Urban Metro System Network Resilience Under Waterlogging Disturbance: Connectivity-Based Measurement and Enhancement

Abstract

Urban metro systems (UMSs) primarily consist of underground structures and are therefore highly susceptible to disasters, such as rainstorms and waterlogging. The damages caused by such events are often substantial and difficult to recover from, highlighting the urgent need to enhance the resilience of metro networks against waterlogging. Based on the principles of urban hydrology, this paper constructs scenarios to analyze the risk of waterlogging under varying rainstorm recurrence intervals and intensities. The ArcGIS geographic information system was employed to improve the existing passive inundation algorithm, enabling more accurate identification of flood-prone areas during heavy rainfall, which supports the topological modeling of UMSs. Structural connectivity was used as an external indicator of network resilience, and tools such as Gephi and NetworkX were applied to evaluate network performance. Using the Nanjing Metro as a case study, the resilience of the UMS under different risk scenarios was assessed by analyzing the impact of waterlogging events. Subsequently, recovery sequences following disruptions were prioritized to optimize post-disaster restoration, and targeted strategies for improving network resilience were proposed. The calculation results indicate that the overall resilience of the Nanjing UMS network is at a relatively high level. When connectivity is used as the performance indicator, the operating network resilience value is between 0.78 and 0.952, while the planned network resilience value is between 0.887 and 0.939.

1. Introduction

Urban metro systems (UMSs) are predominantly underground, with transfer stations being primarily underground spatial structures. These stations are often located in areas of high passenger traffic within the city, and the lines frequently traverse through underground tunnels. Consequently, the operation of a UMS is particularly susceptible to rainstorm flooding events. When faced with urban flooding disturbances, the metro system’s operation exhibits characteristics of “inadequate prevention, weak resistance, and slow recovery.” The growing demand for metro construction and operation, coupled with the frequent disruptions caused by rainstorms due to climate change, has made it imperative for the UMS to improve its resilience and adaptability to waterlogging disasters.

When the UMS operation encounters waterlogging events, the network structure system, platform infrastructure, and electrical equipment facilities are all highly vulnerable. Any interruption in any link can have a decisive impact on the normal operation of the metro system. The occurrence of flooding events is difficult to predict and mainly depends on physical attributes, such as the amount of rainstorm precipitation and the elevation of the urban infrastructure. Starting with the system itself, enhancing the resilience of infrastructure against rainstorm flooding events is a current trend in the field of infrastructure disaster prevention and mitigation. This paper concentrates on the network structure’s performance during the operation of the UMS, with rainstorm and waterlogging events serving as the primary disturbance factors. It defines the network resilience of the UMS operation process as the ability of the network system to maintain or quickly restore its safe operation through processes such as resistance, repair, and adaptation during the UMS operation stage when disturbed by rainstorm and waterlogging disaster events.

Currently, notable progress has been made in the development of research frameworks and computational methodologies related to flood resilience. Ji et al. explored how to establish a scientific evaluation framework to measure and enhance urban flood resilience (UFR) against the backdrop of global climate change and urbanization. Using the Yangtze River Delta urban agglomeration as a case study, a multi-level evaluation framework was proposed [1]. Singh et al. investigated the flood resilience mechanisms, strategies, and outcomes in selected cities—New York (USA), Tokyo (Japan), and Rotterdam (The Netherlands)—as well as their impacts on highway transportation systems. The study explored how institutional, technological, and policy measures can enhance urban capacity to cope with flood disasters in the context of climate change [2]. Fang et al. proposed a systematic approach based on the DPSIR framework for analyzing and evaluating the impact of urban flooding caused by extreme rainfall on traffic performance. The innovation lies in integrating natural systems (such as rainfall and flooding) with traffic systems, describing their interaction through causal chains, and further assessing the performance changes of the traffic system [3]. Dezhi Li et al. established a flood resilience evaluation model based on the 4Rs theory, which comprises 26 indicators. An empirical study was conducted using South China as a case study. The flood resilience of the urban road traffic networks was assessed by comparing conditions before and after pipeline renovation. The findings indicated that a single traditional engineering measure has certain limitations in enhancing the flood resilience of the URTN. Recommendations, such as strengthening public participation and improving the synergistic capabilities of multiple engineering measures, have been suggested to further enhance the flood resilience of the URTN [4]. Ro and Garfin evaluated the urban flood resilience of South Korea by constructing an indicator system for assessing flood resilience at four levels: national, city, regional, and sub-regional. The system included learning ability, regulatory ability, and disaster classification, and proposed corresponding improvement measures [5]. Current research on flood resilience primarily focuses on urban-scale assessments, with the analytical framework still centered around the developmental process of resilience curves and performance characterization. The research methodology predominantly employs indicator system analysis combined with questionnaire surveys to identify the factors significantly impacting flood resilience and to propose targeted enhancement strategies. However, the measurement and improvement of resilience, particularly for specific urban critical infrastructure, such as Urban Metro Systems (UMSs), are still in their early stages. Research on urban waterlogging primarily focuses on cities or large regions, proposing strategies for large-scale disaster prevention and mitigation. These studies employ methods, such as hydrological forecasting, numerical simulation, and big data analysis, to assess current water conditions and to predict future waterlogging intensity. Further research is needed on the safety, operational management, and enhancement of flood resilience in urban metro systems.

This paper, grounded in the fundamental principles of urban hydrology, investigates the evolution and analytical parameters of urban rainstorm flooding disasters. It constructs scenarios to evaluate the risk of flooding across various rainstorm recurrence intervals and intensities. By utilizing actual urban terrain data and the topological characteristics of the metro system, this study employs ArcGIS for topological modeling of the UMS. The research aims to refine the existing passive inundation algorithm, enabling the precise identification of flood-prone areas during urban rainstorms. The utility and precision of the enhanced inundation simulation algorithm are validated through case studies of the Nanjing Metro operating network and the projected 2030 network. Additionally, this study identifies the stations on the Nanjing Metro line affected by the rainstorm event. Utilizing structural connectivity as a measure of network resilience, this study examines the topological structure and operational performance of urban metro systems during rainstorms and flooding disruptions. Gephi 0.9.1 and NetworkX2.2 were used to model and compute the network structure. Based on the impact of flood disasters on Nanjing’s UMS, the resilience of the UMS under various modes was assessed, and the recovery sequences following disturbances were prioritized to achieve the optimal restoration of the UMS post-disaster.

