Maximal Diversity and Zipf's Law

Abstract

Zipf's law describes the empirical size distribution of the components of many systems in natural and social sciences and humanities. We show, by solving a statistical model, that Zipf's law co-occurs with the maximization of the diversity of the component sizes. The law ruling the increase of such diversity with the total dimension of the system is derived and its relation with Heaps' law is discussed. As an example, we show that our analytical results compare very well with linguistics datasets.

arXiv comment5 pages, 3 figures

References in corpus (21)

• Power-law distributions in empirical data
• Power laws, Pareto distributions and Zipf's law
• Are biological systems poised at criticality?
• Power Laws are Logarithmic Boltzmann Laws
• Zipf's Law in Gene Expression
• Interacting Individuals Leading to Zipf's Law
• Zipf's Law Leads to Heaps' Law: Analyzing Their Relation in Finite-Size Systems
• Large-scale analysis of Zipf's law in English texts
• Zipf's law and criticality in multivariate data without fine-tuning
• Understanding scaling through history-dependent processes with collapsing sample space
• On sampling and modeling complex systems
• Scaling laws and fluctuations in the statistics of word frequencies
• Scaling Laws in Human Language
• A scaling law beyond Zipf's law and its relation to Heaps' law
• Statistical Criticality arises in Most Informative Representations
• Dynamical approach to Zipf's law
• The distinct flavors of Zipf's law in the rank-size and in the size-distribution representations, and its maximum-likelihood fitting
• Statistics of shared components in complex component systems
• Condensation for random variables conditioned by the value of their sum
• Development and regression of a large fluctuation
• Dynamical cluster size heterogeneity

Cited by in corpus (11)

• Rényi Entropy of Zeta-Urns
• Entropy and type-token ratio in gigaword corpora
• Critical percolation in the ordering kinetics of twisted nematic phases
• On Random Allocation Models in the Thermodynamic Limit
• Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model
• A branching random-walk model of disease outbreaks and the percolation backbone
• Size distributions reveal regime transition of lake systems under different dominant driving forces
• Is Human Culture Locked by Evolution?
• The Most Dispersed Subset of Random Points in ${\mathbb{R}}^d$
• Yang-Lee zeros for real-space condensation
• Maximal Minimal Spacing for Random Points