional frequency distributions at fixed length display power-law-decay behavior with a fixed exponent     \u03b1 \u2243 1.4     and a characteristic-frequency crossover that scales as an inverse power     \u03b4 \u2243 2.8     of length, which implies the fulfillment of a scaling law analogous to those found in the thermodynamics of critical phenomena. As a by-product, we find a possible model-free explanation for the origin of Zipf\u2019s law, which should arise as a mixture of conditional frequency distributions governed by the crossover length-dependent frequency.<\/jats:p>","DOI":"10.3390\/e22020224","type":"journal-article","created":{"date-parts":[[2020,2,18]],"date-time":"2020-02-18T10:10:25Z","timestamp":1582020625000},"page":"224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["The Brevity Law as a Scaling Law, and a Possible Origin of Zipf\u2019s Law for Word Frequencies"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5280-2692","authenticated-orcid":false,"given":"\u00c1lvaro","family":"Corral","sequence":"first","affiliation":[{"name":"Centre de Recerca Matem\u00e0tica, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain"},{"name":"Departament de Matem\u00e0tiques, Facultat de Ci\u00e8ncies, Universitat Aut\u00f2noma de Barcelona, E-08193 Barcelona, Spain"},{"name":"Barcelona Graduate School of Mathematics, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Stra\u03b2e 39, 1080 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2465-8574","authenticated-orcid":false,"given":"Isabel","family":"Serra","sequence":"additional","affiliation":[{"name":"Centre de Recerca Matem\u00e0tica, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain"},{"name":"Barcelona Supercomputing Center-Centro Nacional de Supercomputaci\u00f3n, Jordi Girona 29, E-08034 Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,17]]},"reference":[{"key":"ref_1","unstructured":"Zipf, G.K. (1949). Human Behavior and the Principle of Least Effort, Addison-Wesley."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Baayen, R.H. (2001). Word Frequency Distributions, Kluwer.","DOI":"10.1007\/978-94-010-0844-0"},{"key":"ref_3","first-page":"803","article-title":"Distributions in text","volume":"Volume 2","year":"2009","journal-title":"Corpus linguistics: An International Handbook"},{"key":"ref_4","unstructured":"Zanette, D. (2014). Statistical patterns in written language. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1112","DOI":"10.3758\/s13423-014-0585-6","article-title":"Zipf\u2019s law in natural language: A critical review and future directions","volume":"21","author":"Piantadosi","year":"2014","journal-title":"Psychon. Bull. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Moreno-S\u00e1nchez, I., Font-Clos, F., and Corral, A. (2016). Large-scale analysis of Zipf\u2019s law in English texts. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0147073"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Corral, A., Serra, I., and Ferrer-i-Cancho, R. (2019). The distinct flavors of Zipf\u2019s law in the rank-size and in the size-distribution representations, and its maximum-likelihood fitting. arXiv.","DOI":"10.1103\/PhysRevE.102.052113"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Jakobson, R. (1961). On the theory of word frequencies and on related Markovian models of discourse. Structure of Language and its Mathematical Aspects, American Mathematical Society.","DOI":"10.1090\/psapm\/012"},{"key":"ref_9","unstructured":"Heaps, H.S. (1978). Information retrieval: Computational and Theoretical Aspects, Academic Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.114.238701","article-title":"Log-log convexity of type-token growth in Zipf\u2019s systems","volume":"114","author":"Corral","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Esposti, M.D., Altmann, E.G., and Pachet, F. (2016). Statistical laws in linguistics. Creativity and Universality in Language. Lecture Notes in Morphogenesis, Springer.","DOI":"10.1007\/978-3-319-24403-7_2"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1093\/biomet\/45.1-2.222","article-title":"The Relation Between the Dictionary Distribution and the Occurrence Distribution of Word Length and its Importance for the Study of Quantitative Linguistics","volume":"45","author":"Herdan","year":"1958","journal-title":"Biometrika"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"191023","DOI":"10.1098\/rsos.191023","article-title":"On the physical origin of linguistic laws and lognormality in speech","volume":"6","author":"Torre","year":"2019","journal-title":"R. Soc. Open Sci."},{"key":"ref_14","unstructured":"Bentz, C., J\u00e4ger, G., and Yanovich, I. (2016). Zipf\u2019s law of abbreviation as a language universal. Proceedings of the Leiden Workshop on Capturing Phylogenetic Algorithms for Linguistics, University of T\u00fcbingen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/42.3-4.425","article-title":"On a class of skew distribution functions","volume":"42","author":"Simon","year":"1955","journal-title":"Biometrika"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1076\/jqul.8.3.165.4101","article-title":"Two regimes in the frequency of words and the origin of complex lexicons: Zipf\u2019s law revisited","volume":"8","year":"2001","journal-title":"J. Quant. Linguist."