The structure of this paper is organized as follows: Section 2 elucidates the formation mechanism of the UMS network resilience in response to waterlogging disturbances; Section 3 examines the vulnerable areas for the operation of the UMS during rainstorms and waterlogging events; Section 4 assesses the resilience of the network structure by measuring connectivity under waterlogging disturbances; Section 5 presents the research conclusions and suggests strategies for enhancing network resilience based on these findings.

2. Formation Progress of the UMS Network Resilience

2.1. Connectivity-Based Evolution Process of the UMS Network Resilience

Flood events resulting from heavy rainfall can disrupt urban infrastructure. Critical traffic junctions, buildings in low-lying areas, and power and communication service equipment are all at risk of being flooded. It is essential to enhance the flood resilience of urban infrastructure projects. Urban Metro Systems (UMSs) operating in tunnels and stations, being primarily underground structures, are particularly vulnerable to waterlogging disruptions. Extensive research has been conducted on the reliability and vulnerabilities of metro systems in disaster scenarios. Zhou et al. studied the safety management of metro system construction, operation, and maintenance. Utilizing an accident case database and complex network theory, they explored the complexity of metro construction accident networks (SCAN) [6]. Hong et al. proposed three types of accessibility measurement methods and conducted an integrated study of metro and high-speed rail systems. Based on accessibility indicators, they examined the vulnerability of China’s integrated railway system in the event of a single high-speed rail station failure and two actual severe weather incidents [7]. Resilience emphasizes the system’s ability to adapt actively and to recover quickly after a disturbance, representing an extension and evolution of traditional safety management. Resilience studies focus on the entire process of a disturbance, encompassing its occurrence, evolution, system feedback, and subsequent post-event backtracking learning, following a chronological logic. To study the resilience of the UMS network structure, it is crucial first to clarify the performance response characteristics of the UMS to waterlogging disturbances. This involves determining whether the subway can maintain normal operations and how operational strategies are adjusted. If operations continue, it is important to evaluate the convenience for passengers and any changes in travel patterns. Based on the UMS’s response characteristics to flooding events, this paper suggests using the connectivity of the network structure as an external indicator of the resilience of UMS networks.

Figure 1 depicts the response process of the UMS to heavy rain and waterlogging disasters, showing the chronological progression from the onset of flooding to the system’s feedback and subsequent reflections. According to risk management theory, the disaster formation process is divided into four stages: preparedness, formation, disaster, and adaptation, corresponding to pre-event, in-event, and post-event phases. During the preparedness and formation stages, rainfall events and terrain conditions, combined with insufficient physical protection measures, contribute to the UMS inundation incidents [8]. Rainstorm events are characterized by heavy rainfall intensity and prolonged duration, which can overwhelm urban municipal drainage systems, leading to excessive surface runoff. The UMS serves diverse urban areas, necessitating network locations that balance community residents’ accessibility with geological safety. Consequently, some stations and lines must traverse low-lying regions or areas near hazardous water bodies, and in some southern cities, subway tunnels pass beneath large lakes and rivers. Terrain conditions present an additional risk factor for disasters. The physical flood prevention measures of the UMS may also be poorly designed, such as the improper layout and sizing of flood barriers and the inadequate design height of metro entrance steps, all of which are risk factors for the flooding of the UMS. The mechanism of disaster formation can be summarized as the cumulative effect of rainfall events and topographic conditions, leading to inundation of the UMS when physical flood control measures are inadequate or poorly designed. During the disaster phase, the response can be categorized into three layers: equipment, network, and service. When confronted with rainstorm flooding, the equipment and facilities of the UMS are the first to be at risk of damage. Gates, electrical boxes, and control equipment within stations are all vulnerable to flooding, and track lines between stations may become partially exposed. Both electrical equipment and tracks face a high risk. Stations and connecting tracks in the network layer can be submerged, causing UMS service performance to plummet instantly. At this juncture, passengers at stations must be evacuated, with some transferred to adjacent, undisturbed stations, and bus connection services must be effectively managed. The mechanism of the disaster phase can be summarized as the cascading failure response across the equipment and facilities layer, the network structure layer, and the social service layer [9,10]. The primary objective of the disaster adaptation phase is to swiftly reinstate the service capabilities of the UMS and to improve system performance to better endure possible future disruptions. The disaster recovery phase necessitates resource integration, organizational management, and post-event improvement. Resource integration primarily involves emergency drainage operations, maintenance of damaged equipment, and detection of submerged networks. Organizational management includes adjusting network operation schedules, timely updating and disseminating new operational information, and swiftly restoring optimal operation performance. Post-event improvement should concentrate on the causes of the accident, the refinement and enhancement of emergency plans, and the monitoring and prevention of flood-prone areas [9,11]. The mechanism of the disaster adaptation stage can be described as achieving enhanced resilience to flooding through measures such as improved disaster event management, coordinated maintenance resource allocation, and the optimization of emergency plans.

2.2. UMS Operation Characteristics Process in Response to Waterlogging

The metro system has entered the era of networked operation, with enhanced connectivity between lines, and the consequences of natural disasters have a cascading effect. Dong et al. proposed a probabilistic model employing the Bayesian method to assess the cascading failure risk of road networks and infrastructure during flood disasters [12]. Various network structures possess distinct topological features and terrain attributes, as well as differing adaptability and recovery capabilities in response to disturbances.

