Y. Xin
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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.
Machine Learning for Multimodal Freight Chain Modelling
Case Study of NEAC’s Mode Chain Builder System
Freight transport plays a critical role in supporting global trade, with multimodal transport systems modelling gaining importance due to their potential to optimize efficiency and sustainability. Accurately modeling these multimodal freight chains is essential for infrastructure planning and policy-making. Yet, it remains a persistent challenge due to fragmented datasets, limited granularity, and the absence of observed multimodal chain-level data. Traditional modeling approaches, particularly heuristic-based methods, often struggle to incorporate real-world operational constraints such as port selection logic and cargo handling requirements. Moreover, these models are typically inflexible and computationally intensive.
This research seeks to develop an adaptive multimodal freight chain model that addresses these limitations. Specifically, it introduces a practical path construction framework that integrates port selection based on geographic and functional suitability, aligning cargo handling requirements with port capabilities during the construction of mode chains. This research also tries to address a gap in the literature by applying machine learning to the estimation of multimodal freight flows, a domain traditionally dominated by heuristic and optimization-based methods. To estimate freight demand distribution across the generated chains, this study explores the use of machine learning, particularly the Expectation-Maximization (EM) algorithm, to leverage the abundant but often unstructured transport data available. The EM model enables demand share prediction without relying on labeled training data, reducing the calibration burden and enhancing model responsiveness to observed transport flows.
The proposed modeling framework is applied to a case study based on the NEAC Mode Chain Builder system for inter-country freight movements between the Netherlands and Belgium, two countries with high multimodal connectivity and the largest ports in Europe. The results demonstrate the model’s ability to generate valid mode chain alternatives and to significantly reduce deviations between predicted and observed freight flows, particularly for sea and rail segments. While some deviation increases occur in other segments, these are outweighed by the overall improvement in prediction accuracy. The EM model also shows stable convergence behavior, confirming its potential under data-limited conditions. However, residual deviations suggest that external factors, such as data incompleteness or behavioral uncertainties, still limit full accuracy.
This study highlights the potential of combining graph search algorithms with unsupervised learning to enhance multimodal freight chain modeling, especially under data-constrained conditions. It contributes both a methodological and practical solution for building data-driven multimodal freight transport models that better reflect operational realities and observed empirical data that can be used to improve the freight transport planning and decision-making process.
...
This research seeks to develop an adaptive multimodal freight chain model that addresses these limitations. Specifically, it introduces a practical path construction framework that integrates port selection based on geographic and functional suitability, aligning cargo handling requirements with port capabilities during the construction of mode chains. This research also tries to address a gap in the literature by applying machine learning to the estimation of multimodal freight flows, a domain traditionally dominated by heuristic and optimization-based methods. To estimate freight demand distribution across the generated chains, this study explores the use of machine learning, particularly the Expectation-Maximization (EM) algorithm, to leverage the abundant but often unstructured transport data available. The EM model enables demand share prediction without relying on labeled training data, reducing the calibration burden and enhancing model responsiveness to observed transport flows.
The proposed modeling framework is applied to a case study based on the NEAC Mode Chain Builder system for inter-country freight movements between the Netherlands and Belgium, two countries with high multimodal connectivity and the largest ports in Europe. The results demonstrate the model’s ability to generate valid mode chain alternatives and to significantly reduce deviations between predicted and observed freight flows, particularly for sea and rail segments. While some deviation increases occur in other segments, these are outweighed by the overall improvement in prediction accuracy. The EM model also shows stable convergence behavior, confirming its potential under data-limited conditions. However, residual deviations suggest that external factors, such as data incompleteness or behavioral uncertainties, still limit full accuracy.
This study highlights the potential of combining graph search algorithms with unsupervised learning to enhance multimodal freight chain modeling, especially under data-constrained conditions. It contributes both a methodological and practical solution for building data-driven multimodal freight transport models that better reflect operational realities and observed empirical data that can be used to improve the freight transport planning and decision-making process.
...
Freight transport plays a critical role in supporting global trade, with multimodal transport systems modelling gaining importance due to their potential to optimize efficiency and sustainability. Accurately modeling these multimodal freight chains is essential for infrastructure planning and policy-making. Yet, it remains a persistent challenge due to fragmented datasets, limited granularity, and the absence of observed multimodal chain-level data. Traditional modeling approaches, particularly heuristic-based methods, often struggle to incorporate real-world operational constraints such as port selection logic and cargo handling requirements. Moreover, these models are typically inflexible and computationally intensive.
