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