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Limit theory for the random on-line nearest-neighbor graph

Penrose, Mathew D.; Wade, Andrew R.

Authors

Mathew D. Penrose



Abstract

In the on-line nearest-neighbor graph (ONG), each point after the first in a sequence of points in ℝd is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large-sample asymptotic behavior of the total power-weighted length of the ONG on uniform random points in (0,1)d. In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbor (directed) graph on uniform random points in the unit interval.

Citation

Penrose, M. D., & Wade, A. R. (2008). Limit theory for the random on-line nearest-neighbor graph. Random Structures and Algorithms, 32(2), 125-156. https://doi.org/10.1002/rsa.20185

Journal Article Type Article
Publication Date Mar 1, 2008
Deposit Date Oct 4, 2012
Journal Random Structures and Algorithms
Print ISSN 1042-9832
Publisher Wiley
Peer Reviewed Peer Reviewed
Volume 32
Issue 2
Pages 125-156
DOI https://doi.org/10.1002/rsa.20185
Keywords Nearest-neighbor graph,Spatial network evolution, Weak convergence, Fixed-point equation, Divide-and-conquer.

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