"Stationary" point processes are uncommon on linear networks

Statistical methodology for analysing patterns of points on a network of lines, such as road traffic accident locations, often assumes that the underlying point process is "stationary" or "correlation-stationary." However, such processes appear to be rare. In this paper, popular procedures for constructing a point process are adapted to linear networks: many of the resulting models are no longer stationary when distance is measured by the shortest path in the network. This undermines the rationale for popular statistical methods such as the K-function and pair correlation function. Alternative strategies are proposed, such as replacing the shortest-path distance by another metric on the network.