Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908919862919168 |
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| author | Rosillo-Rodes, Pablo Hébert-Dufresne, Laurent Dodds, Peter Sheridan |
| author_facet | Rosillo-Rodes, Pablo Hébert-Dufresne, Laurent Dodds, Peter Sheridan |
| contents | 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 $S \sim r^{-α}$ 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 $α\gg 1$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings Rosillo-Rodes, Pablo Hébert-Dufresne, Laurent Dodds, Peter Sheridan Physics and Society Computation and Language 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 $S \sim r^{-α}$ 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 $α\gg 1$. 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. |
| title | Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings |
| topic | Physics and Society Computation and Language |
| url | https://arxiv.org/abs/2511.02069 |