The subway network, due to its unique service attributes, exhibits geometric and passenger flow characteristics, falling under the category of dynamic complex networks. Most existing research on complex networks focuses on the topological characteristics of infrastructure networks [13,14,15,16,17], with few studies integrating the service attributes of the networks themselves. Upon examining the cases of flooding in the metro system and metro operation protocols, it becomes evident that the operational modes of the metro system during flooding disruptions can be categorized into three types: complete shutdown, partial shutdown, and normal operation. From the perspective of the network’s topological characteristics, a complete shutdown converts the network into a purely geometric topological structure devoid of operational attributes, effectively rendering it a static network. Partial shutdowns typically involve the temporary closure of an entire line or a section of it. The network retains its overall operational attributes, and the stations and lines within the network maintain both topological and passenger flow characteristics, thus qualifying as partial dynamic networks. Under normal operating conditions, the structural characteristics of the network remain unchanged; however, operational risks escalate, and passenger flow attributes are correspondingly diminished, rendering it a dynamic network. The unique response characteristics of the UMS network structure to flooding events result in significant variations in the network’s performance levels post-disaster [18,19]. As a result, targeted repair and improvement strategies should be implemented during the recovery phase.

The research innovations presented in this paper include utilizing rainstorm events as the disturbance factor to assess the network resilience of the Urban Metro System (UMS) operational process. Dangerous scenarios are constructed based on various recurrence intervals, rain types, and other parameters. Simulations of these rainstorm events are conducted to calculate the inundation levels of the line network. The urban flood-prone areas are modeled under conditions of 50, 100, 500, and 1000-year recurrence periods. By employing a modified passive inundation algorithm and integrating it with the ArcGIS framework, the inundation levels of the UMS network are examined, and the vulnerable areas within the UMS lines are pinpointed. Connectivity and network efficiency are utilized as metrics to evaluate the load-bearing capacity of the line network structure. The optimal recovery sequence for flood-prone points is determined using NetworkX 2.2 and Python 3 tools, resulting in the calculation of network resilience evaluation values.

3. Vulnerable Areas Analysis of the UMS During Flooding Conditions

By applying the fundamental principles of urban hydrology and utilizing ArcGIS as the modeling tool, this section refines the inundation algorithm for flood disaster assessment. It incorporates the actual surface elevation of the city, ensuring that the algorithm more accurately reflects the real-world progression of urban flooding. Consequently, this enhancement identifies areas within the city that are prone to flooding during heavy rainfall. By integrating the outcomes of the waterlogging assessment with the topographical relationship of the metro network, the vulnerable areas of the UMS operations during rainstorm-induced waterlogging are determined. This provides a foundational model for measuring and enhancing the resilience of the UMS operation process against waterlogging.

3.1. Various Recurrence Periods of Rainstorm Risk Scenarios

The intensity and classification of urban rainstorms are typically assessed using formulas designed to calculate rainstorm intensity. According to the rainstorm intensity formula lookup table released by the Nanjing Urban Management Bureau in 2014, the formula for Nanjing’s rainstorm intensity is as follows:

$$i={\displaystyle \frac{64.300+53.800 \hspace{0.33em}\lg P}{{(t+32.900)}^{1.011}}}$$

(1)

In the formula, i represents the rainfall intensity, measured in millimeters per minute (mm/min); t denotes the duration of rainfall, with the unit of minutes (min); and P signifies the rainfall recurrence period, quantified in years. Common criteria include once every 50 years or once every 100 years, among others.

Nanjing has a subtropical monsoon climate, characterized by abundant rainfall and an average annual precipitation that ranges from 1000 to 1200 mm. In recent years, the city has experienced severe rainstorms during the summer and rainy seasons. These heavy downpours have significantly impacted the functioning of the city’s infrastructure, often leading to the suspension of subway operations. This paper concentrates on rainstorm events, analyzing scenarios of short-duration heavy rainfall with durations of 1 h and 3 h. It simulates precipitation events with return periods of 50 years, 100 years, 500 years, and 1000 years. The selected durations and recurrence intervals of precipitation are sufficient to cover most real-life scenarios, providing valuable insights for the emergency management of urban subway systems during rainstorms.

According to the calculation formula for rainfall intensity in Nanjing City, the calculated values of rainfall intensity for various rainstorm recurrence periods are presented in

Table 1. Calculation table of rainfall intensity under different recurrence periods in Nanjing.

Recurrence Period/Year	1 h Rainfall Intensity (Average)	One-Hour Rainfall (mm)	3 h Rainfall Intensity (Average)	3 h Rainfall (mm)
20a	1.375	82.518	0.595	107.040
50a	1.595	95.673	0.689	124.105
100a	1.760	105.624	0.761	137.013
500a	2.146	128.730	0.928	166.986
1000a	2.311	138.682	0.999	179.895

.

The inundation depth of a city’s critical infrastructure during a rainstorm is primarily influenced by two factors: the variation in rainfall intensity and duration, and the city’s capacity to manage flood events, which includes the development of drainage systems and the permeability of green spaces. The current drainage network in Nanjing is designed with a recurrence interval of 0.5 to 3 years. However, with the evolution of urban drainage network construction standards and shifts in urban rainfall patterns, these existing standards are increasingly unable to meet the contemporary needs of urban construction and management. To accurately evaluate the vulnerable zones of the UMS during rainstorm-induced flooding, it is essential to make informed predictions about the performance of the urban drainage system. This paper calculates the drainage capacity of Nanjing’s pipe network using the 5-year and 10-year return period standards, and in certain areas, the 50-year return period standard is applied. The method for calculating drainage capacity is based on the Nanjing rainstorm intensity formula, with the results presented in

Table 2. Calculation of drainage capacity of urban pipe network in Nanjing.

Recurrence Period/a	1 h Precipitation Intensity	Theoretical Displacement/mm	3 h Precipitation Intensity	Theoretical Displacement/mm
Once in 5 years	1.044	62.615	0.451	81.223
Once in 10 years	1.209	72.567	0.523	94.132
Once in 50 years	1.595	95.673	0.689	124.105

.

