Dynamic vulnerability analysis of multi-modal public transport network using generalized travel costs from a multi-layer perspective

The proper functioning of urban multi-modal public transport networks (MPTNs) is essential for sustainable urban development. However, as these networks become increasingly complex, their dynamic vulnerability to disturbances also rises. This study proposes a cascading failure model based on localized dynamic flow redistribution, aimed at mitigating and controlling the dynamic vulnerability of MPTNs. Firstly, we construct a multi-layered MPTN weighted by both generalized cost and traffic flow attributes, considering the heterogeneity and interdependence among various routes and modes. Building on this structure, we develop a localized user equilibrium traffic redistribution model that accounts for passenger congestion effects, enabling the analysis of dynamic vulnerability from both structural and functional perspectives. The proposed methodology is applied to a case study of the MPTN in Harbin. Simulation results reveal that the propagation of cascading failures in MPTNs is strongly associated with the geographical locations and importance of stations. Increasing station capacity effectively reduces the scale of cascading failure propagation, thereby alleviating network vulnerability. Moreover, dynamic vulnerability analysis shows that network connectivity and generalized travel efficiency deteriorate nonlinearly over time. Failures at critical stations disproportionately accelerate the dynamic vulnerability evolution, leading to nonlinear and compound degradation of network performance, including connectivity loss, increased travel costs, and service efficiency deterioration. This study provides valuable insights for enhancing the resilience of MPTNs, particularly in complex urban environments.