True scale-invariant random spatial networks.

Some aspects of real-world road networks seem to have an approximate scale invariance property, motivating study of mathematical models of random networks whose distributions are exactly invariant under Euclidean scaling. This requires working in the continuum plane, so making a precise definition is not trivial. We introduce an axiomatization of a class of processes we call scale-invariant random spatial networks, whose primitives are routes between each pair of points in the plane. One concrete model, based on minimum-time routes in a binary hierarchy of roads with different speed limits, has been shown to satisfy the axioms, and two other constructions (based on Poisson line processes and on dynamic proximity graphs) are expected also to do so. We initiate study of structure theory and summary statistics for general processes in the class. Many questions arise in this setting via analogies with diverse existing topics, from geodesics in first-passage percolation to transit node-based route-finding algorithms.