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

arXiv: https://arxiv.org/abs/2511.02069 · HTML: https://mlsanity.com/paper/2511.02069/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 33 references that are in the map.
- First posted on arXiv 3 Nov 2025.

## Builds on (most cited first, 3 of 3)

- [Statistical laws in linguistics](https://mlsanity.com/paper/1502.03296/) (2015)
- [Heaps' law and Heaps functions in tagged texts: Evidences of their linguistic relevance](https://mlsanity.com/paper/2001.02178/) (2020)
- [Text mixing shapes the anatomy of rank-frequency distributions: A modern Zipfian mechanics for natural language](https://mlsanity.com/paper/1409.3870/) (2014)

## Abstract (arXiv)

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 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.