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Phys Rev Lett. 2014 Aug 8;113(6):068102. Epub 2014 Aug 7.

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

Author information

1
Department of Physics and Lewis-Sigler Institute, Princeton University, Princeton, New Jersey 08540, USA.
2
Departments of Physics and Biology, Emory University, Atlanta, Georgia 30322, USA.
3
Department of Physics, Boston University, Boston, Massachusetts 02215, USA.

Abstract

The joint probability distribution of states of many degrees of freedom in biological systems, such as firing patterns in neural networks or antibody sequence compositions, often follows Zipf's law, where a power law is observed on a rank-frequency plot. This behavior has been shown to imply that these systems reside near a unique critical point where the extensive parts of the entropy and energy are exactly equal. Here, we show analytically, and via numerical simulations, that Zipf-like probability distributions arise naturally if there is a fluctuating unobserved variable (or variables) that affects the system, such as a common input stimulus that causes individual neurons to fire at time-varying rates. In statistics and machine learning, these are called latent-variable or mixture models. We show that Zipf's law arises generically for large systems, without fine-tuning parameters to a point. Our work gives insight into the ubiquity of Zipf's law in a wide range of systems.

PMID:
25148352
PMCID:
PMC5142845
DOI:
10.1103/PhysRevLett.113.068102
[Indexed for MEDLINE]
Free PMC Article

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