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Background: Zipf's law and Heaps' law are observed in disparate complexsystems. Of particular interests, these two laws often appear together. Manytheoretical models and analyses are performed to understand their co-occurrencein real systems, but it still lacks a clear picture about their relation.Methodology/Principal Findings: We show that the Heaps' law can be consideredas a derivative phenomenon if the system obeys the Zipf's law. Furthermore, werefine the known approximate solution of the Heaps' exponent provided theZipf's exponent. We show that the approximate solution is indeed an asymptoticsolution for infinite systems, while in the finite-size system the Heaps'exponent is sensitive to the system size. Extensive empirical analysis on tensof disparate systems demonstrates that our refined results can better capturethe relation between the Zipf's and Heaps' exponents. Conclusions/Significance:The present analysis provides a clear picture about the relation between theZipf'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. Thepresented numerical method gives considerably better estimation of the Heaps'exponent given the Zipf's exponent and the system size. Our analysis providessome insights and implications of real complex systems, for example, one cannaturally obtained a better explanation of the accelerated growth of scale-freenetworks.

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