Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-24T16:00:15.281Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  01 July 2016

Bhupender Gupta*
Affiliation:
Indian Institute of Information Technology
Srikanth K. Iyer*
Affiliation:
Indian Institute of Science
*
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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let n points be placed independently in d-dimensional space according to the density f(x) = Ade−λ||x||α, λ, α > 0, x ∈ ℝd, d ≥ 2. Let dn be the longest edge length of the nearest-neighbor graph on these points. We show that (λ−1 log n)1−1/α dn - bn converges weakly to the Gumbel distribution, where bn ∼ ((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 dn/ log log n: (d − 1)/αλ ≤ lim infn→∞n ≤ lim supn→∞nd/αλ almost surely. Thus, the exponential rate of decay α = 1 is critical, in the sense that, for α > 1, dn → 0, whereas, for α ≤ 1, dn → ∞ almost surely as n → ∞.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

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

References

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.Google Scholar
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.Google Scholar
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.Google Scholar
Hsing, T. and Rootzén, H. (2005). Extremes on trees. Ann. Prob. 33, 413444.CrossRefGoogle Scholar
Penrose, M. D. (1997). The longest edge of the minimal spanning tree. Ann. Appl. Prob. 7, 340361.Google Scholar
Penrose, M. D. (1998). Extremes for the minimal spanning tree on normally distributed points. Adv. Appl. Prob. 30, 628639.Google Scholar
Penrose, M. D. (1999). A strong law for the largest nearest-neighbour link between random points. J. London Math. Soc. 60, 951960.CrossRefGoogle Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.CrossRefGoogle Scholar
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.Google Scholar