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Maximal Diversity and Zipf's Law.

Onofrio Mazzarisi1,2, Amanda de Azevedo-Lopes3, Jeferson J Arenzon3,4

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Zipf's law, describing component size distributions, is mathematically linked to maximizing diversity in system components. This finding explains the prevalence of Zipf's law across various scientific fields.

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Area of Science:

  • Complex Systems Science
  • Statistical Physics
  • Quantitative Linguistics

Background:

  • Zipf's law is an empirical observation of component size distributions in diverse systems.
  • Understanding the underlying principles governing these distributions is crucial for various scientific disciplines.

Purpose of the Study:

  • To demonstrate that Zipf's law arises from the maximization of component size diversity.
  • To derive the relationship between diversity increase and system dimension.
  • To explore the connection between Zipf's law and Heaps's law.

Main Methods:

  • Solving a statistical model to establish the co-occurrence of Zipf's law and diversity maximization.
  • Deriving analytical laws for diversity increase with system dimension.

Main Results:

  • Zipf's law is shown to be a consequence of maximizing the diversity of component sizes.
  • A new law governing the increase of diversity with system dimension is derived.
  • Analytical results show strong agreement with empirical linguistics and population data.

Conclusions:

  • The maximization of component size diversity provides a theoretical basis for Zipf's law.
  • The derived laws offer insights into the scaling properties of complex systems.
  • The findings have broad applicability across natural and social sciences.