The stretch-length tradeoff in geometric networks: average case and worst case study

Abstract

Consider a network linking the points of a rate-1 Poisson point process on the plane. Write ave (s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function ave (s), and on the analogous "worst-case" function worst (s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent α such that each function has (s) (s − 1) −α as s ↓ 1.

Route lengths in invariant spatial tree networks

“…Figure 1 (right) shows the 5 maximal paths and the 1 maximal circuit in that example. Note that, by definition, there is a single-color path 4 immediately inside and a single-opposite-color path immediately outside each maximal path or circuit. Moreover the colors of these immediately-inside paths are the same (say •) for each component, because a path in G between a vertex in each component must cross component boundaries an even number of times.…”

“…Substantial recent literature, surveyed in the 2018 monograph [9], concerns toy models of more specific types of real-world spatial network, studied in statistical physics style rather than theorem-proof style. Intermediate between those styles, and envisioning examples such as inter-city road networks, one can model the city positions as a Poisson point process, and one can study the trade-off between a network's cost (taken as network length) and its effectiveness at providing short routes [4,5,6]. It is often remarked that tree networks are obviously very ineffective at providing short routes, and the purpose of this article is to give one formalization, as Theorem 1.2.…”

Route Lengths in Invariant Spatial Tree Networks

“…Substantial recent literature, surveyed in the 2018 monograph [9], concerns toy models of more specific types of real-world spatial network, studied in statistical physics style rather than theorem-proof style. Intermediate between those styles, and envisioning examples such as inter-city road networks, one can model the city positions as a Poisson point process, and one can study the trade-off between a network's cost (taken as network length) and its effectiveness at providing short routes [4,5,6]. It is often remarked that tree networks are obviously very ineffective at providing short routes, and the purpose of this article is to give one formalization, as Theorem 2.…”