Understanding the connectivity and interactions within complex systems is crucial for managing disruptions. Public transport networks (PTNs), critical infrastructures, are increasingly studied through a network science lens to understand their topology and dynamics. Ensuring reliable and attractive services requires minimizing the impact of disruptions during both design and operation. While research extensively covers PTN vulnerability and robustness, topological aspects of the recovery process and the impact on passenger travel remain understudied. Previous work often adopts a strictly topological perspective, neglecting service aspects and passenger travel consequences. This study focuses on recoverability – the ability of a PTN to return to its original performance level after disruptions – using a topological approach adapted from optical network research. The study uses a dataset of 42 global metro networks, created using General Transit Feed Specification (GTFS) data. This dataset surpasses previous work in both size and detail, including weighted network representations of both infrastructure and service layers, incorporating in-vehicle and waiting times. This more comprehensive approach allows for a more nuanced understanding of recoverability than previous studies which have often relied on unlabeled networks.

A significant body of research exists on the robustness and vulnerability of PTNs to disruptions, often simulated through random or targeted attacks on nodes or links. However, the topological aspects of the recovery process and its impact on passenger travel have received less attention. Past studies frequently adopt a purely topological viewpoint, overlooking the service characteristics of public transportation and the consequences of disruptions for passengers' travel itineraries. This study addresses this gap by focusing on the recoverability of PTNs, a crucial aspect of their resilience to various disruptions.

The study models disruptions by successively removing random links from the network. A greedy heuristic is employed to recover the network to its original state. This heuristic iteratively adds the removed link that yields the largest performance improvement at each step. Performance is defined as the mean of the reciprocals of the generalized travel cost (GTC) between all node pairs. The GTC considers in-vehicle travel times, waiting times, transfers, and a penalty cost per transfer. The failure/recovery process is simulated for each network, measuring performance at each step. The retained performance ratio (P) is calculated by normalizing the performance at each step by the original network's performance. The cumulative performance loss (F) during failure and the cumulative performance gain (R) during recovery are quantified as areas above and below the performance curve respectively. Recoverability is characterized using F, R, R/F, and R-F. These indicators are then correlated with several topological characteristics (size, connectivity, efficiency, hierarchy) to identify relationships between network topology and recoverability.

The study reveals that the greedy recovery strategy effectively rebounds from performance losses during failures. The retained performance curves show asymmetry, with the recovery phase exhibiting a steeper slope than the failure phase. Variability in the retained performance ratio decreases with increased link removal. The analysis of four recoverability indicators (F, R, R/F, R-F) across different link-removal thresholds shows consistent performance for F across scenarios, while R shows moderate positive correlation across different removal thresholds. Rankings of cities based on recoverability indicators reveal variations in performance. Small networks (Marseille, Lyon, Warsaw) exhibit the best performance in terms of F (cumulative performance loss), while large networks (London, Santiago, Paris) show better F values than New York, Madrid, Berlin, and Chicago. The worst-performing networks in terms of F are Oslo, Stockholm, and Rome. Santiago stands out with the best R (cumulative performance gain) values across all thresholds. Networks like New York, London, Paris, Madrid, and Valencia consistently rank high in R. Oslo, Rome, Stockholm, Boston, and Prague consistently perform poorly in terms of R. Combined indicators (R/F and R-F) rank Santiago, London, New York, Paris, and Lyon as top performers; Boston, Chicago, Rome, Stockholm, and Oslo perform poorly. High correlation is observed between F and topological indicators: diameter (D), average shortest path (SP), and density (γ). Positive correlation with D and SP indicates that networks with longer travel times experience greater performance loss. Negative correlation with density suggests that denser networks withstand disruptions better. The R indicator exhibits strong positive correlations with meshedness (α) and network size (number of nodes and links). This suggests redundant networks (with cycles) and larger networks recover faster. Combined indicators show moderate positive correlation with meshedness, low correlation with network size, and moderate-to-low correlation with average (weighted) shortest path. Assortativity and the rate of the exponential distribution of node betweenness centrality show negligible correlation with recoverability indicators. Scatter plots further illustrate the relationship between F and SP, and R and meshedness, supporting the findings of the correlation analysis.

The findings demonstrate a strong link between PTN topology and recoverability. More efficient networks (shorter travel times), denser networks, and networks with higher redundancy exhibit superior recoverability. This suggests that network design strategies aiming to reduce average travel time and increase redundancy can significantly enhance network resilience. The results have practical implications for both strategic (network design) and operational (timetable planning) decision-making. Planners can leverage the methodology to assess the recoverability implications of various network designs or operational changes. The simplicity of the greedy heuristic provides a tractable approach for practical applications.

This study introduces a novel topological approach for assessing PTN recoverability, applying it to a large, detailed dataset of global metro networks. The results highlight the importance of network efficiency, density, and redundancy in enhancing resilience. The methodology offers valuable insights for planning, design, and operational improvement of PTNs. Future research could involve expanding the dataset, exploring advanced recovery algorithms, incorporating additional performance metrics, and analyzing multi-modal networks.

The study's focus on service topology neglects detailed infrastructure features (e.g., rail junctions, extra tracks) that might influence failure/recovery operations. The greedy heuristic, while computationally efficient, is a simplified representation of recovery strategies. The use of GTFS data introduces potential limitations due to data quality and variations across datasets. The current performance metric (GTC) may not fully capture all aspects of passenger experience.