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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 $\tau-$ 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., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2011.04479

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. $\lambda$ and $\lambda^2 a_{\lambda},$ where $\lambda$ 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 MacMillian Theorem for the stochastic SNIR network processes. Note, for tupical empirical connectivity measure, $q\pi\otimes\pi,$ we can deduce from the LLDP a bound on the cardinality of the space of SINR networks to be approximately equal to $\displaystyle e^{\lambda^2 a_{\lambda}\|q\pi\otimes\pi\|H\big(q\pi\otimes\pi/\|q\pi\otimes\pi\|\big)},$ where the connectivity probability of the network, $Q^{z^\lambda} ,$ satisfies $ a_{\lambda}^{-1}Q^{z^\lambda} \to q.$ Observe, the LDP for the empirical measures of the stochastic SINR network were obtained on spaces of measures equipped with the $\tau-$ topology, and the LLDPs were obtained in the space of SINR network process without any topological restrictions., Comment: 12 pages

For a \emph{ powered Poisson process}, we define \emph{Signal-to-Interference-plus-Noise Ratio}(SINR) and thesinr network as a Telecommunication Network. We define the Empirical Measures (\emph{empirical powered measure}, \emph{empirical link measure} and \emph{empirical sinr measure}) of a class of Telecommunication Networks. For this class of Telecommunication Network we prove a joint large deviation principle for the empirical measures of the Telecommunication Networks. All our rate functions are expressed in terms of relative entropies., Comment: 12 pages

Given devices space $D$, an intensity measure $\lambda m\in(0,\infty)$, a transition kernel $Q$ from the space $D$ to positive real numbers $(0,\infty,$ a path-loss function (which depends on the Euclidean distance between the devices and a positive constant $\alpha$), we define a Marked Poisson Point process (MPPP). For a given MPPP and technical constants $\tau_{\lambda},\gamma_{\lambda}:(0,\,\infty)\to (0,\infty),$ we define a Marked Signal-to- Interference and Noise Ratio (SINR) graph, and associate with it two empirical measures; the \emph{empirical marked measure} and the \emph{empirical connectivity measure}. For a class of marked SINR graphs, we prove a joint \emph{ large deviation principle}(LDP) for these empirical measures, with speed $\lambda$ in the $\tau$-topology. From the joint large deviation principle for the empirical marked measure and the empirical connectivity measure, we obtain an Asymptotic Equipartition Property(AEP) for network structured data modelled as a marked SINR graph. Specifically, we show that for large dense marked SINR graph one require approximately about $\lambda^{2}H(Q\times Q)/\log 2$ bits to transmit the information contained in the network with high probability, where $ H(Q\times Q)$ is a properly defined entropy for the exponential transition kernel with parameter $c$. Further, we prove a \emph {local large deviation principle} (LLDP) for the class of marked SINR graphs on $D,$ where $\lambda[\tau_{\lambda}(a)\gamma_{\lambda}(a)+\lambda\tau_{\lambda}(b)\gamma_{\lambda}(b)]\to \beta(a,b),$ $ a,b\in (0,\infty)$, with speed $\lambda$ from a \emph{ spectral potential} point. From the LLDP we derive a conditional LDP for the marked SINR graphs., Comment: 17 pages

In this article we study the stochastic block model also known as the multi-type random networks (MRNs). For the stochastic block model or the MRNs we define the empirical group measure, empirical cooperative measure and the empirical locality measure. We derive large deviation principles for the empirical measures in the weak topology. These results will form the basis of understanding asymptotics of the evolutionary and co-evolutionary processes on the stochastic block model., Comment: 8 pages

For a $d-$regular random model, we assign to vertices $q-$state spins. From this model, we define the \emph{empirical co-operate measure}, which enumerates the number of co-operation between a given couple of spins, and \emph{ empirical spin measure}, which enumerates the number of sites having a given spin on the $d-$regular random graph model. For these empirical measures we obtain large deviation principle(LDP) in the weak topology., Comment: 5 pages

