Zipf's Law Leads to Heaps' Law: Analyzing Their Relation in Finite-Size Systems

TL;DR

This paper demonstrates that Heaps' law can be derived from Zipf's law in finite systems, providing refined estimates of their relationship and empirical validation across various complex systems.

Contribution

It clarifies the theoretical connection between Zipf's and Heaps' laws and offers a numerical method for better estimating the Heaps' exponent from Zipf's exponent and system size.

Findings

Heaps' law is a derivative of Zipf's law in finite systems.

Refined approximation improves estimation of Heaps' exponent.

Empirical analysis confirms the theoretical relationship across diverse systems.

Abstract

Background: Zipf's law and Heaps' law are observed in disparate complex systems. Of particular interests, these two laws often appear together. Many theoretical models and analyses are performed to understand their co-occurrence in real systems, but it still lacks a clear picture about their relation. Methodology/Principal Findings: We show that the Heaps' law can be considered as a derivative phenomenon if the system obeys the Zipf's law. Furthermore, we refine the known approximate solution of the Heaps' exponent provided the Zipf's exponent. We show that the approximate solution is indeed an asymptotic solution for infinite systems, while in the finite-size system the Heaps' exponent is sensitive to the system size. Extensive empirical analysis on tens of disparate systems demonstrates that our refined results can better capture the relation between the Zipf's and Heaps' exponents. Conclusions/Significance: The present analysis provides a clear picture about the relation between the Zipf's law and Heaps' law without the help of any specific stochastic model, namely the Heaps' law is indeed a derivative phenomenon from Zipf's law. The presented numerical method gives considerably better estimation of the Heaps' exponent given the Zipf's exponent and the system size. Our analysis provides some insights and implications of real complex systems, for example, one can naturally obtained a better explanation of the accelerated growth of scale-free networks.