Scale-Invariant Random Spatial Networks

TL;DR

This paper introduces a new class of mathematically defined random spatial networks that are exactly scale-invariant, modeling real-world road networks, and provides a concrete example based on minimum-time routes in a hierarchical road system.

Contribution

It formalizes the concept of scale-invariant random spatial networks, introduces axioms for their properties, and presents a concrete model satisfying these axioms, advancing the mathematical understanding of such networks.

Findings

Axiomatization of scale-invariant random spatial networks

Construction of a concrete model based on minimum-time routes

Expected applicability to other models like Poisson line processes

Abstract

Real-world road networks have an approximate scale-invariance property; can one devise mathematical models of random networks whose distributions are {\em exactly} invariant under Euclidean scaling? This requires working in the continuum plane. We introduce an axiomatization of a class of processes we call {\em scale-invariant random spatial networks}, whose primitives are routes between each pair of points in the plane. We prove that one concrete model, based on minimum-time routes in a binary hierarchy of roads with different speed limits, satisfies the axioms, and note informally that two other constructions (based on Poisson line processes and on dynamic proximity graphs) are expected also to satisfy the axioms. We initiate study of structure theory and summary statistics for general processes in this class.