Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

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 $1-2\varphi$ where $0<\varphi <1/2$, which for the standard Cantor distribution is 1/3. We also show that $\Vert R_n-(1-2\varphi ){\Vert }_1\sim 2C(\varphi )n^{-1/d_{\varphi }}$ where $C(\varphi )>0$ is a constant and $d_{\varphi }:=-\log 2/\log \varphi$ is the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $\varphi$.

8 pages