In this article, we prove a local large deviation principle (LLDP) for the empirical locality measure of typed random networks on $n$ nodes conditioned to have a given \emph{ empirical type measure} and \emph{ empirical link measure.} From the LLDP, we deduce a full large deviation principle for the typed random graph, and the classical Erdos-Renyi graphs, where $nc/2$ links are inserted at random among $n$ nodes. No topological restrictions are required for these results., Comment: 5 pages

For a finite typed graph on $n$ nodes and with type law $\mu,$ we define the so-called spectral potential $\rho_{\lambda}(\,\cdot,\,\mu),$ of the graph.From the $\rho_{\lambda}(\,\cdot,\,\mu)$ we obtain Kullback action or the deviation function, $\mathcal{H}_{\lambda}(\pi\,\|\,\nu),$ with respect to an empirical pair measure, $\pi,$ as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure $\pi$ and empirical type measure $\mu$, we prove a Local large deviation principle (LLDP), with rate function $\mathcal{H}_{\lambda}(\pi\,\|\,\nu)$ 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, $\lambda\mu\otimes\mu,$ the number of typed random graphs is approximately equal $e^{n\|\lambda\mu\otimes\mu\|H\big(\lambda\mu\otimes\mu/\|\lambda\mu\otimes\mu\|\big)}.$ Note that we do not require any topological restrictions on the space of finite graphs for these LLDPs., Comment: 8 pages

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 $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding to the leading \emph{Perron-Frobenius eigen value } $\1$ of the transition matrix $\skria(\cdot,\,\cdot)$ defined from ${\skrik},$ the transition kernel. We show that the Kullback action or the deviation function, $J(\pi,\rho),$ with respect to an empirical offspring measure, $\rho,$ is the Legendre dual of $U_{\skrik}(\,\cdot,\,\pi).$ 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 $\varpi,$ we show that the number of critical multitype Galton-Watson processes on $n$ vertices is approximately $e^{n\langle \skrih_{\varpi},\,\pi\rangle},$ where $\skrih_{\varpi}$ is a suitably defined entropy., Comment: 8 pages

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.., Comment: 6 pages. arXiv admin note: text overlap with arXiv:1609.05252

In this article we prove a Generalized Asypmtotic Equipartition Property for Networked Data Structures modelled as coloured random graphs. The main techniques in this article remains large deviation principles for suitably defined empirical measures on coloured random graphs. We apply our main result to a concrete example from the field of Biology., Comment: 9 pages

This paper presents a rate-distortion theory for hierarchical networked data structures modelled as tree-indexed multitype process. To be specific, this paper gives a generalized Asymptotic Equipartition Property (AEP) for the Process. The general methodology of proof of the AEP are process level large deviation principles for suitably defined empirical measures for muiltitype Galton-Watson trees., Comment: 7 pages

In this article, we prove Shannon-MacMillan-Breiman Theorem for Wireless Sensor Networks modelled as coloured geometric random graphs. For large $n,$ we show that a Wireless Sensor Network consisting of $n$ sensors in $[0,1]^d$ connected by an average number of links of order $n\log n $ can be coded by about $[n(\log n )^2\pi^{d/2}/(d/2)!]\,\mathcal{H}$ bits, where $\mathcal{H}$ is an explicitly defined entropy. In the process, we derive a joint large deviation principle (LDP) for the \emph{empirical sensor measure} and \emph{the empirical link measure} of coloured random geometric graph models., Comment: 5 pages

For fitness preferential attachment random networks, we define the empirical degree and pair measure, which counts the number of vertices of a given degree and the number of edges with given fits, and the sample path empirical degree distribution. For the empirical degree and pair distribution for the fitness preferential attachment random networks, we find a large deviation upper bound. From this result we obtain a weak law of large numbers for the empirical degree and pair distribution, and the basic information theorem or an asymptotic equipartition property for fitness preferential attachment random networks., Comment: 11 pages

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 $nH$ bits, where $H$ is an explicitly defined entropy. The main technique in our proofs are large deviation principles for suitably defined empirical measures., Comment: 25 pages