Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings
Year: 2025
Type: article
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.
Source: Physical Review Research
DOI: 10.1103/q9w5-7k3j
Language: English
Subfield: Economics and Econometrics
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $2,625
Awards: G1175115578, G1722358650, G2262748287, G294895050, G300979063, G3480869486, G425668148, G5080475149, RII Track-1: Harnessing the Data Revolution for Vermont: The Science of Online Corpora, Knowledge, and Stories (SOCKS), Collaborative Research: HNDS-R: Altruistic stress, economic networks, and endogenous organizational change, G5812444529, G6118605381, G7051782649, G7084143925, G779402943, G8345352404
OpenAlex ID: W7117558629