Local Large Deviation Principle, Large Deviation Principle and Information theory for the Signal -to- Interference -Plus- Noise Ratio Graph Models

Given devices space $D$, an intensity measure $λm\in(0,\infty)$, 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 $α$), we define a Marked Poisson Point process (MPPP). For a given MPPP and technical constants $τ_λ,γ_λ:(0,\,\infty)\to (0,\infty),$ 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 $λ$ in the $τ$-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 $λ^{2}H(Q\times Q)/\log 2$ 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 $λ[τ_λ(a)γ_λ(a)+λτ_λ(b)γ_λ(b)]\to β(a,b),$ $a,b\in (0,\infty)$, with speed $λ$ from a \emph{ spectral potential} point. From the LLDP we derive a conditional LDP for the marked SINR graphs.

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