Asymptotic equipartition properties for simple hierarchical and networked structures

Doku-Amponsah, Kwabena

doi:10.1051/ps/2010016

Cited by 15 publications

(12 citation statements)

References 11 publications

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“…i.e. [15], [2], [9], [3], [7], [5]. The main technique use to prove our main result is rooted in spectral potential theory.…”

Section: Introductionmentioning

confidence: 94%

“…Observe from the variational formulation of the relative entropy, see Dembo et al [12], and Lemma 3.1(i) that 1 2 sup g∈C g, π − ρ λ (g, µ) reduces to equation 2.2 above. Now the relative entropy lower semi-continuous, and by [7,Remark 4] all its level sets are compact. Hence it holds H λ (π µ) is lower semi-continuous, and all its level sets are weakly compact in the weak topology which ends the proof of the Lemma.…”

Section: Properties Of the Kullback Actionmentioning

confidence: 99%

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Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

Doku-Amponsah¹

2017

Journal of Mathematics and Statistics

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For a finite typed graph on n nodes and with type law µ, we define the socalled spectral potential ρ λ ( ·, µ), of the graph.From the ρ λ ( ·, µ) we obtain Kullback action or the deviation function, H λ (π ν), with respect to an empirical pair measure, π, as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure π and empirical type measure µ, we prove a Local large deviation principle (LLDP), with rate function H λ (π ν) and speed n. We deduce from this LLDP, a full conditional large deviation principle and a weak variant of the classical McMillian Theorem for the typed random graphs. Given the typical empirical link measure, λµ ⊗ µ, the number of typed random graphs is approximately equal e n λµ⊗µ H λµ⊗µ/ λµ⊗µ . Note that we do not require any topological restrictions on the space of finite graphs for these LLDPs. Now, some Large deviation principles(LDPs) and Coding Theorems exists for networked data structures modelled as the typed random graph (TRG) models.See,[15], [2], [9], [3],[7], [5] and the reference therein. O'Connor [15] proved large deviation principle (LDP) for the relative size of the largest connected component in the random graph with small edge probability. Biggin and Penman [2] have found LDPs for the number of edges of TRG models, where the link probabilities are independent of the number of nodes, using the Garten-Ellis Theorem, see [12].[9] proved LDPs for the empirical measures of the TRG where the link probabilities are dependent on the number of nodes of the graph. In [3], LDP for the empirical neighborhood distribution in sparse random graphs was proved using a technique that relies on the typical behavior within the framework of the local weak convergence of finite graph sequences. Asymptotic Equipartition Properties including the Lossy version have been found in [7] and [5], by the techniques of exponential change of measure and random allocation, respectively. Mathematics Subject Classification : 94A15, 94A24, 60F10, 05C80

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“…i.e. [15], [2], [9], [3], [7], [5]. The main technique use to prove our main result is rooted in spectral potential theory.…”

Section: Introductionmentioning

confidence: 94%

Section: Properties Of the Kullback Actionmentioning

confidence: 99%

Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

Doku-Amponsah¹

2017

Journal of Mathematics and Statistics

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“…W×W the solution can be found, see example [4], would obviously reduces to the good rate function as such…”

Section: Proof Of Theorem 21(i)mentioning

confidence: 99%

Local large deviation principle, large deviation principle and information theory for the signal-to-interference-plus-noise ratio graph models

Sakyi-Yeboah

Asiedu

Doku-Amponsah

2020

Journal of Information and Optimization Sciences

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The article obtains large deviation asymptotic for sub-critical communication networks modelled as signal-interference-noise-ratio(SINR) random networks. To achieve this, we define the empirical power measure and the empirical connectivity measure, as well as prove joint large deviation principles(LDPs) for the two empirical measures on two different scales. Using the joint LDPs, we prove an Asymptotic equipartition property(AEP) for wireless telecommunication Networks modelled as the subcritical SINR random networks. Further, we prove a Local Large deviation principle(LLDP) for the sub-critical SINR random network. From the LLDPs, we prove the large deviation principle, and a classical McMillan Theorem for the stochastic SINR model processes. Note that, the LDPs for the empirical measures of this stochastic SINR random network model were derived on spaces of measures equipped with the τ − topology, and the LLDPs were deduced in the space of SINR model process without any topological limitations. We motivate the study by describing a possible anomaly detection test for SINR random networks.

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“…Presently there exists some Large deviation Principle(LDP) and Basic Information Theory for the multitype Galton-Watson Process.See, example Dembo et al [5], Doku-Amponsah [4], Doku-Amponsah [3], Doku-Amponsah [2]. In [5], LDPs were proved for empirical measures of the multitype Galton-Watson trees with the exponential moments of offspring transition kernel being finite, in a topology stronger than weak topology.…”

Section: Introductionmentioning

confidence: 99%

“…6]. [3] proved an Asymptotic Equipatition Property for hierarchical Data Structures, using [5,Theorem 2.2], for bounded offspring transition kernels. Recently, Doku-Amponsah [2] found a Lossy version of the result [2, Theorem 2.1].…”

Section: Introductionmentioning

confidence: 99%

LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS

Doku-Amponsah¹

2017

FJMS

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In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential U K ( ·, π) for the Galton-Watson process, where π is the normalized eigen vector corresponding to the leading Perron-Frobenius eigen value 1l of the transition matrix A(·, ·) defined from K, the transition kernel. We show that the Kullback action or the deviation function, J(π, ρ), with respect to an empirical offspring measure, ρ, is the Legendre dual of U K ( ·, π). From the LLDP we deduce a conditional large deviation principle and a weak variant of the classical McMillian Theorem for the multitype Galton-Watson process. To be specific, given any empirical offspring measure ̟, we show that the number of critical multitype Galton-Watson processes on n vertices is approximately e n H̟, π , where H ̟ is a suitably defined entropy.Mathematics Subject Classification : 94A15, 94A24, 60F10, 05C80

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Asymptotic equipartition properties for simple hierarchical and networked structures

Cited by 15 publications

References 11 publications

Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

Local large deviation principle, large deviation principle and information theory for the signal-to-interference-plus-noise ratio graph models

LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS

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