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

摘要

Given devices space D, an intensity measure lambda m epsilon (0, infinity), a transition kernel Q from the space D to positive real numbers R+, 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, infinity) - (0, infinity), we define a Marked Signal-to- Interference and Noise Ratio (SINR) graph, and associate with it two empirical measures; the empirical marked measure and the empirical connectivity measure.For a class of marked SINR graphs, we prove a joint 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 requires approximately about lambda H-2(Q x Q)/log 2 bits to transmit the information contained in the network with high probability, where H(Q x Q) is a properly defined entropy for the exponential transition kernel with parameter c.Further, we prove a 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)]-beta(a, b), a, b epsilon (0, infinity), with speed lambda from a spectral potential point. From the LLDP we derive a conditional LDP for the marked SINR graphs.Note that, while the joint LDP is established in the tau-topology, the LLDP assume no topological restriction on the space of marked SINR graphs. Observe also that all our rate functions are expressed in terms of the relative entropy or the kullback action or divergence function of the marked SINR on the devices space D.