Connected Spatial Networks over Random Points and a Route-Length Statistic

Details Connected Spatial Networks over Random Points and a Route-Length Statistic Connected Spatial Networks over Random Points and a Route-Length Statistic

2010-03-19

arxiv.org/abs/1003.3700v2

Statistical Science 2010, Vol. 25, No. 3, 275-288

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical Science (http://www.imstat.org/sts/) by the Institute of M

Scientific paper

10.1214/10-STS335

We review mathematically tractable models for connected networks on random points in the plane, emphasizing the class of proximity graphs which deserves to be better known to applied probabilists and statisticians. We introduce and motivate a particular statistic $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ in a one-parameter family of proximity graphs. How close this family comes to the optimal trade-off over all possible networks remains an intriguing open question. The paper is a write-up of a talk developed by the first author during 2007--2009.

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