3.2. UMS Network Modeling Within Geographic Information

As of December 2022, the Nanjing Metro operates 12 lines: Lines 1, 2, 3, 4, 7, 10, S1, S3, S6, S7, S8, and S9. It features a total of 208 stations, including transfer stations that are counted multiple times. The combined length of these metro lines spans 427.1 km, creating a network that serves all 11 districts of Nanjing and its surrounding cities. Additionally, more than ten lines are under construction, with the expectation that they will be completed and operational before 2026. According to the rail transit planning, by 2030, Nanjing’s rail transit network is projected to include 17 lines, with a total length of 617 km.

The operational network of the Nanjing Metro, along with the planned network for 2030, was modeled using ArcGIS. By employing the actual longitude and latitude coordinates of the stations and lines within the metro system, the current operational network map of Nanjing and the planned network for 2030 were created, as illustrated in Figure 2. Compared to the operational network, the density of the planned network and stations is significantly increased, and the complex network characteristics of the system are further highlighted.

3.3. Flooding Scenario Simulation of the UMS Network Operation

3.3.1. Selection of Algorithm for Urban Flooding Simulation

Numerical simulation methods for flood and rainstorm events are characterized by their strong universality, high accuracy, and fast calculation speed. They have been widely applied in urban flood prevention and drainage system designs. This section summarizes several common flood simulation algorithms and modifies them based on existing algorithms to provide algorithmic support for subsequent flood simulations [20,21,22,23,24].

The characteristics of several common types of rainstorm flooding simulation algorithms are summarized in

Table 3. Summary of characteristics of rainstorm flooding simulation algorithm.

Algorithm Name	Main Idea	Simulation Results	Main Advantages	Main Disadvantages	Common Software
One-dimensional pipe network simulation	Solve the one-dimensional Saint-Venant equations to calculate the flow state of water in the network, and subsequently allocate the water volume.	Check well overflow, as well as the flow rate, velocity, and section fill of water in the network.	The calculation speed is fast and the accuracy is relatively high.	It is impossible to calculate the flow and diffusion of water on the surface.	SWMM
Two-dimensional pipe network simulation	Solve the two-dimensional shallow water wave equation to diffuse the overflow from the inspection well along the flow path.	The spatio-temporal distribution of waterlogged water bodies along the urban road network.	It is currently the most computationally accurate method for simulating waterlogging.	The model calculation takes a lot of time and requires a high level of computing power.	MIKE+/ Fluent
Active flooding	Starting from the water diffusion point within the basin, flow and diffuse along the basin elevation to simulate events such as dam failures.	The flow path of the water body along the basin, the inundation area and depth of each sub-basin.	The algorithm is clear, the modeling process is simple, and the calculation speed is fast.	It is not applicable to the simulation of waterlogging events in small watersheds. Basin topographic data have a significant impact on the calculation results.	ArcGIS/ SWMM
Passive flooding	Determine the fixed water level values and compare the elevations of all grids within the basin, which is suitable for simulating uniform precipitation events.	The size and depth of all inundation points within the basin.	The algorithm is clear in thinking and fast in calculation, suitable for simulating rainstorm flooding in urban plain areas.	It does not satisfy the fluidity and continuity of water bodies.	ArcGIS/ SWMM
One-dimensional expansion method	Simplify the allocation of overflow water bodies based on urban road network data or regional V-H curves.	It solves the problem that one-dimensional simulation methods cannot allocate water volume to water bodies.	High calculation accuracy and faster calculation speed compared to pure two-dimensional simulation methods.	Compared with the shallow water wave equation, the calculation accuracy is insufficient and it is a simplified algorithm.	ArcGIS/ SWMM/ CAD

.

A method for calculating pipe overflow can simulate the identification of weak areas in urban drainage systems, proving highly effective for sponge city planning and the formulation of urban water supply and drainage schemes. However, urban rainstorms are characterized by their short duration and high intensity, which can overwhelm the urban drainage network, preventing it from promptly draining the accumulated water. The effectiveness of the pipe network in draining water during rainstorms is relatively limited and, as a result, short-term flooding often flows into low-lying areas via the urban road network. The process of urban flooding caused by rainstorm events is one of uniform precipitation across the urban surface. Consequently, a flood simulation method that incorporates rainfall calculations and urban topographic information more accurately reflects the actual conditions of rainstorm-induced flooding.

Therefore, building on the concept of passive inundation, this paper enhances the existing algorithm. It considers the actual conditions of the precipitation process and employs urban topographic data to simulate the flooding caused by short-duration rainstorms. As a result, it generates a risk map that depicts the potential for rainstorm-induced flooding across various urban areas.

3.3.2. Numerical Simulation in UMS Operations Under Waterlogging

The topographic hydrological analysis process typically includes main steps, such as DEM data correction, depression filling, flow direction analysis, flow accumulation, and basin division of catchment areas. The algorithms for inundation simulation mainly fall into two categories: active inundation and passive inundation. This paper corrects the passive inundation algorithm based on the principle of water balance, identifies low-lying flood-prone points in the study area, calculates the volume of flood-prone points and the overflow drainage area, solves the precipitation and rainfall intensity required to fill each flood-prone point, and conducts an actual scenario analysis of flood-prone points. The algorithm takes into account both the urban surface infiltration and the drainage effect of the pipe network.

The principle of water distribution is based on the water balance equation, which is expressed as follows:

P = I + E + A_O + A_U + Q

(2)

In the formula, rainfall P equals vegetation interception I, plus evaporation E, plus surface runoff A_O, plus soil infiltration and sewerage system drainage A_U, plus local reservoir settlement Q. During heavy rainfall events, it is assumed that vegetation interception, evaporation, and soil infiltration are negligible. The drainage capacity of Nanjing’s drainage network is calculated using the 5-year and 10-year return period standards, with some areas using the 50-year return period standard. The calculation of drainage capacity still employs the Nanjing rainstorm intensity formula, and thus the values for A_U are presented in Table 2. “Bluespots” refer to areas that are prone to waterlogging or flooding during heavy rainstorms, posing a threat to buildings and infrastructure in or near these zones. When a “bluespot” is filled to its spout point, surface runoff, A_O will flow into the downstream confluence, lake, river, or sea adjacent to it. The local reservoir subsidence Q represents the “bluespots” issue that needs to be addressed [25,26,27,28].

