The Problem of Localization in Networks of Randomly Deployed Nodes: Asymptotic and Finite Analysis, and Thresholds

TL;DR

This paper analyzes the probability of node localization in randomly deployed networks, deriving asymptotic thresholds for network localization success in dense and expanding domains, highlighting the critical role of transmission range.

Contribution

It introduces new asymptotic and finite analysis methods for network localization probabilities, identifying thresholds based on node density and transmission range.

Findings

Identifies thresholds for network localization in dense networks.

Derives probability functions for individual and network-wide localization.

Highlights the importance of maximum transmission range in large-scale networks.

Abstract

We derive the probability that a randomly chosen NL-node over $S$ gets localized as a function of a variety of parameters. Then, we derive the probability that the whole network of NL-nodes over $S$ gets localized. In connection with the asymptotic thresholds, we show the presence of asymptotic thresholds on the network localization probability in two different scenarios. The first refers to dense networks, which arise when the domain $S$ is bounded and the densities of the two kinds of nodes tend to grow unboundedly. The second kind of thresholds manifest themselves when the considered domain increases but the number of nodes grow in such a way that the L-node density remains constant throughout the investigated domain. In this scenario, what matters is the minimum value of the maximum transmission range averaged over the fading process, denoted as $d_{max}$, above which the network of NL-nodes almost surely gets asymptotically localized.