Department of Mathematics
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.
Point process, random graph, convex hull, degree
Source Publication Title
Advances in Applied Probability
Applied Probability Trust
© Applied Probability Trust 2003
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).
Link to Publisher's Edition
Chiu, S. N., and I. S. Molchanov. "A new graph related to the directions of nearest neighbours in a point process." Advances in Applied Probability 35.1 (2003): 49-55.