Log-log Convexity of Type-Token Growth in Zipf's Systems

Abstract

It is traditionally assumed that Zipf's law implies the power-law growth of the number of different elements with the total number of elements in a system - the so-called Heaps' law. We show that a careful definition of Zipf's law leads to the violation of Heaps' law in random systems, and obtain alternative growth curves. These curves fulfill universal data collapses that only depend on the value of the Zipf's exponent. We observe that real books behave very much in the same way as random systems, despite the presence of burstiness in word occurrence. We advance an explanation for this unexpected correspondence.

References in corpus (8)

• Power-law distributions in empirical data
• Beyond word frequency: Bursts, lulls, and scaling in the temporal distributions of words
• Languages cool as they expand: Allometric scaling and the decreasing need for new words
• Zipf's Law Leads to Heaps' Law: Analyzing Their Relation in Finite-Size Systems
• Understanding scaling through history-dependent processes with collapsing sample space
• Zipf's law for word frequencies: word forms versus lemmas in long texts
• Scaling laws and fluctuations in the statistics of word frequencies
• A practical recipe to fit discrete power-law distributions

Cited by in corpus (6)

• Linguistic laws in biology
• The brevity law as a scaling law, and a possible origin of Zipf's law for word frequencies
• Heaps' Law and Vocabulary Richness in the History of Classical Music Harmony
• Corrections of Zipf's and Heaps' Laws Derived from Hapax Rate Models
• Range-Limited Heaps' Law for Functional DNA Words in the Human Genome
• Quadratic Term Correction on Heaps' Law