Abstract
BackgroundZipf'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 FindingsWe 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/SignificanceThe 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 the 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.
Highlights
Giant strides in Complexity Sciences have been the direct outcome of efforts to uncover the universal laws that govern disparate systems
As a signature of complex systems, the Zipf’s law is observed everywhere [3]: these include the distributions of firm sizes [4], wealths and incomes [5], paper citations [6], gene expressions [7], sizes of blackouts [8], family names [9], city sizes [10], personal donations [11], chess openings [12], traffic loads caused by YouTube videos [13], and so on
We use the language of word statistics in text, where zðrÞ denotes the frequency of the word with rank r
Summary
Giant strides in Complexity Sciences have been the direct outcome of efforts to uncover the universal laws that govern disparate systems. Ranking all the words in descending order of occurrence frequency and denoting by zðrÞ the frequency of the word with rank r, the Zipf’s law reads zðrÞ~zmax:r{a, where zmax is the maximal frequency and a is the so-called Zipf’s exponent. This power-law frequency-rank relation indicates a power-law probability distribution of the frequency itself, say pðzÞ*z{b with b equal to 1z1=a (see Materials and Methods). 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
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