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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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TYPICAL VERTEX DEGREES

TYPICAL VERTEX DEGREES

Chapter:
(p.74) 4 TYPICAL VERTEX DEGREES
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0004

This chapter is concerned with the empirical process of k-nearest neighbour distances for n random points, where k=k(n) is specified and is either fixed or grows with n. That is, the proportion of k-nearest neighbour distances amongst n random points which are less than t, is considered. This is shown to obey a law of large numbers, and after appropriate centring and scaling, to converge to a Gaussian process in t. The number of k-nearest neighbour distances less than t can be re-interpreted as the number of vertices of G(n,t) of degree at least k.

Keywords:   empirical process, k-nearest neighbour distances, vertex degrees, random points, large numbers, Gaussian process

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