HC
H. Cheng
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European long-distance passenger travel is carried largely by two networks, a nationally fragmented rail system and a partially liberalised air system. The robustness of both is conventionally assessed on a topological representation, as the capacity of the graph to maintain connectivity under the removal of nodes and links. Such an assessment omits the service-level characteristics on which passengers experience a disruption, the timetable that determines when a connection exists, the transfer window that determines whether it can be used, and the operator that decides whether a stranded passenger may be rebooked. A robustness analysis of the European rail and air service networks is therefore conducted to determine to what extent these networks are robust to disruptions when service-level characteristics are incorporated into their network representation.
To answer this question two complementary multilayer transport network representations are constructed. The structural representation keeps rail and air as coupled but distinct layers whose edges are the segments between consecutive stations, the non-stop flights, and the public-transport links among the terminals of the same urban area, while the functional representation uses the structural network as its topological layer and extends it with the scheduled services that realise each connection, carrying their operator, timetable, and generalised travel cost. The two are deliberately kept apart rather than merged into one graph, so that a disruption can be aimed at the structural or the supply level independently while the shared node set keeps the two sets of results comparable. On the functional network a path-enumeration algorithm constructs the 482,947,150 feasible itineraries between 1,674 rail stations and 197 airports in 30 countries, and robustness is measured mainly through the retained efficiency R(x) and the share of baseline itinerary quality that survives a disruption. Three experiment families are evaluated. Structural stress tests remove edges by unweighted topological betweenness, by a service-aware betweenness set by the itineraries each edge carries, or at random. Operator withdrawals and cooperation-tier restrictions act on the 82 rail and air operators. Two documented macro-shocks, the Iceland 2010 volcanic eruption and the July 2021 Bernd flood, test the network's response to real hazards.
Incorporating service-level information does not merely lower the measured robustness, it changes which edges are deemed critical. The two rankings share not a single rail edge among their top fifty, since topology elevates the international cut-edges that keep the continental graph connected while service load concentrates 88 per cent of the top two hundred edges on the German Rhine-Ruhr and Frankfurt-Mannheim corridors. Removing the top five per cent of edges leaves retained efficiency at R(x) = 0.168 in service-aware order against 0.557 in topology order, and against the random benchmark the topology failure is statistically indistinguishable (z_R = -0.47) while the service-aware failure is an extreme outlier (z_R = -8.51). The damage falls as outright disconnection rather than delay, so the service layer collapses long before the largest connected component does. Rail-air co-existence does not offset this concentration. The robustness of the hub-and-spoke air layer does not transfer to the combined network, since an air leg reaches it through rail-side airport-feeder edges whose removal severs the intermodal itinerary at its rail end. The two macro-shocks test this in both directions. An ash closure grounding 77.7 per cent of active airports still leaves R(x) = 0.875 through the immunity of the rail-only baselines, with not a single air baseline replaced by a within-tolerance rail substitute. The Bernd flood lowers intra-German retained efficiency to R(x) = 0.510, with approximately 81,000 structurally feasible air substitutes arriving outside the reroute tolerance. In neither direction does cross-modal rescue materialise.
In sum, robustness is not a fixed property of the European air and rail networks. It depends on how the network is represented, on how severe and how targeted the disruption is, and on how service is organised across operators, from the concentration of supply in a few of them to the cooperation rules that decide whose services a passenger may combine. With service-level characteristics incorporated, they prove substantially less robust to disruptions than their topology suggests. ...