The flowchart of the algorithm for identifying “bluespots” is shown in Figure 3.

At this juncture, we have identified the bluespot layer within the study area, representing the regions at potential flood risk, as well as the metro network and stations that could be impacted by flooding. However, the presence of bluespots does not inherently indicate that flooding will occur. Broadly speaking, deeper bluespots can accumulate more rainfall and are less prone to flooding compared to shallower ones. Buildings located in bluespots that fill rapidly are at a greater risk of flooding than those in areas with a slower filling rate. Consequently, to more accurately evaluate flood risk, it is essential to calculate the volume of rainfall needed to fill each bluespot during periods of heavy rain, which is quantified in terms of hours based on rainfall intensity.

Based on the fundamental principles of hydrology, each sink (bluespot) within a basin has an associated catchment area or local basin (where precipitation flows exclusively to that particular bluespot). By determining the volume of the bluespot and the area of its basin, one can ascertain the quantity of rainfall required to fill the bluespot, which is calculated by dividing the volume by the area. After acquiring the bluespot layers, the flowchart depicting the algorithm for calculating the volume of water needed to fill the bluespots is presented in Figure 4.

3.3.3. UMS Vulnerable Points Identification Under Waterlogging Disruption

Utilizing the aforementioned bluespot identification algorithm, identify the areas at risk of waterlogging in Nanjing during various periods of rainstorm recurrence, and evaluate whether the UMS stations are situated in flood-prone zones using scenarios of continuous rainstorms lasting for 1 h and 3 h, with recurrence intervals of 50-year, 100-year, 500-year, and 1000-year durations.

The inundation scenario for Nanjing under a specified rainfall condition was depicted using ArcGIS. As illustrated in Figure 5, the simulation results of the UMS operating network during various one-hour rainfall recurrence intervals are displayed. The blue regions in the figure depict enlarged views of the inundation points, illustrating potential inundation patterns at the location. During the actual rainstorm event in the Nanjing UMS, the Daming Road Station was recorded to have been flooded several times. Under the simulation of the “Bluespot” identification algorithm, the Daming Road Station was flooded in the scenario of 1 h–50 a. The high consistency between the actual event and the simulation result indicates the accuracy of the algorithm.

Based on the recurrence period, the risk areas of the UMS operations that may be affected by flooding events are organized, as shown in

Table 4. Identification results of waterlogging risk points during the operation of the UMS (1 h).

1 h Rainfall	Operating Network	Node Degree Distribution	Planned Network	Node Degree Distribution
50 a	Xinghuo Road, Daming Road, Dongliu (2)	0 + 3 Degree 4:0 Degree 2:3	Xinting Road, Luotang Road (4) Daming Road, Dongliu, October Square, Xinghuo Road (2)	2 + 4 Degree 4:2 Degree 2:4
100 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4:1 Degree 2:3	Xinting Road, Luotang Road, Taifeng Road (4) Daming Road, Dongliu, October Square, Xinghuo Road (2) Fangjiaying (1)	3 + 4 + 1 Degree 4:3 Degree 2:4 Degree 1:1
500 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4:1 Degree 2:3	Xinting Road, Luotang Road, Taifeng Road (4) Daming Road, Dongliu, October Square, Xinghuo Road (2) Fangjiaying (1)	3 + 4 + 1 Degree 4:3 Degree 2:4 Degree 1:1
1000 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4:1 Degree 2:3	Xinting Road, Luotang Road, Taifeng Road (4) Daming Road, Dongliu, October Square, Xinghuo Road (2) Fangjiaying (1)	3 + 4 + 1 Degree 4:3 Degree 2:4 Degree 1:1

and

Table 5. Identification results of waterlogging risk points during the operation of the UMS (3 h).

3 h Rainfall	Operating Network	Node Degree Distribution	Planned Network	Node Degree Distribution
50 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4:1 Degree 2:3	Xinting Road, Luotang Road, Taifeng Road (4) Daming Road, Dongliu, October Square, Xinghuo Road (2) Fangjiaying (1)	3 + 4 + 1 Degree 4:3 Degree 2:4 Degree 1:1
100 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4:1 Degree 2:3	Xinting Road, Luotang Road, Taifeng Road (4) Daming Road, Dongliu, October Square, Xinghuo Road, Yanjiang New Town (2) Fangjiaying (1)	3 + 5 + 1 Degree 4:3 Degree 2:5 Degree 1:1
500 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4: 1 Degree 2: 3	Xinting Road, Luotang Road, Taifeng Road, Yangzhuang (4) Daming Road, Dongliu, October Square, Xinghuo Road, Yanjiang New Town, Nantieyuan (2) Fangjiaying (1)	4 + 6 + 1 Degree 4:4 Degree 2:6 Degree 1:1
1000 a	Taifeng Road (4) Xinghuo Road, Daming Road, Dongliu (2)	1 + 3 Degree 4: 1 Degree 2: 3	Xinting Road, Luotang Road, Taifeng Road, Yangzhuang (4) Daming Road, Dongliu, October Square, Xinghuo Road, Yanjiang New Town, Nantieyuan, Xiangfeng Road (2) Fangjiaying (1)	4 + 7 + 1 Degree 4:4 Degree 2:7 Degree 1:1

. Table 4 depicts the 1-h flooding scenario, whereas Table 5 corresponds to the 3-h flooding scenario. The numbers in parentheses following the site names indicate the site node degrees. Under varying recurrence periods of rainfall conditions, the operation of the Nanjing UMS stations was disrupted to varying extents, with the number of affected stations ranging from 3 to 12, including some transfer stations. In the actual operation of the UMS, various operational adjustment plans must be implemented based on the assessment results of the extent of damage to the stations.

