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Approximate maximizers of intricacy functionals
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  • Published: 02 March 2011

Approximate maximizers of intricacy functionals

  • J. Buzzi1 &
  • L. Zambotti2 

Probability Theory and Related Fields volume 153, pages 421–440 (2012)Cite this article

  • 157 Accesses

  • 4 Citations

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Abstract

G. Edelman, O. Sporns, and G. Tononi introduced in theoretical biology the neural complexity of a family of random variables. A previous work showed that this functional is a special case of intricacy, i.e., an average of the mutual information of subsystems with specific mathematical properties. Moreover, its maximum value grows at a definite speed with the size of the system. In this work, we compute exactly this speed of growth by building “approximate maximizers” subject to an entropy condition. These approximate maximizers work simultaneously for all intricacies. We also establish some properties of arbitrary approximate maximizers, in particular the existence of a threshold in the size of subsystems of approximate maximizers: most smaller subsystems are almost equidistributed, most larger subsystems determine the full system. The main ideas are a random construction of almost maximizers with a high statistical symmetry and the consideration of entropy profiles, i.e., the average entropies of sub-systems of a given size. The latter gives rise to interesting questions of probability and information theory.

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Author information

Authors and Affiliations

  1. Laboratoire de Mathématique d’Orsay, CNRS (UMR 8628), Université Paris-Sud, 91405, Orsay Cedex, France

    J. Buzzi

  2. Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR 7599), Université Paris 6, Pierre et Marie Curie, UFR Mathematiques, Case 188, 4 Place Jussieu, 75252, Paris Cedex 05, France

    L. Zambotti

Authors
  1. J. Buzzi
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  2. L. Zambotti
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Correspondence to L. Zambotti.

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Cite this article

Buzzi, J., Zambotti, L. Approximate maximizers of intricacy functionals. Probab. Theory Relat. Fields 153, 421–440 (2012). https://doi.org/10.1007/s00440-011-0350-y

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  • Received: 14 September 2010

  • Revised: 15 February 2011

  • Published: 02 March 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0350-y

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Keywords

  • Entropy
  • Complexity
  • Maximization
  • Discrete probability

Mathematics Subject Classification (2000)

  • 94A17
  • 92B20
  • 60C05
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