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

Abstract

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.