4. Network Resilience Measurement Under Waterlogging Disturbance

Connectivity is a crucial indicator in the study of complex networks, characterizing the connections between nodes and links within a network system and the network’s overall topological performance [29,30,31]. In this section, connectivity serves as an external measure of flood resilience during the operations of the UMS. The network characteristics and operational capacities of the UMS are assessed under the topology disrupted by urban flooding, and an optimal network repair strategy is proposed.

4.1. Definition of Connectivity in the UMS Network

The UMS represents a complex network, and connectivity serves as a fundamental metric for an evaluation of the performance. In the context of transportation networks, the presence of a direct route between two points is the cornerstone for assessing network performance. This paper defines the extent of the connection between any two stations within the UMS as the network’s connectivity. If any OD (origin-destination) pair can be reached, it is considered connected; if it cannot be reached, it is considered disconnected. The network connectivity is optimal when any two sites in the network can be connected, and the value is expressed as 1.0. The ratio of OD pairs connected in the UMS after a disturbance to the number of OD pairs before disturbance is taken as the connectivity of the UMS, as follows:

C_ums = N_a-dis/N_b-dis

(3)

Here, C_ums represents the connectivity of the UMS under the waterlogging disturbance, N_a-dis is the number of effective OD pairs after the waterlogging disturbance, and N_b-dis is the number of effective OD pairs before the disturbance occurs.

4.2. Recovery Strategy and Resilience Calculation for the UMS

Urban metro systems exhibit resilience evolution processes that are characterized by resistance, absorption, repair, and adaptation in response to disturbances from rainstorms. Currently, the methods for measuring resilience primarily fall into two categories: performance–time curve integration and capacity representation index systems [32,33,34,35,36]. The evolution of resilience in urban metro systems under waterlogging disturbances displays significant temporal characteristics. In this paper, PRF curves are used to characterize the network resilience of metro systems.

Disturbances to the network structure typically manifest in two forms: deliberate attacks and random attacks. The formation of flooding damage primarily depends on the intensity of rainfall, as well as the topography and drainage conditions of the city. There is no direct correlation between the performance of the network and the occurrence of flooding events. Consequently, the attack pattern of flooding events on complex networks more closely resembles random attacks. This section has been programmed using the NetworkX platform to simulate the response behavior of the UMS under waterlogging disturbances, to calculate the performance parameters of the network, and to integrate the performance calculation results with the PRF method for resilience measurement. The detailed algorithm is illustrated in Figure 6.

The impact of flooding on urban infrastructure networks seems random, with no specific targets for flooding. Using Nanjing’s UMS operating network and planned network as subjects for analysis, and incorporating identified urban flood-prone areas as foundational data for algorithm development, two indicators, network connectivity and network efficiency, are selected to represent the network’s performance under the influence of waterlogging events. The detailed procedures for determining the UMS network structure repair sequence and calculating resilience are as follows:

(1)

Calculate the global connectivity and network efficiency of the network. When the UMS network is operating normally, the network performance index value can serve as the baseline before any disturbance, corresponding to the starting point of the PRF curve. Define the global connectivity of the UMS as the connection status of nodes in the system to all other nodes, with a value of one indicating full connectivity. Network efficiency is a metric defined based on the shortest distance d_ij between nodes. The sum of the reciprocals of the shortest distances between all node pairs in the network is the network efficiency. The shorter the distance between each node pair, the higher the efficiency value. In NetworkX, functions can be directly called to calculate both connectivity and network efficiency.

Upon calculation, the initial network efficiency of the Nanjing UMS operating line was found to be 0.09559, with 33,438 globally connected node pairs. The planned network efficiency is 0.09899, and it is expected to have 155,912 globally connected node pairs. The planned network is significantly denser than the operational one, with a greater number of nodes and lines.

(2)

Remove flood-prone areas and calculate the minimal impact on network performance. During heavy rain flooding, certain locations and lines of the UMS become submerged, leading to the cessation of line operations and a significant reduction in service capacity. As a result, connectivity and network efficiency plummet to the lowest point on the resilience PRF curve. To ascertain the network’s minimum performance, flood-prone areas within the UMS topology were removed, creating a connected subgraph of the network. The all-node-connectivity and global-efficiency commands were subsequently employed to compute the network’s connectivity and global efficiency. The findings of the minimum performance calculations for both the operational and planned networks in Nanjing are detailed in

Table 6. The UMS Line network performance calculation after disturbance (one of six sequences).

Rainfall intensity: 50 a Rainfall Duration: 1 h Inundation risk sites: Xinghuo Road, Daming Road, Dongliu	Delete Node	Connectivity	Connectivity Rate	Network Efficiency
	Dongliu	32,064	0.95891	0.09363
	Xinghuo Road	31,386	0.93863	0.09309
	Daming Road	29,220	0.87386	0.09145
	Delete node order: [‘Dongliu’, ‘Xinghuo Road’, ‘Daming Road’]
	Connectivity set: [32,064, 31,386, 29,220]
	Connectivity rate set: [0.95891, 0.93863, 0.87386]
	Network efficiency set: [0.09363, 0.09309, 0.09145]

and

Table 7. The UMS Line network performance calculation after disturbance (1 of 24 sequences).

Rainfall intensity: 50 a/100 a/500 a/1000 a Rainfall Duration: 1 h/3 h Flooding risk sites: Xinghuo Road, Taifeng Road, Daming Road, Dongliu	Delete Node	Connectivity	Connectivity Rate	Network Efficiency
	Tai Feng Road	27,074	0.80968	0.08598
	Daming Road	24,936	0.74573	0.08436
	Dongliu	23,730	0.70967	0.08246
	Xinghuo Road	23,724	0.70949	0.08326
	Delete node order: [‘Taifeng Road’, ‘Daming Road’, ‘Dongliu’, ‘Xinghuo Road’]
	Connectivity set: [27,074, 24,936, 23,730, 23,724]
	Connectivity rate set: [0.80968, 0.74573, 0.70967, 0.70949]
	Network efficiency set: [0.08598, 0.08436, 0.08246, 0.08326]

.

a.

