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  • Letter
  • Open Access

Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings

Pablo Rosillo-Rodes1,2,*, Laurent Hébert-Dufresne3,4,†, and Peter Sheridan Dodds2,3,4,‡

  • *Contact author: prosillo@ifisc.uib-csic.es
  • †Contact author: laurent.hebert-dufresne@uvm.edu
  • ‡Contact author: peter.dodds@uvm.edu
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Phys. Rev. Research 8, L012029 – Published 2 February, 2026

DOI: https://doi.org/10.1103/q9w5-7k3j

Abstract

The growth dynamics of complex systems often exhibit statistical regularities involving power-law relationships. For real finite complex systems formed by countable tokens (animals, words) as instances of distinct types (species, dictionary entries), an inverse power-law scaling S∼r−α between type count S and type rank r, widely known as Zipf’s law, is widely observed to varying degrees of fidelity. A secondary, summary relationship is Heaps’ law, which states that the number of types scales sublinearly with the total number of observed tokens present in a growing system. Here, we propose an idealized model of a growing system that (1) deterministically produces arbitrary inverse power-law count rankings for types and (2) allows us to determine the exact asymptotics of the type-token relationship. Our argument improves upon and remedies earlier work. We obtain a unified asymptotic expression for all values of α, which corrects the special cases of α=1 and α≫1. Our approach relies solely on the form of count rankings, avoids unnecessary approximations, and does not involve any stochastic mechanisms or sampling processes. We thereby demonstrate that a general type-token relationship arises solely as a consequence of Zipf’s law.

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References (33)

  1. G. K. Zipf, Human Behaviour and the Principle of Least Effort (Addison-Wesley, Cambridge, MA, 1949).
  2. M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law, Contemp. Phys. 46, 323 (2005).
  3. E. G. Altmann and M. Gerlach, Statistical laws in linguistics, in Creativity and Universality in Language, edited by M. Degli Esposti, E. G. Altmann, and F. Pachet (Springer International Publishing, Cham, 2016), pp. 7–26.
  4. I. Arnon, S. Kirby, J. A. Allen, C. Garrigue, E. L. Carroll, and E. C. Garland, Whale song shows language-like statistical structure, Science 387, 649 (2025).
  5. G. U. Yule, A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S., Philos. Trans. B 213, 21 (1925).
  6. B. B. Mandelbrot, An informational theory of the statistical structure of languages, in Communication Theory, edited by W. Jackson (Butterworth, Woburn, MA, 1953), pp. 486–502.
  7. H. A. Simon, On a class of skew distribution functions, Biometrika 42, 425 (1955).
  8. T. Maillart, D. Sornette, S. Spaeth, and G. von Krogh, Empirical tests of Zipf’s law mechanism in open source linux distribution, Phys. Rev. Lett. 101, 218701 (2008).
  9. R. Ferrer-i-Cancho and B. Elvevåg, Random texts do not exhibit the real Zipf’s law-like rank distribution, PLoS One 5, e9411 (2010).
  10. J. R. Williams, J. P. Bagrow, C. M. Danforth, and P. S. Dodds, Text mixing shapes the anatomy of rank-frequency distributions, Phys. Rev. E 91, 052811 (2015).
  11. J. Camacho and R. V. Solé, Scaling in ecological size spectra, Europhys. Lett. 55, 774 (2001).
  12. S. Redner, How popular is your paper? An empirical study of the citation distribution, Eur. Phys. J. B 4, 131 (1998).
  13. A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286, 509 (1999).
  14. M. A. Serrano and M. Boguñá, Topology of the world trade web, Phys. Rev. E 68, 015101(R) (2003).
  15. C. M. Pinto, A. M. Lopes, and J. T. Machado, A review of power laws in real life phenomena, Commun. Nonlinear Sci. Numer. Simul. 17, 3558 (2012).
  16. X. Gabaix, Power laws in economics: An introduction, J. Econ. Perspect. 30, 185 (2016).
  17. A. Gaillard, P. Wangner, C. Hellwig, and N. Werquin, Consumption, wealth, and income inequality: A tale of tails, available at SSRN 4636704 (2023), https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4636704.
  18. G. Herdan, Type-Token Mathematics, Janua linguarum: Series maior, Studia Memoriae Nicolai van Wijk Dedicata (Mouton, The Hague, 1960), Vol. 4, p. 448.
  19. H. Heaps, Information Retrieval, Computational and Theoretical Aspects, Library and Information Science (Academic Press, Florida, 1978).
  20. A. Chacoma and D. H. Zanette, Heaps’ law and Heaps’ functions in tagged texts: Evidences of their linguistic relevance, R. Soc. Open Sci. 7, 200008 (2020).
  21. R. W. Benz, S. J. Swamidass, and P. Baldi, Discovery of power-laws in chemical space, J. Chem. Inf. Model. 48, 1138 (2008).
  22. H. Zhang, Discovering power laws in computer programs, Inf. Process. Manage. 45, 477 (2009).
  23. F. Simini and C. James, Testing Heaps’ law for cities using administrative and gridded population data sets, EPJ Data Sci. 8, 24 (2019).
  24. R. Baeza-Yates and G. Navarro, Block addressing indices for approximate text retrieval, J. Am. Soc. Inf. Sci. 51, 69 (2000).
  25. D. Zanette and M. Montemurro, Dynamics of text generation with realistic Zipf’s distribution, J. Quant. Linguist. 12, 29 (2005).
  26. M. Á. Serrano, A. Flammini, and F. Menczer, Modeling statistical properties of written text, PLoS One 4, e5372 (2009).
  27. L. Hébert-Dufresne, A. Allard, J.-G. Young, and L. J. Dubé, Constrained growth of complex scale-independent systems, Phys. Rev. E 93, 032304 (2016).
  28. D. van Leijenhorst and T. van der Weide, A formal derivation of Heaps’ law, Inf. Sci. 170, 263 (2005).
  29. I. Eliazar, The growth statistics of Zipfian ensembles: Beyond Heaps’ law, Physica A 390, 3189 (2011).
  30. S. Bernhardsson, L. E. C. da Rocha, and P. Minnhagen, The meta book and size-dependent properties of written language, New J. Phys. 11, 123015 (2009).
  31. L. Lü, Z.-K. Zhang, and T. Zhou, Zipf’s law leads to Heaps’ law: Analyzing their relation in finite-size systems, PLoS One 5, e14139 (2010).
  32. T. M. Apostol, An elementary view of Euler’s summation formula, Am. Math. Mon. 106, 409 (1999).
  33. J. I. Perotti and O. V. Billoni, On the emergence of Zipf’s law in music, Physica A 549, 124309 (2020).

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Phys. Rev. Research 8, L012029– Published 2 February, 2026
  • Received 5 November 2025
  • Accepted 30 December 2025
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