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Asymptotic equipartition properties for simple hierarchical and networked structures

Published online by Cambridge University Press:  03 July 2012

Kwabena Doku-Amponsah
Kwabena Doku-Amponsah*
Affiliation:
Statistics Department, University of Ghana, Box LG 115, Legon, Ghana. kdoku@ug.edu.gh
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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.

Keywords

Asymptotic equipartition propertylarge deviation principlerelative entropyrandom graphmultitype Galton-Watson treerandomly coloured random graphtyped graphtyped tree.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 114 - 138
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

R. Arking, Biology of Aging, 2nd edition. Sinauer, Sunderland, MA (1998).
Biggins, J.D., Large deviations for mixtures. Electron. Commun. Probab. 9 (2004) 60–71. Google Scholar
Biggins, J.D. and Penman, D.B., Large deviations in randomly coloured random graphs. Electron. Commun. Probab. 14 (2009) 290–301. Google Scholar
C. Cannings and D.B. Penman, Models of random graphs and their applications, Handbook of Statistics. Stochastic Processes : Modeling and Simulation, edited by D.N. Shanbhag and C.R. Rao. Elsevier 21 (2003) 51–91.
T.M. Cover and J.A. Thomas, Elements of Information Theory, in Wiley Series in Telecommunications (1991).
Dembo, A. and Kontoyiannis, I., Source Coding, Large deviations and Approximate Pattern. IEEE Trans. Inf. Theory 48 (2002) 1590–1615. Google Scholar
A. Dembo and O. Zeitouni, Large deviations techniques and applications. Springer, New York (1998).
Dembo, A., Mörters, P. and Sheffield, S., Large deviations of Markov chains indexed by random trees. Ann. Inst. Henri Poincaré : Probab. Stat. 41 (2005) 971–996. Google Scholar
K. Doku-Amponsah, Large deviations and basic information theory for hierarchical and networked data structures. Ph.D. thesis, Bath (2006).
Doku-Amponsah, K. and Mörters, P., Large deviation principle for empirical measures of coloured random graphs. Ann. Appl. Prob. 20 (2010) 1989–2021. Google Scholar
M.Kimmel and D.E.Axelrod, Branching Processes with Biology. Springer, New York (2002).
C.J. Mode, Multitype Branching Processes Theory and Applications. American Elsevier, New York (1971).
M.E. Newman, Random graphs as models of networks. http://arxiv.org/abs/cond-mat/0202208
Olofsson, P. and Shaw, C.A., Exact sampling formulas for multitype Galton-Watson processes. J. Math. Biol. 45 (2002) 279–293. Google Scholar
D.B. Penman, Random graphs with correlation structure. Ph.D. thesis, Sheffield (1998).