Operating network: (once in 50 years, continuous rainfall for 1 h)

b.

Operating network: (100, 500, 1000 years once, continuous rainfall 1 h/50, 100, 500, 1000 years once, continuous rainfall 3 h)

The calculation method for the planned network is identical to that for the operating network, and the sequence in which stations are removed is unrelated to the network’s minimum performance value. The results of the minimum performance value calculation for the planned network of the Nanjing Metro are displayed in

Table 8. The UMS Planning network performance statistics after disturbance.

Rainfall Situation	Number of Affected Nodes	Connectivity	Connectivity Rate	Network Efficiency
1 h–50 a	2 + 4	125,602	0.80560	0.09063
1 h–100, 500, 1000 a 3 h–50 a	3 + 4 + 1	113,872	0.73036	0.08611
3 h–100 a	3 + 5 + 1	112,176	0.71948	0.08519
3 h–500 a	4 + 6 + 1	109,585	0.70286	0.08344
3 h–1000 a	4 + 7 + 1	108,978	0.69897	0.08297

.

Figure 7 illustrates the response of system functions in the Nanjing UMS when operations were disrupted by various types of rainstorm flooding events. The connectivity, connectivity rate, and network efficiency of the system were assessed individually. The operational line connectivity damage value was minor in the 1-h–50 a scenario and further diminished in the 1-h–100 a scenario, after which connectivity did not significantly change with increased rainfall intensity. In the 3-h scenario, the network system’s connectivity reached a minimum at 50 a and remained unchanged with further increases in rainfall intensity. This indicates that the Nanjing Metro network is highly susceptible to flooding disturbances, with network connectivity dropping to its lowest level within one hour during a 100 a once rainfall event. The connectivity rate is defined as the ratio of the number of effective OD pairs to the number of initial OD pairs of the network at a specific moment. The value of the connectivity rate visually represents the extent of the network connectivity decline. When the operational network connectivity rate drops to 0.87 in the 1-h–50 a scenario and then to approximately 0.70 under the remaining rainy conditions, it signifies that 30% of the nodes in the entire network are disconnected. At this point, the network must be shut down for maintenance and is typically fully shut down. The trend of network efficiency parallels that of connectivity, with an efficiency value of 0.091 for the 1-h–50 a scenario and around 0.083 for the remainder of the time.

The performance of the planned network differs from that of the operating network, with response times ranging from 1 to 3 h during rainfall. The network’s connectivity rate was 0.81 at 1 h–50 a, decreasing to approximately 0.73 before stabilizing. The network sustained more severe damage during the 3-h rainfall scenario, with the connectivity rate falling from 0.73 to 0.69 and stabilizing there, while network efficiency decreased from 0.09 to around 0.082. Notably, the network performance of the planned network in the 3-h disturbance scenario declined more than in the 1-h scenario, and the minimum values of connectivity rate and network efficiency were not significantly different from those of the existing operational network. This suggests that, at the structural level, the increase in network complexity does not impact the network’s invulnerability, and when network performance drops to a certain threshold, there is no substantial change in the structural-level response state. When faced with flooding disturbances, the metro network’s connectivity rate drops to 0.7 and the network efficiency drops to 0.08, with no further significant changes observed.

(3)

Calculate the connectivity and network efficiency after the restoration of flood-prone areas. The performance of stations in the UMS networks significantly declines when disrupted, and the sequence of station recovery and the selection of repair strategies have varying impacts on the evolution of network resilience. Different flood-prone points have varying node degrees and terrain attributes, and the restoration of the UMS nodes and lines requires substantial material resources, as well as human and time investments. The station restoration sequence is calculated based on the following fundamental assumptions:

(1)

The resources required for each site to resume operations are identical;

(2)

The time to complete restoration at different sites is also uniform;

(3)

Regardless of the regional differences in resource distribution where the repair nodes are located, the maintenance team has equal capabilities and repairs each site sequentially.

When numerous points require sorting, the calculation of the repair sequence follows the “node degree” priority principle, which dictates that nodes with a larger node degree are repaired first, as an attack on nodes with a high degree value could potentially trigger a significant cascading failure response.

Based on the identification results of the simulated urban flood hazard zones, three nodes (Xinghuo Road, Daming Road, Dongliu) are at risk in the operational network during a rainfall event with a 1 h–50 a return period. Consequently, there are six distinct scenarios for the network recovery sequence. In the operating network, there are four flood-prone points in the 1-h scenarios with return periods of 100, 500, and 1000 years, as well as in the 3-h scenarios with return periods of 50, 100, 500, and 1000 years. Repairing these four points sequentially results in 24 different repair sequences. The network connectivity rate and the network efficiency change in a similar pattern over time. The PRF approximation curve for the resilience of the UMS operating network in Nanjing is illustrated in Figure 8.

As the complexity of the network increases and the number of flood-prone points multiplies, the network’s recovery strategy must be adapted accordingly. For example, with a degree distribution of “3 + 4 + 1”, the three nodes with a degree of 4 should be prioritized for repair, leading to six possible repair schemes. Subsequently, the four nodes with a degree of 2 should be repaired, resulting in 24 sequences. Nodes with a degree of 1 are located at the network’s endpoints and have minimal impact on network performance, thus they should be placed last in the repair sequence. In this scenario, there are 144 actual repair sequences, which is the product of 6 multiplied by 24, in the offline network structure. If the degree-first principle is not implemented, the network would have 8 factorial possible sequences, equating to 40,320 sequences. This significantly complicates the algorithm and results in a substantial waste of computing resources. Algorithms that prioritize node degree can ensure the accuracy of the algorithm while reducing the computational workload. When network disruptions are more severe, the algorithm requires further optimization to enhance computational efficiency. Intelligent optimization algorithms, such as genetic algorithms, particle swarm algorithms, or simulated annealing algorithms, are introduced to assess the performance of complex networks.

