# Connections between spectral properties of asymptotic mappings and solutions to wireless network problems

## TL;DR

This paper links spectral properties of asymptotic mappings to solutions of wireless network optimization problems, revealing how spectral analysis can predict utility behavior and feasibility conditions in power control scenarios.

## Contribution

It introduces a class of asymptotic mappings, showing their spectral properties explain solution behaviors and provide criteria for feasibility in wireless network problems.

## Key findings

- Optimal utility is approximately linear at low power regimes.
- A transition point from utility gains to marginal returns is identified via spectral radius.
- Spectral radius of asymptotic mappings determines fixed point existence and constraint satisfaction.

## Abstract

In this study we establish connections between asymptotic functions and properties of solutions to important problems in wireless networks. We start by introducing a class of self-mappings (called asymptotic mappings) constructed with asymptotic functions, and we show that spectral properties of these mappings explain the behavior of solutions to some maxmin utility optimization problems. For example, in a common family of max-min utility power control problems, we prove that the optimal utility as a function of the power available to transmitters is approximately linear in the low power regime. However, as we move away from this regime, there exists a transition point, easily computed from the spectral radius of an asymptotic mapping, from which gains in utility become increasingly marginal. From these results we derive analogous properties of the transmit energy efficiency. In this study we also generalize and unify existing approaches for feasibility analysis in wireless networks. Feasibility problems often reduce to determining the existence of the fixed point of a standard interference mapping, and we show that the spectral radius of an asymptotic mapping provides a necessary and sufficient condition for the existence of such a fixed point. We further present a result that determines whether the fixed point satisfies a constraint given in terms of a monotone norm.