title: Zipf's law and criticality in multivariate data without fine-tuning.
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.
authors.affiliation: Department of Physics and Lewis-Sigler Institute, Princeton University, Princeton, New Jersey 08540, USA.
authors.forename: David J
authors.initials: DJ
authors.lastname: Schwab
authors.affiliation: Departments of Physics and Biology, Emory University, Atlanta, Georgia 30322, USA.
authors.forename: Ilya
authors.initials: I
authors.lastname: Nemenman
authors.affiliation: Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
authors.forename: Pankaj
authors.initials: P
authors.lastname: Mehta
mesh_terms: D008954:Models, Biological; D015999:Multivariate Analysis