Asymptotic critical transmission radii in random geometry graphs over three-dimensional regions

2025-11-21

AI Summary

This study addresses the asymptotic distribution of two critical transmission radii in three-dimensional random geometric graphs: one defined by $k$-connectivity and the other by minimum vertex degree. Existing theory lacks precise asymptotic characterizations for the 3D case. To bridge this gap, we model node deployment via a spatial Poisson point process and employ rigorous probabilistic analysis combined with extreme-value asymptotics. We derive, for the first time, the exact asymptotic distributions of both radii under the high-density limit—both converge to the Gumbel extreme-value distribution, with explicitly characterized centering and scaling constants. This result reveals the fine-grained structure of the connectivity phase transition in 3D wireless networks and establishes, for the first time, the asymptotic equivalence between the $k$-connectivity and minimum-degree thresholds in three dimensions. The findings provide foundational theoretical support and principled guidelines for topology-aware design and parameter selection in 3D sensor networks and UAV communication systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Security and Privacy: Large-scale security measurements

Abstract

This article presents the precise asymptotical distribution of two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, for random geometry graphs distributed over three-dimensional regions.

Problem

• Analyzing critical transmission radii in 3D random geometry graphs
• Studying k-connectivity and minimum vertex degree requirements
• Deriving asymptotic distribution for wireless network connectivity

Innovation

• Analyzes critical transmission radii asymptotically
• Focuses on k-connectivity and minimum vertex degree
• Models random geometry graphs in 3D regions