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

David J. Schwab*, Ilya Nemenman, Pankaj Mehta

*Corresponding author for this work

Research output: Contribution to journalArticle

48 Scopus citations

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.

Original languageEnglish (US)
Article number068102
JournalPhysical review letters
Volume113
Issue number6
DOIs
StatePublished - Aug 7 2014

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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