To answer this question two complementary multilayer transport network representations are constructed. The structural representation keeps rail and air as coupled but distinct layers whose edges are the segments between consecutive stations, the non-stop flights, and the public-transport links among the terminals of the same urban area, while the functional representation uses the structural network as its topological layer and extends it with the scheduled services that realise each connection, carrying their operator, timetable, and generalised travel cost. The two are deliberately kept apart rather than merged into one graph, so that a disruption can be aimed at the structural or the supply level independently while the shared node set keeps the two sets of results comparable. On the functional network a path-enumeration algorithm constructs the 482,947,150 feasible itineraries between 1,674 rail stations and 197 airports in 30 countries, and robustness is measured mainly through the retained efficiency R(x) and the share of baseline itinerary quality that survives a disruption. Three experiment families are evaluated. Structural stress tests remove edges by unweighted topological betweenness, by a service-aware betweenness set by the itineraries each edge carries, or at random. Operator withdrawals and cooperation-tier restrictions act on the 82 rail and air operators. Two documented macro-shocks, the Iceland 2010 volcanic eruption and the July 2021 Bernd flood, test the network's response to real hazards.
Incorporating service-level information does not merely lower the measured robustness, it changes which edges are deemed critical. The two rankings share not a single rail edge among their top fifty, since topology elevates the international cut-edges that keep the continental graph connected while service load concentrates 88 per cent of the top two hundred edges on the German Rhine-Ruhr and Frankfurt-Mannheim corridors. Removing the top five per cent of edges leaves retained efficiency at R(x) = 0.168 in service-aware order against 0.557 in topology order, and against the random benchmark the topology failure is statistically indistinguishable (z_R = -0.47) while the service-aware failure is an extreme outlier (z_R = -8.51). The damage falls as outright disconnection rather than delay, so the service layer collapses long before the largest connected component does. Rail-air co-existence does not offset this concentration. The robustness of the hub-and-spoke air layer does not transfer to the combined network, since an air leg reaches it through rail-side airport-feeder edges whose removal severs the intermodal itinerary at its rail end. The two macro-shocks test this in both directions. An ash closure grounding 77.7 per cent of active airports still leaves R(x) = 0.875 through the immunity of the rail-only baselines, with not a single air baseline replaced by a within-tolerance rail substitute. The Bernd flood lowers intra-German retained efficiency to R(x) = 0.510, with approximately 81,000 structurally feasible air substitutes arriving outside the reroute tolerance. In neither direction does cross-modal rescue materialise.
In sum, robustness is not a fixed property of the European air and rail networks. It depends on how the network is represented, on how severe and how targeted the disruption is, and on how service is organised across operators, from the concentration of supply in a few of them to the cooperation rules that decide whose services a passenger may combine. With service-level characteristics incorporated, they prove substantially less robust to disruptions than their topology suggests. ...
European long-distance passenger travel is carried largely by two networks, a nationally fragmented rail system and a partially liberalised air system. The robustness of both is conventionally assessed on a topological representation, as the capacity of the graph to maintain connectivity under the removal of nodes and links. Such an assessment omits the service-level characteristics on which passengers experience a disruption, the timetable that determines when a connection exists, the transfer window that determines whether it can be used, and the operator that decides whether a stranded passenger may be rebooked. A robustness analysis of the European rail and air service networks is therefore conducted to determine to what extent these networks are robust to disruptions when service-level characteristics are incorporated into their network representation.
To answer this question two complementary multilayer transport network representations are constructed. The structural representation keeps rail and air as coupled but distinct layers whose edges are the segments between consecutive stations, the non-stop flights, and the public-transport links among the terminals of the same urban area, while the functional representation uses the structural network as its topological layer and extends it with the scheduled services that realise each connection, carrying their operator, timetable, and generalised travel cost. The two are deliberately kept apart rather than merged into one graph, so that a disruption can be aimed at the structural or the supply level independently while the shared node set keeps the two sets of results comparable. On the functional network a path-enumeration algorithm constructs the 482,947,150 feasible itineraries between 1,674 rail stations and 197 airports in 30 countries, and robustness is measured mainly through the retained efficiency R(x) and the share of baseline itinerary quality that survives a disruption. Three experiment families are evaluated. Structural stress tests remove edges by unweighted topological betweenness, by a service-aware betweenness set by the itineraries each edge carries, or at random. Operator withdrawals and cooperation-tier restrictions act on the 82 rail and air operators. Two documented macro-shocks, the Iceland 2010 volcanic eruption and the July 2021 Bernd flood, test the network's response to real hazards.
