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Zipf's law and criticality in multivariate data without fine-tuning.

David J Schwab1, Ilya Nemenman2, Pankaj Mehta3

  • 1Department of Physics and Lewis-Sigler Institute, Princeton University, Princeton, New Jersey 08540, USA.

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Zipf's law in biological systems, like neural networks, arises from unobserved fluctuating variables. This finding explains the widespread occurrence of Zipf's law without needing fine-tuning.

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Area of Science:

  • Complex Systems Biology
  • Statistical Physics
  • Computational Neuroscience

Background:

  • Biological systems often exhibit joint probability distributions following Zipf's law.
  • This power-law behavior suggests systems are near a critical point where entropy and energy balance.
  • Zipf's law is observed in diverse areas like neural firing patterns and antibody sequences.

Purpose of the Study:

  • To analytically and numerically demonstrate how Zipf-like distributions emerge in complex systems.
  • To identify the underlying mechanisms responsible for the ubiquity of Zipf's law.
  • To provide a theoretical framework for understanding systems near critical points.

Main Methods:

  • Analytical derivations of probability distributions.
  • Numerical simulations of systems with fluctuating unobserved variables.
  • Investigation of latent-variable and mixture models.

Main Results:

  • Zipf-like probability distributions naturally arise in large systems with fluctuating unobserved variables.
  • The presence of latent variables, such as common stimuli, drives Zipf's law.
  • The phenomenon occurs generically, without requiring fine-tuning of system parameters.

Conclusions:

  • Fluctuating unobserved variables are a key mechanism generating Zipf's law in biological and other complex systems.
  • This provides a unified explanation for the prevalence of Zipf's law across various scientific domains.
  • The findings offer insights into the statistical properties of systems operating near critical states.