 
Summary: Statistical Science
2010, Vol. 25, No. 3, 275288
DOI: 10.1214/10STS335
© Institute of Mathematical Statistics, 2010
Connected Spatial Networks over Random
Points and a RouteLength Statistic
David J. Aldous and Julian Shun
Abstract. We review mathematically tractable models for connected net
works on random points in the plane, emphasizing the class of proximity
graphs which deserves to be better known to applied probabilists and statisti
cians. We introduce and motivate a particular statistic R measuring shortness
of routes in a network. We illustrate, via Monte Carlo in part, the tradeoff
between normalized network length and R in a oneparameter family of prox
imity graphs. How close this family comes to the optimal tradeoff over all
possible networks remains an intriguing open question.
The paper is a writeup of a talk developed by the first author during 2007
2009.
Key words and phrases: Proximity graph, random graph, spatial network,
geometric graph.
1. INTRODUCTION