Incorporating service-level information does not merely lower the measured robustness, it changes which edges are deemed critical. The two rankings share not a single rail edge among their top fifty, since topology elevates the international cut-edges that keep the continental graph connected while service load concentrates 88 per cent of the top two hundred edges on the German Rhine-Ruhr and Frankfurt-Mannheim corridors. Removing the top five per cent of edges leaves retained efficiency at R(x) = 0.168 in service-aware order against 0.557 in topology order, and against the random benchmark the topology failure is statistically indistinguishable (z_R = -0.47) while the service-aware failure is an extreme outlier (z_R = -8.51). The damage falls as outright disconnection rather than delay, so the service layer collapses long before the largest connected component does. Rail-air co-existence does not offset this concentration. The robustness of the hub-and-spoke air layer does not transfer to the combined network, since an air leg reaches it through rail-side airport-feeder edges whose removal severs the intermodal itinerary at its rail end. The two macro-shocks test this in both directions. An ash closure grounding 77.7 per cent of active airports still leaves R(x) = 0.875 through the immunity of the rail-only baselines, with not a single air baseline replaced by a within-tolerance rail substitute. The Bernd flood lowers intra-German retained efficiency to R(x) = 0.510, with approximately 81,000 structurally feasible air substitutes arriving outside the reroute tolerance. In neither direction does cross-modal rescue materialise.
In sum, robustness is not a fixed property of the European air and rail networks. It depends on how the network is represented, on how severe and how targeted the disruption is, and on how service is organised across operators, from the concentration of supply in a few of them to the cooperation rules that decide whose services a passenger may combine. With service-level characteristics incorporated, they prove substantially less robust to disruptions than their topology suggests.
To answer this question two complementary multilayer transport network representations are constructed. The structural representation keeps rail and air as coupled but distinct layers whose edges are the segments between consecutive stations, the non-stop flights, and the public-transport links among the terminals of the same urban area, while the functional representation uses the structural network as its topological layer and extends it with the scheduled services that realise each connection, carrying their operator, timetable, and generalised travel cost. The two are deliberately kept apart rather than merged into one graph, so that a disruption can be aimed at the structural or the supply level independently while the shared node set keeps the two sets of results comparable. On the functional network a path-enumeration algorithm constructs the 482,947,150 feasible itineraries between 1,674 rail stations and 197 airports in 30 countries, and robustness is measured mainly through the retained efficiency R(x) and the share of baseline itinerary quality that survives a disruption. Three experiment families are evaluated. Structural stress tests remove edges by unweighted topological betweenness, by a service-aware betweenness set by the itineraries each edge carries, or at random. Operator withdrawals and cooperation-tier restrictions act on the 82 rail and air operators. Two documented macro-shocks, the Iceland 2010 volcanic eruption and the July 2021 Bernd flood, test the network's response to real hazards.
Incorporating service-level information does not merely lower the measured robustness, it changes which edges are deemed critical. The two rankings share not a single rail edge among their top fifty, since topology elevates the international cut-edges that keep the continental graph connected while service load concentrates 88 per cent of the top two hundred edges on the German Rhine-Ruhr and Frankfurt-Mannheim corridors. Removing the top five per cent of edges leaves retained efficiency at R(x) = 0.168 in service-aware order against 0.557 in topology order, and against the random benchmark the topology failure is statistically indistinguishable (z_R = -0.47) while the service-aware failure is an extreme outlier (z_R = -8.51). The damage falls as outright disconnection rather than delay, so the service layer collapses long before the largest connected component does. Rail-air co-existence does not offset this concentration. The robustness of the hub-and-spoke air layer does not transfer to the combined network, since an air leg reaches it through rail-side airport-feeder edges whose removal severs the intermodal itinerary at its rail end. The two macro-shocks test this in both directions. An ash closure grounding 77.7 per cent of active airports still leaves R(x) = 0.875 through the immunity of the rail-only baselines, with not a single air baseline replaced by a within-tolerance rail substitute. The Bernd flood lowers intra-German retained efficiency to R(x) = 0.510, with approximately 81,000 structurally feasible air substitutes arriving outside the reroute tolerance. In neither direction does cross-modal rescue materialise.
