Skip to main content Accessibility help
×
Home

Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs

  • Bhupender Gupta (a1) and Srikanth K. Iyer (a2)

Abstract

Let n points be placed independently in d-dimensional space according to the density f(x) = A d e−λ||x||α , λ, α > 0, x ∈ ℝ d , d ≥ 2. Let d n be the longest edge length of the nearest-neighbor graph on these points. We show that (λ−1 log n)1−1/α d n - b n converges weakly to the Gumbel distribution, where b n ∼ ((d − 1)/λα) log log n. We also prove the following strong law for the normalized nearest-neighbor distance n = (λ−1 log n)1−1/α d n / log log n: (d − 1)/αλ ≤ lim inf n→∞ n ≤ lim sup n→∞ n d/αλ almost surely. Thus, the exponential rate of decay α = 1 is critical, in the sense that, for α > 1, d n → 0, whereas, for α ≤ 1, d n → ∞ almost surely as n → ∞.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs
      Available formats
      ×

Copyright

Corresponding author

Postal address: Department of Computer Science and Engineering, Indian Institute of Information Technology, Jabalpur 482011, India.
∗∗ Postal address: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. Email address: skiyer@math.iisc.ernet.in

Footnotes

Hide All

Research supported in part by UGC SAP IV and a grant from the DRDO-IISc program on Mathematical Engineering.

Footnotes

References

Hide All
Appel, M. J. B. and Russo, R. P. (1997). The minimum vertex degree of a graph on the uniform points in 0,1 d . Adv. Appl. Prob. 29, 582594.
Dette, H. and Henze, N. (1989). The limit distribution of the largest nearest-neighbour link in the unit d-cube. J. Appl. Prob. 26, 6780.
Gupta, B., Iyer, S. K. and Manjunath, D. (2005). On the topological properties of one dimensional exponential random geometric graphs. Random Structures and Algorithms 32, 181204.
Hsing, T. and Rootzén, H. (2005). Extremes on trees. Ann. Prob. 33, 413444.
Penrose, M. D. (1997). The longest edge of the minimal spanning tree. Ann. Appl. Prob. 7, 340361.
Penrose, M. D. (1998). Extremes for the minimal spanning tree on normally distributed points. Adv. Appl. Prob. 30, 628639.
Penrose, M. D. (1999). A strong law for the largest nearest-neighbour link between random points. J. London Math. Soc. 60, 951960.
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
Steele, J. M. and Tierney, L. (1986). Boundary domination and the distribution of the largest nearest-neighbor link in higher dimensions. J. Appl. Prob. 23, 524528.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed