Multi-modal and multi-layer robustness analysis of the European rail and air networks.

The robustness of long-distance transport services is paramount for ensuring network connectivity under disruptions. We conduct a comparative analysis of the European rail and air networks of 124 main metropolitan areas, assessing their ability to withstand successive network degradation. We undertake a multi-modal and multi-layer approach in our analysis of the robustness of long-distance transport networks. In particular, we are interested in the role of individual nodes for both air and rail networks as well as for the integrated multi-modal network. Given the hierarchical nature of long-distance transport services, we adopt a multi-layer perspective by means of clustering nodes based on their criticality in order to identify common performance profiles. Original metrics are formulated to measure the impact of nodes on network fragmentation, both at the individual level and as cluster members. Additionally, a new metric is introduced to assess cities' reliance on air transportation, when considering the integrated multi-layer air-rail network during disruptions in air connections. Our findings indicate that the air network exhibits significantly greater robustness compared to rail, i.e. 10% versus 71% performance loss in the worst case scenario, respectively. Furthermore, primary sub-graph of nodes whose protection from attacks can greatly enhance network's overall robustness is identified. We discuss our findings in terms of the relationship between network structure, robustness, and the role of critical nodes as well as propose potential mitigation measures.