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The connectivity threshold of random geometric graphs with Cantor distributed vertices

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  • Bandyopadhyay, Antar
  • Sajadi, Farkhondeh

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.

Suggested Citation

  • Bandyopadhyay, Antar & Sajadi, Farkhondeh, 2012. "The connectivity threshold of random geometric graphs with Cantor distributed vertices," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2103-2107.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:12:p:2103-2107
    DOI: 10.1016/j.spl.2012.07.015
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    References listed on IDEAS

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    1. Hosking, J. R. M., 1994. "Moments of order statistics of the Cantor distribution," Statistics & Probability Letters, Elsevier, vol. 19(2), pages 161-165, January.
    2. Knopfmacher, Arnold & Prodinger, Helmut, 1996. "Explicit and asymptotic formulae for the expected values of the order statistics of the Cantor distribution," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 189-194, April.
    3. Appel, Martin J. B. & Russo, Ralph P., 2002. "The connectivity of a graph on uniform points on [0,1]d," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 351-357, December.
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