  Railway systems provide pivotal support to modern societies, making their
efficiency and robustness important to ensure. However, these systems are
susceptible to disruptions and delays, leading to accumulating economic damage.
The large spatial scale of delay spreading typically make it difficult to
distinguish which regions will ultimately affected from an initial disruption,
creating uncertainty for risk assessment. In this paper, we identify
geographical structures that reflect how delay spreads through railway
networks. We do so by proposing a graph-based, hybrid schedule and
empirical-based model for delay propagation and apply spectral clustering. We
apply the model to four European railway systems: the Netherlands, Germany,
Switzerland and Italy. We characterize geographical structures in the railway
systems of these countries and interpret these regions in terms of delay
severity and how dynamically disconnected they are from the rest. The method
also allows us to point out important differences between these countries'
railway systems. For practitioners, this geographical characterization of
railways provide natural boundaries for local decision-making structures and a
first-order prioritization on which regions are at risk, given an initial
disruption.
