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

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.

💡 Research Summary

The paper investigates the probability that a randomly placed non‑localized node (NL‑node) can determine its position in a two‑dimensional region populated by two independent Poisson point processes: one representing nodes with known positions (L‑nodes) and the other representing nodes that need to localize themselves (NL‑nodes). The authors assume that an NL‑node can uniquely recover its (x, y) coordinates once it has direct communication links to at least three distinct L‑nodes, which is the standard trilateration requirement in two dimensions.

System Model

• The region S ⊂ ℝ² is a disk of radius R, area |S| = πR².
• L‑nodes are distributed with density ρ_L, NL‑nodes with density ρ_NL; the expected numbers are ρ_L|S| and ρ_NL|S| respectively.
• All nodes share the same transceiver hardware and operate under a shadow‑fading channel.
• Two nodes can communicate if the received power exceeds a threshold P_w,th.

Channel and Connectivity Models
Three models are considered:

1. Random Geometric Graph (RGG) – a deterministic range r: an edge exists if Euclidean distance ≤ r.
2. Path‑loss without shadowing – the maximum distance d_max at which the received power equals P_w,th. In this case d_max plays the same role as r.
3. Path‑loss with log‑normal shadowing – the received power in dB is a Gaussian random variable with mean μ_d = 10 log₁₀