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

Part of: Geometric probability and stochastic geometry Stochastic processes Graph theory

Published online by Cambridge University Press:  01 July 2016

Bhupender Gupta  and
Srikanth K. Iyer
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
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Abstract

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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 d̃n = (λ−1 log n)1−1/α dn/ log log n: (d − 1)/αλ ≤ lim infn→∞d̃n ≤ lim supn→∞d̃n ≤ d/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 → ∞.

Keywords

Random geometric graphnearest-neighbor graphPoisson point processlargest nearest-neighbor linkvertex degree

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 05C05: Trees 90C27: Combinatorial optimization
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 631 - 658
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.

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