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

TL;DR

This paper reviews models of connected networks on random points, introduces a route-length statistic, and explores the trade-off between network length and route efficiency in proximity graphs.

Contribution

It introduces a new route-length statistic and analyzes the trade-off between network length and route efficiency in proximity graphs.

Findings

Monte Carlo simulations illustrate the trade-off between normalized network length and route statistic R.

The proximity graph family approaches an optimal trade-off, but the question remains open.

The paper emphasizes the importance of proximity graphs in applied probabilistic modeling.

Abstract

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.