Local Large Deviation Principle, Large Deviation Principle for the Signal -to- Interference and Noise Ratio Graph Models
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Given devices space D, an intensity measure λ∈(0,∞), transition kernel from the positive reals to the space D, a path -loss function which depends on some positive constant α, and some technical constants τ_λ,γ_λ:(0, ∞)→ (0,∞) we define a Marked Poisson Point Process(MPPP) and, a Signal-to- Interference and Noise Ratio (SINR) graph model. For the SINR graph model we define the empirical marked measure and the empirical connectivity measure. For a class of SINR graphs model, we prove joint Large Deviation Principle(LDP) for the empirical marked measure and the empirical connectivity measure with speed λ in the τ-topology. In particular if D=^d-2×[-T, T]^2, for T>0, we obtain a much explicit expression for the rate function. From the joint large deviation principle we obtain an Asymptotic Equipartition Property(AEP) for network structured data modelled as an SINR graph. In particular if D=^d-2×[-T, T]^2, for T>0, we obtain a explicit function. We also prove a Local Large Deviation Principle(LLDP) for the class of SINR graphs on D=^d-2×[-T, T]^2, with speed λ from spectral potential point. Given, an empirical marked measure ω, we define the so-called spectral potential U_R_T,d(ω, ·), for the SINR graph process, where R_T,d is a properly defined constant function which depends on the device locations and the marks. We show that the Kullback action or the divergence function I_ω(π), with respect to the empirical connectivity measure π, is the legendre dual of the spectral potential.
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