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nemenman>Ilya at 17:26, 19 August 2014

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D Schwab, I Nemenman, and P Mehta. Zipf’s law and criticality in multivariate data without fine-tuning. ''Phys Rev Lett'' '''113''', 068102, 2014. [[media:schwab-etal-14.pdf |PDF]], [http://arxiv.org/abs/1310.0448 arXiv].
;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.

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nemenman>Ilya at 17:26, 19 August 2014