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"052811","DOI":"10.1103\/PhysRevE.91.052811","article-title":"Text mixing shapes the anatomy of rank-frequency distributions","volume":"91","author":"Williams","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"066119","DOI":"10.1103\/PhysRevE.81.066119","article-title":"Statistical mechanics of letters in words","volume":"81","author":"Stephens","year":"2010","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Corral, A., and Garc\u00eda del Muro, M. (2020). From Boltzmann to Zipf through Shannon and Jaynes. Entropy, 22.","DOI":"10.3390\/e22020179"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gerlach, M., and Font-Clos, F. (2020). A standardized Project Gutenberg Corpus for statistical analysis of natural language and quantitative linguistics. Entropy, 22.","DOI":"10.3390\/e22010126"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"P11030","DOI":"10.1088\/1742-5468\/2010\/11\/P11030","article-title":"Universality of rain event size distributions","volume":"11","author":"Peters","year":"2010","journal-title":"J. Stat. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.2478\/s11600-013-0154-9","article-title":"Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions","volume":"61","author":"Deluca","year":"2013","journal-title":"Acta Geophys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1029\/2018EA000479","article-title":"Power law distributions in geoscience revisited","volume":"6","author":"Corral","year":"2019","journal-title":"Earth Space Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Corral, A., Boleda, G., and Ferrer-i-Cancho, R. (2015). Zipf\u2019s law for word frequencies: Word forms versus lemmas in long texts. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0129031"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1137\/070710111","article-title":"Power-law distributions in empirical data","volume":"51","author":"Clauset","year":"2009","journal-title":"SIAM Rev."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"066103","DOI":"10.1103\/PhysRevE.83.066103","article-title":"Non-characteristic half-lives in radioactive decay","volume":"83","author":"Corral","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"033034","DOI":"10.1103\/PhysRevResearch.1.033034","article-title":"Scale-free networks well done","volume":"1","author":"Voitalov","year":"2019","journal-title":"Phys. Rev. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"555","DOI":"10.5194\/npg-21-555-2014","article-title":"Scale invariant events and dry spells for medium-resolution local rain data","volume":"21","author":"Deluca","year":"2014","journal-title":"Nonlinear Proc. Geophys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.chaos.2015.01.011","article-title":"Scaling in the timing of extreme events","volume":"74","author":"Corral","year":"2015","journal-title":"Chaos Solitons Fract."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"093033","DOI":"10.1088\/1367-2630\/15\/9\/093033","article-title":"A scaling law beyond Zipf\u2019s law and its relation to Heaps\u2019 law","volume":"15","author":"Boleda","year":"2013","journal-title":"New J. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"022318","DOI":"10.1103\/PhysRevE.96.022318","article-title":"Dependence of exponents on text length versus finite-size scaling for word-frequency distributions","volume":"96","author":"Corral","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_32","unstructured":"Bhattacharyya, P., and Chakrabarti, B.K. (2007). Statistical features of earthquake temporal occurrence. Modelling Critical and Catastrophic Phenomena in Geoscience, Springer."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"022134","DOI":"10.1103\/PhysRevE.97.022134","article-title":"Increasing power-law range in avalanche amplitude and energy distributions","volume":"97","author":"Serra","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Aitchison, L., Corradi, N., and Latham, P.E. (2016). Zipf\u2019s law arises naturally when there are underlying, unobserved variables. PLoS Comput. Biol., 12.","DOI":"10.1371\/journal.pcbi.1005110"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1002\/cplx.21820","article-title":"Compression and the origins of Zipf\u2019s law for word frequencies","volume":"21","year":"2016","journal-title":"Complexity"},{"key":"ref_36","unstructured":"Ferrer-i-Cancho, R., Bentz, C., and Seguin, C. (2015). Compression and the origins of Zipf\u2019s law of abbreviation. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/224\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:58:22Z","timestamp":1760173102000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/224"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,17]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["e22020224"],"URL":"https:\/\/doi.org\/10.3390\/e22020224","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,17]]}}}