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Random Geometric Graphs$
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Mathew Penrose

Print publication date: 2003

Print ISBN-13: 9780198506263

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506263.001.0001

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MINIMUM DEGREE: CONVERGENCE IN DISTRIBUTION

MINIMUM DEGREE: CONVERGENCE IN DISTRIBUTION

Chapter:
(p.155) 8 MINIMUM DEGREE: CONVERGENCE IN DISTRIBUTION
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0008

This chapter demonstrates certain convergence in distribution results for the number of vertices of degree k in G(n,r) from which convergence in distribution results are derived for the largest k-nearest neighbour link on n random points (denoted Mk ). In the case of uniformly distributed points in the cube, the value of Mk d (scaled and centred appropriately) converges in distribution to the double exponential distribution. A similar result is demonstrated for M1 (with different scaling and centring constants) in the case of normally distributed points.

Keywords:   largest k-nearest neighbour link, convergence in distribution, double exponential distribution, uniformly distributed points, normally distributed points

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