IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v82y2012i12p2103-2107.html
   My bibliography  Save this article

The connectivity threshold of random geometric graphs with Cantor distributed vertices

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715212002878
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2012.07.015?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Hosking, J. R. M., 1994. "Moments of order statistics of the Cantor distribution," Statistics & Probability Letters, Elsevier, vol. 19(2), pages 161-165, January.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Arnold, Barry C., 2011. "The generalized Cantor distribution and its corresponding inverse distribution," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1098-1103, August.
    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. Grabner, P. J. & Prodinger, H., 1996. "Asymptotic analysis of the moments of the Cantor distribution," Statistics & Probability Letters, Elsevier, vol. 26(3), pages 243-248, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:82:y:2012:i:12:p:2103-2107. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.