Interconnectedness enhances network resilience of multimodal public transportation systems for Safe-to-Fail urban mobility

Modern urban infrastructures are increasingly interconnected and interdependent, altering system behavior and challenging traditional fail-safe and risk-based approaches. Catastrophic failures can still occur under unexpected disruptions, prompting the adoption of resilience science with a safe-to-fail philosophy. The paper situates resilience in two complementary approaches: resilience-by-design (system topology and inherent response) and resilience-by-intervention (capacity enhancements, contingency, and recovery). The authors focus on resilience-by-design using network science to design safe-to-fail systems, where failures are allowed but constrained to avoid catastrophic outcomes and enable graceful degradation. The study addresses cities’ growing reliance on extensive interconnected networks and the need to understand resilience impacts of interconnection. It specifically examines how resilience changes when multiple public transportation systems interconnect into a multimodal public transportation network (MPTN), asking: How does resilience change when different modes of public transport networks (PTNs) are interconnected, and do insights from monomodal systems hold in multimodal systems? Hong Kong, with ~90% public transport mode share and six fixed-route modes, provides the case study.

The paper contrasts resilience-by-design versus resilience-by-intervention and argues for their complementarity. Prior work has proposed a safe-to-fail resilience framework with preparedness, robustness, and interoperability indicators. Recent literature increasingly examines interconnected and interdependent systems, recognizing interdependency as a potential vulnerability while acknowledging that interconnectedness can enhance functionality and flexibility in logistics, communication, and transportation. In transportation, substantial research has analyzed single-mode networks (e.g., metro robustness) and developed multilayer metrics (multiplex and interconnected networks). Interconnection is often seen as beneficial by offering complementary paths and services, but past multimodal studies have rarely incorporated a resilience perspective or provided quantitative comparative assessments isolating the effect of interconnectedness on resilience. The authors also note the importance of geospatial factors and walkability in defining intermodal transfer, arguing that interconnections extend beyond formal interchanges to any walkable transfer opportunities between modes.

Case study: Hong Kong’s fixed-route public transport comprising six subsystems: Mass Transit Railway (MTR), Light Rail (LR), Franchised Bus (FB), Green Minibus (GMB), Ferry (FERRY), and Tram (TRAM). Flexible/ad hoc modes (e.g., cycling, taxis) are excluded for modeling consistency. Modeling approach: Directed multilayer topological model focusing on resilience-by-design. Each subsystem is modeled as a simple digraph G_m = (V_m, E_m) where nodes are stops/stations/piers and directed edges represent at least one service connection. The complete MPTN includes all layers and intermodal transfer (IMT) edges E_IMT connecting nodes across different modes based on spatial proximity (haversine distance). While earlier text references L-space, the key assumptions emphasize an unweighted topology (I-space) representation; nodes are stops, edges represent connectivity. Interconnection identification: Compute pairwise haversine distances between nodes of different subsystems to add IMT edges when distance ≤ D_IMT. D_IMT varied from 0 up to 1600 m (0 indicates isolated systems; 100 m approximates typical public transport interchange scale). Intermodal edges are assumed topologically equivalent to within-mode edges. Sequential integration: Integrate subsystems in order of real-world capacity importance: MTR → FB → GMB → LR → FERRY → TRAM. At each step, compare isolated (no IMT edges) vs interconnected (with IMT edges) states and track observables and resilience indicators. Resilience indicators (safe-to-fail principle): - Preparedness: Gini coefficient of node degree and of node betweenness centrality to quantify inequality (sources of vulnerability). Gini ranges 0–1 (homogeneous to heterogeneous). - Robustness: Area under the degradation curve r_b = ∫_0^1 S(c) dc during node-percolation dismantling, where S is the size of the largest strongly connected component normalized by original node count; scenarios include random failures and targeted attacks by highest node degree (ND) and highest betweenness centrality (BC), recalculated after removals. Percolation repetitions: 1000 (random), 100 (targeted). To reduce computation in BC-targeted attacks, removals proceed in 5% node batches per step, recalculating centralities each step. - Interoperability: Node relocation rate RI(v) considering proximity and reachability after the failure of node v. For each originally reachable destination n, find a nearest neighbor u within distance l ≤ d_max that reconnects to n; apply linear distance-decay df(l) = 1 − l/d_max for 0 ≤ l ≤ d_max. Network-level relocation R_I is the global average over nodes. Two relocation distance limits analyzed: 750 m and 1600 m. Additional network efficiency metrics: Global efficiency E and geospatial efficiency E_geospatial measuring detour between straight-line and network path lengths using haversine distances. Null model benchmark: To control for scale sensitivity (e.g., robustness, efficiency), develop a geospatially constrained Erdős–Rényi null model. Generate networks with the same node set and edge count as the observed network, and approximate the observed edge-length distribution (mean and standard deviation) by rewiring edges to match a discrete distribution of haversine lengths. Compute Z-scores of observed indicators relative to 50 null-model realizations. Data and implementation: Public GTFS-like and transport department datasets for all six modes (various years 2020–2022). Modeling and analysis primarily in NetworkX, with code and data archived on Zenodo. Statistics: standard errors targeted below 0.01; no data excluded.

