Geodesics and flows in a Poissonian city

TL;DR

This paper studies the geometry, fluctuations, and efficiency of near-geodesic paths in a Poissonian city network, introducing new asymptotic results and comparing them with grid-based networks.

Contribution

It introduces the concept of a Poissonian city, analyzes near-geodesic paths, and derives asymptotics for their geometric features and flow characteristics.

Findings

Near-geodesics have efficiency comparable to true geodesics.

Asymptotic behavior of geometric features is characterized.

Network flow characteristics are computed and compared to grid networks.

Abstract

The stationary isotropic Poisson line process was used to derive upper bounds on mean excess network geodesic length in Aldous and Kendall [Adv. in Appl. Probab. 40 (2008) 1-21]. The current paper presents a study of the geometry and fluctuations of near-geodesics in the network generated by the line process. The notion of a "Poissonian city" is introduced, in which connections between pairs of nodes are made using simple "no-overshoot" paths based on the Poisson line process. Asymptotics for geometric features and random variation in length are computed for such near-geodesic paths; it is shown that they traverse the network with an order of efficiency comparable to that of true network geodesics. Mean characteristics and limiting behavior at the center are computed for a natural network flow. Comparisons are drawn with similar network flows in a city based on a comparable rectilinear grid. A concluding section discusses several open problems.