Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

Abstract

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

Keywords

Point process, random graph, convex hull, degree

Publication Date

3-2003

Source Publication Title

Advances in Applied Probability

Volume

35

Issue

1

Start Page

49

End Page

55

Publisher

Applied Probability Trust

Peer Reviewed

1

Copyright

© Applied Probability Trust 2003

Funder

We acknowledge the support of the UK Engineering and Physical Sciences Research Council. SNC was also partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKBU/2048/02P).

DOI

10.1017/S0001867800012076

Link to Publisher's Edition

http://dx.doi.org/10.1017/S0001867800012076

ISSN (print)

00018678

ISSN (electronic)

14756064

Included in

Mathematics Commons

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