- Interconnectedness improves resilience metrics across robustness, interoperability, and efficiency compared to isolated subsystems. Topological and efficiency impacts (Table 2): - With interconnection at D_IMT = 100 m by the final step (all six subsystems), average shortest path length drops from ~72.89 (isolated) to ~11.79 (interconnected); l_max reduces from 107 to 74. - Global efficiency E increases from ~0.03 (isolated) to ~0.10 (interconnected); geospatial efficiency E_geospatial rises from ~0.24–0.26 (isolated) to ~0.83 (interconnected). - Largest strongly connected component proportion S_0 improves markedly: ~0.49–0.51 (isolated) to ~0.99 (interconnected) after full integration. Preparedness (inequality): - Gini(ND) slightly increases with interconnection (about +5.5%), indicating modestly more heterogeneous degree distribution, likely due to more IMT edges attached to high-degree nodes. - Gini(BC) remains roughly unchanged (≈ −0.7%), implying minimal effect on betweenness inequality. Robustness (Fig. 3): - Across random failures and targeted attacks (ND and BC), degradation curves show higher tolerance and larger area-under-curve for interconnected MPTN versus isolated networks. Z-scores over the null model confirm advantages of interconnection across scenarios. - Robustness gains are larger for ND-targeted attacks than for BC-targeted attacks, consistent with the relative stability of Gini(BC). - Integrating vulnerable subsystems can substantially raise overall robustness: notable improvements observed when adding GMB (step 3) and TRAM (step 6), especially when many IMT edges are created and when bus-rail complementarities reduce topological vulnerabilities. Conversely, adding a robust but localized subsystem (LR, step 4) can yield limited or counterproductive impact on overall robustness. - Robustness increases with IMT distance D_IMT; marginal benefits are highest at short distances, with diminishing returns at larger D_IMT. Random-failure robustness approaches the theoretical upper bound as D_IMT reaches 1600 m. Interoperability (Fig. 4): - All subsystems show higher relocation capability when interconnected. MTR’s global average relocation rate increases from 0.06 (isolated) to 0.91 at 750 m when interconnected; FERRY also gains substantially at both 750 m and 1600 m, indicating reliance on other modes post-disruption. - Bus-based modes (FB, GMB) and LR show smaller gains due to dense stop spacing. - Fully integrated MPTN achieves overall relocation rate ≈ 0.93 at 750 m, implying that about 93% of passengers can continue travel via nearby alternatives under single-station disruptions. General insights: - Interconnecting systems enhances safe-to-fail design by enabling graceful degradation and rapid redistribution of flows. - There are trade-offs: improved robustness and efficiency come with slightly more centralized (less prepared) node-degree structure. - Planners can prioritize short-distance IMT enhancements (e.g., passageways, footbridges, interchange sites) for high marginal resilience gains at lower cost.

The study directly addresses how interconnecting multiple public transport modes changes resilience. Results demonstrate that interconnection reduces topological vulnerabilities, boosts robustness to random failures and targeted attacks on hubs/interchange stations, and markedly improves interoperability and efficiency. The findings support a resilience-by-design perspective: topology-level interventions (adding intermodal edges and enhancing walkable transfers) can deliver safe-to-fail behavior, allowing networks to maintain connectivity and redistribute flows during disruptions. The benefits are particularly strong for degree-targeted attacks, suggesting a practical approach to mitigate vulnerabilities at high-degree hubs common in PTNs. The work also reveals trade-offs: interconnection slightly increases degree inequality, potentially complicating preparedness from an equity perspective. Consequently, investments in satellite stations and diversified transfer points may be warranted to balance preparedness, robustness, and efficiency. The observed concave-down relationship between robustness and IMT distance highlights the high value of improving short-range intermodal walkability. The results generalize to other network-like infrastructures while cautioning that inter-sectoral interdependencies may behave differently and exhibit emergent effects beyond topological models.

Interconnecting multimodal public transport systems provides a resilience-by-design pathway to safe-to-fail urban mobility. Using Hong Kong’s six-mode network, the paper shows that interconnection improves robustness (to random failures and targeted attacks), enhances post-disruption interoperability (e.g., MTR relocation 0.06→0.91 at 750 m), and increases efficiency (E and E_geospatial), while slightly increasing degree centralization. The study offers actionable insights: prioritize intermodal transfers and short-distance walkability to realize high marginal robustness gains; consider integrating vulnerable systems to bolster overall resilience; and balance trade-offs across preparedness, robustness, efficiency, and cost. Future research should extend to weighted networks (demand, capacity), refined pedestrian and walkability data, alternative weighting methodologies, and exploration of inter-sectoral connections (e.g., energy-transport) where interdependencies and emergent phenomena may alter outcomes. The use of standardized GTFS data and publicly available code facilitates replication and adaptation to other cities.

- Topological abstraction: Unweighted, topology-focused model does not incorporate capacities, demands, service frequencies, or operational constraints; results may differ under weighted/operational models. - Distance simplifications: Use of haversine distances and linear decay to approximate walkability and transfer convenience; real pedestrian networks, barriers, and hazards (e.g., flooding) are not modeled. - Disruption scope: Analysis centers on single-node disruptions and node percolation; multi-hazard, simultaneous, or correlated failures and edge failures were not the focus. - Connectivity metric choice: Robustness measured via largest strongly connected component in directed graphs; alternative connectivity definitions could yield different robustness values. - Preparedness metrics: Gini indices based on degree and betweenness capture limited aspects of vulnerability; other centralities or functional indicators might reveal different preparedness patterns. - Mode coverage: Excludes flexible/ad hoc modes (taxis, cycling) which could influence real-world resilience. - Generalizability and emergence: Findings from a single case study and topological model may not fully capture inter-sectoral interdependencies or emergent behaviors in real contexts.