Large deviations, asymptotic equipartition property for super-critical SINR random networks

Abstract

In this article, we obtain large deviation asymptotic for supercritical social or communication networks modelled as signal-interference-noise ratio (SINR) graphs. 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. λ and λ²a_λ, where λ is the intensity measure of the Poisson point process (PPP) which defines the SINR random network and a_λ real-valued sequence such that λ²a_λ → ∞ as λ → ∞Using these joint LDPs we prove an asymptotic equipartition property for the super-critical networks modelled as the SINR random networks. Furthermore, we prove a local large deviation principle (LLDP) for the SINR random network. From the LLDP we prove a large deviation principle, and a classical McMillian theorem for the SINR random network processes. Note that, for a given empirical power measure and typical empirical connectivity measure, qπ ⊗ π, we can deduce from the LLDP a bound on the cardinality of the space of SINR networks to be approximately equal to , where the connectivity probability of the network, , satisfies and qπ ⊗ π is the typical behavior of the empirical connectivity measure. Observe, the LDPs for the empirical measures of SINR random networks were obtained on spaces of measures equipped with the τ-topology, and the LLDPs were obtained in the space of SINR network process without any topological restrictions.

Subject Classification:

• 60F10
• 05C80
• 68Q87

Keywords:

• Super-critical SINR networks
• Poisson point process
• Empirical power measure
• Empirical connectivity measure
• Large deviations
• Relative entropy
• Entropy
• Asymptotic equiparttion property