In sum, robustness is not a fixed property of the European air and rail networks. It depends on how the network is represented, on how severe and how targeted the disruption is, and on how service is organised across operators, from the concentration of supply in a few of them to the cooperation rules that decide whose services a passenger may combine. With service-level characteristics incorporated, they prove substantially less robust to disruptions than their topology suggests.
Accessibility reflects the ease with which different individuals can overcome travel impedance to reach spatially distributed opportunities. Since accessibility of public transport networks is jointly determined by network topology and service attributes, this study applies access graphs derived from time-weighted L-space and frequency-weighted P-space graphs as the standardized framework. For previously underresearched East Asia region, a new dataset is compiled from open-source data for 61 systems and access graphs are constructed over increasing generalized travel-time budgets, and reachability and equity indicators at critical time points are benchmarked against systems in other world regions. Finally, based on the temporal evolution of average degree, East Asian and metro networks worldwide are classified into four clusters using k-means clustering. The findings show that East Asian metro systems vary widely in sizes. Influenced by the accessibility growth pattern, most medium- to mega-sized networks follow logistic (S-shaped) curves of degree growth consistent with a core–branch structure, whereas smaller or systems with degraded service have irregular degree growth curves. In terms of performance, large systems show greater spatial disparity and less uniform service quality, resulting in lower equity and reachability at 30 minutes. Regardless of size, East Asian metros tend to underperform in the medium and late stages of accessibility growth due to early decay in degree-growth rate and limited reachability at time of maximum degree growth rate. When controlling for size, East Asian metros remain less reachable than European systems and experience earlier growth-rate decay than both European and North American counterparts. Based on the timing of the peak and subsequent decay in degree growth rate, four distinctive accessibility growth patterns are identified across the worldwide systems. Clustering analysis further reveals that large East Asian networks experience an earlier decline in accessibility growth, similar to networks lacking direct cross-line connections, whereas most medium and small networks sustain growth longer on par with those in other regions. The presence of networks of similar size in different clusters suggests that improving service attributes can enhance overall accessibility performance.
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Accessibility reflects the ease with which different individuals can overcome travel impedance to reach spatially distributed opportunities. Since accessibility of public transport networks is jointly determined by network topology and service attributes, this study applies access graphs derived from time-weighted L-space and frequency-weighted P-space graphs as the standardized framework. For previously underresearched East Asia region, a new dataset is compiled from open-source data for 61 systems and access graphs are constructed over increasing generalized travel-time budgets, and reachability and equity indicators at critical time points are benchmarked against systems in other world regions. Finally, based on the temporal evolution of average degree, East Asian and metro networks worldwide are classified into four clusters using k-means clustering. The findings show that East Asian metro systems vary widely in sizes. Influenced by the accessibility growth pattern, most medium- to mega-sized networks follow logistic (S-shaped) curves of degree growth consistent with a core–branch structure, whereas smaller or systems with degraded service have irregular degree growth curves. In terms of performance, large systems show greater spatial disparity and less uniform service quality, resulting in lower equity and reachability at 30 minutes. Regardless of size, East Asian metros tend to underperform in the medium and late stages of accessibility growth due to early decay in degree-growth rate and limited reachability at time of maximum degree growth rate. When controlling for size, East Asian metros remain less reachable than European systems and experience earlier growth-rate decay than both European and North American counterparts. Based on the timing of the peak and subsequent decay in degree growth rate, four distinctive accessibility growth patterns are identified across the worldwide systems. Clustering analysis further reveals that large East Asian networks experience an earlier decline in accessibility growth, similar to networks lacking direct cross-line connections, whereas most medium and small networks sustain growth longer on par with those in other regions. The presence of networks of similar size in different clusters suggests that improving service attributes can enhance overall accessibility performance.