Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

Details Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

2012-04-03

arxiv.org/abs/1204.0667v1

Mathematics

Probability

6 pages

Scientific paper

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 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 a the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $\phi$.

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