Zipf's Law and Criticality in Multivariate Data without FineTuning
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 rankfrequency 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 Zipflike 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 timevarying rates. In statistics and machine learning, these are called latentvariable or mixture models. We show that Zipf's law arises generically for large systems, without finetuning parameters to a point. Our work gives insight into the ubiquity of Zipf's law in a wide range of systems.
 Publication:

Physical Review Letters
 Pub Date:
 August 2014
 DOI:
 10.1103/PhysRevLett.113.068102
 arXiv:
 arXiv:1310.0448
 Bibcode:
 2014PhRvL.113f8102S
 Keywords:

 87.19.ll;
 64.60.De;
 87.10.e;
 89.75.Fb;
 Models of single neurons and networks;
 Statistical mechanics of model systems;
 General theory and mathematical aspects;
 Structures and organization in complex systems;
 Quantitative Biology  Neurons and Cognition;
 Condensed Matter  Statistical Mechanics;
 Quantitative Biology  Quantitative Methods
 EPrint:
 5 pages, 3 figures