This research seeks to develop an adaptive multimodal freight chain model that addresses these limitations. Specifically, it introduces a practical path construction framework that integrates port selection based on geographic and functional suitability, aligning cargo handling requirements with port capabilities during the construction of mode chains. This research also tries to address a gap in the literature by applying machine learning to the estimation of multimodal freight flows, a domain traditionally dominated by heuristic and optimization-based methods. To estimate freight demand distribution across the generated chains, this study explores the use of machine learning, particularly the Expectation-Maximization (EM) algorithm, to leverage the abundant but often unstructured transport data available. The EM model enables demand share prediction without relying on labeled training data, reducing the calibration burden and enhancing model responsiveness to observed transport flows.
The proposed modeling framework is applied to a case study based on the NEAC Mode Chain Builder system for inter-country freight movements between the Netherlands and Belgium, two countries with high multimodal connectivity and the largest ports in Europe. The results demonstrate the model’s ability to generate valid mode chain alternatives and to significantly reduce deviations between predicted and observed freight flows, particularly for sea and rail segments. While some deviation increases occur in other segments, these are outweighed by the overall improvement in prediction accuracy. The EM model also shows stable convergence behavior, confirming its potential under data-limited conditions. However, residual deviations suggest that external factors, such as data incompleteness or behavioral uncertainties, still limit full accuracy.
This study highlights the potential of combining graph search algorithms with unsupervised learning to enhance multimodal freight chain modeling, especially under data-constrained conditions. It contributes both a methodological and practical solution for building data-driven multimodal freight transport models that better reflect operational realities and observed empirical data that can be used to improve the freight transport planning and decision-making process.
This research seeks to develop an adaptive multimodal freight chain model that addresses these limitations. Specifically, it introduces a practical path construction framework that integrates port selection based on geographic and functional suitability, aligning cargo handling requirements with port capabilities during the construction of mode chains. This research also tries to address a gap in the literature by applying machine learning to the estimation of multimodal freight flows, a domain traditionally dominated by heuristic and optimization-based methods. To estimate freight demand distribution across the generated chains, this study explores the use of machine learning, particularly the Expectation-Maximization (EM) algorithm, to leverage the abundant but often unstructured transport data available. The EM model enables demand share prediction without relying on labeled training data, reducing the calibration burden and enhancing model responsiveness to observed transport flows.
The proposed modeling framework is applied to a case study based on the NEAC Mode Chain Builder system for inter-country freight movements between the Netherlands and Belgium, two countries with high multimodal connectivity and the largest ports in Europe. The results demonstrate the model’s ability to generate valid mode chain alternatives and to significantly reduce deviations between predicted and observed freight flows, particularly for sea and rail segments. While some deviation increases occur in other segments, these are outweighed by the overall improvement in prediction accuracy. The EM model also shows stable convergence behavior, confirming its potential under data-limited conditions. However, residual deviations suggest that external factors, such as data incompleteness or behavioral uncertainties, still limit full accuracy.
This study highlights the potential of combining graph search algorithms with unsupervised learning to enhance multimodal freight chain modeling, especially under data-constrained conditions. It contributes both a methodological and practical solution for building data-driven multimodal freight transport models that better reflect operational realities and observed empirical data that can be used to improve the freight transport planning and decision-making process.
A Model-Based Characterisation of Delay Propagation in Metro Networks
Developing a Data-Driven Framework for Discovering and Quantifying Delay Behaviour
The reliability of a metro system is an important factor in building user trust and ridership, as it helps maintain efficient and sustainable urban mobility. Within the high-frequency environment of metro networks, even minor disturbances can propagate across the network, resulting in secondary delays that reduce schedule stability and passenger satisfaction. With the need for analysis approaches that balance predictive flexibility with analytical interpretability, this thesis develops a data-driven framework for modelling and quantifying delay propagation in metro networks.
The proposed framework reduces the modelling of delay propagation to the statistical fitting of relationship functions between network elements using subsets of available variable data. The methodology follows five general steps: defining relationship structures, setting analysis dimensions and their granularity, selecting the method for fitting the relationship functions, fitting the functions using available data, and quantifying the residual variability. This framework is applied to the Washington D.C. Metro using schedule and operational train movement data to analyse and characterise delay propagation behaviour.
