Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

TL;DR

This paper investigates the connectivity threshold of random geometric graphs with vertices distributed according to a Cantor distribution, revealing convergence properties and explicit formulas involving the Hausdorff dimension.

Contribution

It establishes the almost sure convergence of the connectivity threshold for Cantor distributed vertices and derives its asymptotic behavior related to the Hausdorff dimension.

Findings

Connectivity threshold converges to 1 - 2φ almost surely.

The difference between the threshold and its limit scales as n^{-1/d_φ}.

Explicit formula for the limit involving the Cantor set parameter.

Abstract

For connectivity of \emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_{n}$, converges almost surely to a constant $1-2\phi$ where $0 < \phi < 1/2$, which for the standard Cantor distribution is 1/3. We also show that $\| R_n - (1 - 2 \phi) \|_1 \sim 2 \, C(\phi) \, n^{-1/d_{\phi}}$ where $C(\phi) > 0$ is a constant and $d_{\phi} := - {\log 2}/{\log \phi}$ is the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $\phi$.