Nearest-neighbour Markov point processes on graphs with Euclidean edges
M. N. M. van Lieshout
TL;DR
This work develops a rigorous framework for nearest-neighbour Markov point processes on graphs with Euclidean edges, extending renewal-process ideas to linear networks. It defines a Delaunay-type neighbourhood $\sim_{\mathbf{x}}$ on configurations and proves Baddeley--Møller consistency for trees, with a complete Markov-function characterisation via a Hammersley--Clifford factorisation in terms of an interaction function $\gamma$ depending on the weighted shortest-path distance $d_G$. To cover general graphs, the authors introduce a local Delaunay relation $\sim^E_{\mathbf{z}}$ that preserves consistency, enabling tractable likelihoods and conditional intensities on arbitrary graphs. The results provide a principled, geometrically adapted approach to renewal-like point processes on networks, with explicit expressions for Papangelou intensities and a clear pathway to model fitting 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--Møller 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.