Two model implementations are developed. The first explores the full breadth of variables available in the delay data to identify those offering statistically significant delay relationships, revealing that propagation occurs predominantly between directly connected stations and along shared lines. The second, more targeted model quantifies these relationships, demonstrating that propagation strength is independent of delay magnitude and mostly consistent across different time periods. Further cross-examination revealed minimal sensitivity to operational and network design variables, such as headways and connectivity, suggesting that other variables like scheduling regimes might be the cause of the different observed delay propagation behaviours.
The results highlight that localised delay effects dominate over network-wide influences, suggesting that metro operators can improve service reliability by focusing mitigation efforts on key inter-station connections rather than entire lines. The framework offers a versatile and multiplicative foundation for future applications in delay propagation prediction and analysis. ...
The proposed framework reduces the modelling of delay propagation to the statistical fitting of relationship functions between network elements using subsets of available variable data. The methodology follows five general steps: defining relationship structures, setting analysis dimensions and their granularity, selecting the method for fitting the relationship functions, fitting the functions using available data, and quantifying the residual variability. This framework is applied to the Washington D.C. Metro using schedule and operational train movement data to analyse and characterise delay propagation behaviour.
Two model implementations are developed. The first explores the full breadth of variables available in the delay data to identify those offering statistically significant delay relationships, revealing that propagation occurs predominantly between directly connected stations and along shared lines. The second, more targeted model quantifies these relationships, demonstrating that propagation strength is independent of delay magnitude and mostly consistent across different time periods. Further cross-examination revealed minimal sensitivity to operational and network design variables, such as headways and connectivity, suggesting that other variables like scheduling regimes might be the cause of the different observed delay propagation behaviours.
The results highlight that localised delay effects dominate over network-wide influences, suggesting that metro operators can improve service reliability by focusing mitigation efforts on key inter-station connections rather than entire lines. The framework offers a versatile and multiplicative foundation for future applications in delay propagation prediction and analysis. ...
The reliability of a metro system is an important factor in building user trust and ridership, as it helps maintain efficient and sustainable urban mobility. Within the high-frequency environment of metro networks, even minor disturbances can propagate across the network, resulting in secondary delays that reduce schedule stability and passenger satisfaction. With the need for analysis approaches that balance predictive flexibility with analytical interpretability, this thesis develops a data-driven framework for modelling and quantifying delay propagation in metro networks.
The proposed framework reduces the modelling of delay propagation to the statistical fitting of relationship functions between network elements using subsets of available variable data. The methodology follows five general steps: defining relationship structures, setting analysis dimensions and their granularity, selecting the method for fitting the relationship functions, fitting the functions using available data, and quantifying the residual variability. This framework is applied to the Washington D.C. Metro using schedule and operational train movement data to analyse and characterise delay propagation behaviour.
Two model implementations are developed. The first explores the full breadth of variables available in the delay data to identify those offering statistically significant delay relationships, revealing that propagation occurs predominantly between directly connected stations and along shared lines. The second, more targeted model quantifies these relationships, demonstrating that propagation strength is independent of delay magnitude and mostly consistent across different time periods. Further cross-examination revealed minimal sensitivity to operational and network design variables, such as headways and connectivity, suggesting that other variables like scheduling regimes might be the cause of the different observed delay propagation behaviours.
The results highlight that localised delay effects dominate over network-wide influences, suggesting that metro operators can improve service reliability by focusing mitigation efforts on key inter-station connections rather than entire lines. The framework offers a versatile and multiplicative foundation for future applications in delay propagation prediction and analysis.
The proposed framework reduces the modelling of delay propagation to the statistical fitting of relationship functions between network elements using subsets of available variable data. The methodology follows five general steps: defining relationship structures, setting analysis dimensions and their granularity, selecting the method for fitting the relationship functions, fitting the functions using available data, and quantifying the residual variability. This framework is applied to the Washington D.C. Metro using schedule and operational train movement data to analyse and characterise delay propagation behaviour.
Two model implementations are developed. The first explores the full breadth of variables available in the delay data to identify those offering statistically significant delay relationships, revealing that propagation occurs predominantly between directly connected stations and along shared lines. The second, more targeted model quantifies these relationships, demonstrating that propagation strength is independent of delay magnitude and mostly consistent across different time periods. Further cross-examination revealed minimal sensitivity to operational and network design variables, such as headways and connectivity, suggesting that other variables like scheduling regimes might be the cause of the different observed delay propagation behaviours.
The results highlight that localised delay effects dominate over network-wide influences, suggesting that metro operators can improve service reliability by focusing mitigation efforts on key inter-station connections rather than entire lines. The framework offers a versatile and multiplicative foundation for future applications in delay propagation prediction and analysis.