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Asymptotic equipartition properties for simple hierarchical and networked structures

Abstract

We prove asymptotic equipartition properties for simple hierarchical structures (modelled as multitype Galton-Watson trees) and networked structures (modelled as randomly coloured random graphs). For example, for large n, a networked data structure consisting of n units connected by an average number of links of order n / log n can be coded by about H × n bits, where H is an explicitly defined entropy. The main technique in our proofs are large deviation principles for suitably defined empirical measures.

We prove asymptotic equipartition properties for simple hierarchical structures (modelled as multitype Galton-Watson trees) and networked structures (modelled as randomly coloured random graphs). For example, for large n, a networked data structure consisting of n units connected by an average number of links of order n / log n can be coded by about H × n bits, where H is an explicitly defined entropy. The main technique in our proofs are large deviation principles for suitably defined empirical measures.

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J.D. Biggins , Large deviations for mixtures. Electron. Commun. Probab. 9 (2004) 6071.

J.D. Biggins and D.B. Penman , Large deviations in randomly coloured random graphs. Electron. Commun. Probab. 14 (2009) 290301.

A. Dembo and I. Kontoyiannis , Source Coding, Large deviations and Approximate Pattern. IEEE Trans. Inf. Theory 48 (2002) 15901615.

A. Dembo , P. Mörters and S. Sheffield , Large deviations of Markov chains indexed by random trees. Ann. Inst. Henri Poincaré : Probab. Stat. 41 (2005) 971996.

K. Doku-Amponsah and P. Mörters , Large deviation principle for empirical measures of coloured random graphs. Ann. Appl. Prob. 20 (2010) 19892021.

P. Olofsson and C.A. Shaw , Exact sampling formulas for multitype Galton-Watson processes. J. Math. Biol. 45 (2002) 279293.

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ESAIM: Probability and Statistics
  • ISSN: 1292-8100
  • EISSN: 1262-3318
  • URL: /core/journals/esaim-probability-and-statistics
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