Asymptotic Critical Radii in Random Geometric Graphs over Higher-dimensional Regions

TL;DR

This paper derives the exact asymptotic distribution of critical radii for connectivity and degree in high-dimensional random geometric graphs, advancing understanding of their phase transition behavior in complex regions.

Contribution

It provides the first precise asymptotic analysis of critical radii for k-connectivity and minimum degree in high-dimensional regions, extending previous results to higher dimensions.

Findings

Asymptotic distribution formulas for critical radii are established.

Results apply to regions in  with d, improving understanding of phase transitions.

The analysis covers complex, higher-dimensional geometric regions.

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 a random geometry graph distributed over a unit-volume region $\Omega\subset \mathbb{R}^d (d\geq 3)$.