The connectivity threshold of random geometric graphs with Cantor distributed vertices

Antar Bandyopadhyay and Farkhondeh Sajadi

Statistics & Probability Letters, 2012, vol. 82, issue 12, 2103-2107

Abstract: For the connectivity of random geometric graphs, where there is no density for the underlying distribution of the vertices, we consider n i.i.d. Cantor distributed points on [0,1]. We show that for such a random geometric graph, the connectivity threshold, Rn, converges almost surely to a constant 1−2ϕ where 0<ϕ<1/2, which for the 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≔−log2/logϕ is the Hausdorff dimension of the generalized Cantor set with parameter ϕ.

Keywords: Cantor distribution; Connectivity threshold; Random geometric graph; Singular distributions (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spl.2012.07.015

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