{"ID":2846876,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.02069","arxiv_id":"2511.02069","title":"Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings","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 $S \\sim r^{-α}$ 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.","short_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 $S \\sim r^{-α}$ between type count $S$ a...","url_abs":"https://arxiv.org/abs/2511.02069","url_pdf":"https://arxiv.org/pdf/2511.02069v2","authors":"[\"Pablo Rosillo-Rodes\",\"Laurent Hébert-Dufresne\",\"Peter Sheridan Dodds\"]","published":"2025-11-03T21:07:33Z","proceeding":"physics.soc-ph","tasks":"[\"physics.soc-ph\",\"cs.CL\"]","methods":"[]","has_code":false}