The connectivity threshold of random geometric graphs with Cantor distributed vertices
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.
Volume (Year): 82 (2012)
Issue (Month): 12 ()
|Contact details of provider:|| Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description|
|Order Information:|| Postal: http://www.elsevier.com/wps/find/supportfaq.cws_home/regional|
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- 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.
- 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.
- Hosking, J. R. M., 1994. "Moments of order statistics of the Cantor distribution," Statistics & Probability Letters, Elsevier, vol. 19(2), pages 161-165, January.
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.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Shamier, Wendy)
If references are entirely missing, you can add them using this form.