Nearest-neighbour Markov point processes on graphs with Euclidean edges

TL;DR

This paper introduces nearest-neighbour Markov point processes on graphs with Euclidean edges, establishing their properties and consistency conditions, and providing characterizations for Markov functions based on local graph geometry.

Contribution

It defines a new class of Markov point processes on Euclidean edge graphs, characterizes their Markov functions, and verifies consistency conditions for these processes.

Findings

Delaunay neighbourhood relation on trees satisfies consistency conditions

Modified relation based on local geometry satisfies conditions for all Euclidean edge graphs

Provides a theoretical framework for Markov processes on linear networks

Abstract

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as the analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley--M{\o}ller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges.