Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

For connectivity of random geometric graphs, where there is no density for underlying distribution of the vertices, we consider n i.i.d. Cantor distributed points on [0, 1]. We show that for this random geometric graph, the connectivity threshold Rn, converges almost surely to a constant 1−2φ where 0 < φ < 1/2, which for standard Cantor distribution is 1/3. We also show that ‖Rn − (1− 2φ)‖1 ∼ 2C (φ) n−1/dφ where C (φ) > 0 is a constant and dφ := −log 2/log φ is a the Hausdorff dimension of the generalized Cantor set with parameter φ.