Lossy Asymptotic Equipartition Property for Networked Data Structures

Doku-Amponsah, Kwabena

doi:10.3844/jmssp.2017.152.158

Cited by 4 publications

(6 citation statements)

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“…This article might may be regarded as a first step in the proof of a Lossy asymptotic equipartition property for the SINR networks. See, [6] and [7] for similar results for the networked data structures modelled as colored random graph process and for the hierarchical data structure modelled as Galton-Watson tree process.…”

Section: Discussionmentioning

confidence: 78%

Large Deviations, Sharron-McMillan-Breiman Theorem for Super-Critical Telecommunication Networks

Sakyi-Yeboah,

Andam,

Asiedu

et al. 2020

Preprint

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In this article we obtain large deviation asymptotics for supercritical communication networks modelled as signal-interference-noise ratio networks. To do this, we define the empirical power measure and the empirical connectivity measure, and prove joint large deviation principles(LDPs) for the two empirical measures on two different scales i.e. λ and λ 2 a λ , where λ is the intensity measure of the poisson point process (PPP) which defines the SINR random network.Using this joint LDPs we prove an asymptotic equipartition property for the stochastic telecommunication Networks modelled as the SINR networks. Further, we prove a Local large deviation principle(LLDP) for the SINR Network. From the LLDP we prove the a large deviation principle, and a classical McMillian Theorem for the stochastic SNIR network processes. Note, for tupical empirical connectivity measure, qπ ⊗ π, we can deduce from the LLDP a bound on the cardinality of the space of SINR networks to be approximately equal to e λ 2 a λ qπ⊗π H qπ⊗π/ qπ⊗π , where the connectivity probability of the network, Q z λ , satisfies a −1 λ Q z λ → q. Observe, the LDP for the empirical measures of the stochastic SINR network were obtained on spaces of measures equipped with the τ − topology, and the LLDPs were obtained in the space of SINR network process without any topological restrictions.

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Section: Discussionmentioning

confidence: 78%

Large Deviations, Sharron-McMillan-Breiman Theorem for Super-Critical Telecommunication Networks

Sakyi-Yeboah,

Andam,

Asiedu

et al. 2020

Preprint

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“…To design and implement simplex (Linear programming) algorithm for the solution of generalized network flow problems of the geometric structured network data ,see example [1], or to find an efficient coding scheme or an approximate pattern matching algorithms, see example [2], we need an information theory for such data structures, and the lossy Asymptotic Equipartition Property (AEP) for the geometric networked data structures is key to finding an information theory for the data structure. See [6] and [7] for similar results for other types of data structures.…”

Section: Introductionmentioning

confidence: 63%

“…If we take σ(s, r) = (s−r) 2 then, by Theorem 2.1 we have the rate-distortion of a, b))π(a)π(b). See, [6] for the relationship between the connectivity radius and λ [d] . We refer to [13] for more on modelling of the physical environment using the Wireless Sensor Network.…”

Section: Application: Wireless Sensor Network For Monitoring Water Qumentioning

confidence: 99%

“…The aim of this article is to extend the Lossy AEP for Networked Data Structures modelled as Coloured Random Graph (CRG), see [6,Theorem 2.1], to cover the WSN. To be specific we model the Geometric Networked Data Structures ( WSN) as a CGRG and use some of the large deviation techniques developed in [7] to prove a strong law of large numbers (SLLN), see Lemma 3.3, for the random network.…”

Section: Introductionmentioning

confidence: 99%

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Lossy asymptotic equipartition property for geometric networked data structures

Doku-Amponsah

2019

Journal of Information and Optimization Sciences

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This article extends the Generalized Asypmtotic Equipartition Property of Networked Data Structures to cover the Wireless Sensor Network modelled as coloured geometric random graph (CGRG). The main techniques used to prove this result remains large deviation principles for properly defined empirical measures on CGRGs. As a motivation for this article, we apply our results to some data from Wireless Sensor Network for Monitoring Water Quality from a Lake.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: 92%

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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Lossy Asymptotic Equipartition Property for Networked Data Structures

Cited by 4 publications

References 12 publications

Large Deviations, Sharron-McMillan-Breiman Theorem for Super-Critical Telecommunication Networks

Large Deviations, Sharron-McMillan-Breiman Theorem for Super-Critical Telecommunication Networks

Lossy asymptotic equipartition property for geometric networked data structures

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

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