(4)

Resilience value Calculation and result analysis. There are two primary methods for measuring infrastructure resilience: capacity representation and index system evaluation. These methods are based on dimensions such as resistance, absorption, repair, and adaptation. Furthermore, mathematical measurement methods exist that depend on PRF curves and probabilistic reasoning [37,38,39,40]. The resilience calculation of the UMS under waterlogging disturbances relies on the PRF curve, with connectivity and network efficiency serving as performance indicators for the resilience evolution process.

The performance response function (PRF) curve, also known as the performance response curve, characterizes the trend of performance parameters over time in response to disturbances within a system. Curves are categorized into two types: continuous and discontinuous, based on the response characteristics of the subject under analysis. The methods for calculating toughness also vary slightly between these curve types. The various shapes of the toughness PRF curves are illustrated in Figure 9.

For continuous curves, resilience is measured using the integral method. In the case depicted in Figure 9a, the calculation expression is as follows:

$$R={\displaystyle \frac{{\displaystyle {\int }_{t_1}^{t_4}p(t)}dt}{E_{f1}\cdot (t_4-t_1)}}$$

For discrete curves, the method of resilience measurement is the same as for continuous curves. Assuming that each node takes the same amount of time from the start of repair to the restoration of function, the calculation expression is as follows:

$$\begin{array}{l}R={\displaystyle \frac{\left[(E_{fo}+E_{f1})\times t\times \frac{1}{2}\right]+\left[(E_{f1}+E_{f2})\times t\times \frac{1}{2}\right]+\left[(E_{f2}+E_{f3})\times t\times \frac{1}{2}\right]+\left[(E_{f3}+E_{f4})\times t\times \frac{1}{2}\right]}{E_{f4}\times 4\times t}}\\ ={\displaystyle \frac{E_{f0}+2E_{f1}+2E_{f2}+2E_{f3}+E_{f4}}{8E_{f4}}}\end{array}$$

When the number of discrete nodes increases and there are S points that need to be repaired, the expression can be further modified as follows:

$$\begin{array}{l}R={\displaystyle \frac{\left[(E_{fo}+E_{f1})\times t\times \frac{1}{2}\right]+\left[(E_{f1}+E_{f2})\times t\times \frac{1}{2}\right]+\left[(E_{f2}+E_{f3})\times t\times \frac{1}{2}\right]+\left[(E_{f3}+E_{f4})\times t\times \frac{1}{2}\right]+\dots +\left[(E_{fr-1}+E_{fr})\times t\times \frac{1}{2}\right]}{E_{fr}\times S\times t}}\\ ={\displaystyle \frac{\frac{t}{2}\left\{{\displaystyle \sum _{r=1}^{r=S}\left[E_{f(r-1)}+E_{fr}\right]}\right\}}{S\times t\times E_{fr}}}={\displaystyle \frac{{\displaystyle \sum _{r=1}^{r=s}\left[E_{f(r-1)}+E_{fr}\right]}}{2\times S\times E_{fr}}}\end{array}$$

Utilizing the resilience calculation formula derived from the PRF curve, whether through the continuous curve integration method or the discrete polyline area dissection method, resilience is defined as the ratio of the system’s remaining and restored performance post-disturbance to its original performance level. The outcomes of the resilience calculations for both the Nanjing UMS operating network and the planned network under waterlogging disturbance are presented in

Table 9. Resilience values of the Nanjing UMS network under different performance indicators.

Performance Indicators: Connectivity/Connectivity Rate/Network Efficiency		A: Range of Resilience Values	B: Range of Resilience Values	C: Range of Resilience Values
Operating network	1 h–50 a	[0.922–0.952]	[0.922–0.952]	[0.973–0.983]
	1 h–100, 500, 1000 a 3 h–50, 100, 500, 1000 a	[0.780–0.912]	[0.780–0.912]	[0.894–0.966]
Planned network	1 h–50 a	[0.901–0.939]	[0.901–0.939]	[0.947–0.969]
	1 h–100, 500, 1000 a 3 h–50 a	[0.887–0.922]	[0.887–0.922]	[0.940–0.965]

.

The calculation results indicate that the overall UMS network resilience in Nanjing is at a relatively high level. When connectivity is used as the performance metric, the operating line network resilience value ranges from 0.922 to 0.952 under a 1 h–50 a disturbance. With an increase in the severity of the disturbance, the resilience values’ range decreases slightly, maintaining an overall level between 0.780 and 0.912. When network efficiency is considered as the performance metric, the resilience values’ distribution range is higher than that of connectivity, reaching up to 0.983 under minor perturbations. The resilience characteristics for the remaining perturbations are similar. Notably, the topological complexity of the UMS planned network is significantly higher than that of the operating network. Faced with identical disturbances, the resilience level of the planned network can surpass that of the operating network. This suggests that, for random attack patterns, such as those represented by rainstorm flooding disturbances, increased network structural complexity does not necessarily lead to diminished network performance. This insight is valuable for the design and planning of the UMS network structures, as well as for line adjustments during operation. In the event of disturbances, complex networks can achieve optimal performance and resilience through the implementation of reasonable repair strategies. The distribution of network resilience in the Nanjing UMS is depicted in Figure 10.

5. Conclusions

The UMS is crucial city infrastructure for most citizens’ daily commute. It has a dense distribution and networked operations, with lines mainly running in urban tunnels and stations mostly underground, making it more vulnerable to rainstorms, which may cause station waterlogging or network-wide flooding. This paper refines the inundation algorithm in flood disaster assessment to accurately reflect urban flooding progression. It identifies rainstorm-prone areas and examines the metro system’s topological structure and operational efficiency under waterlogging. Taking the Nanjing Metro as a case, it measures the UMS’s network resilience under different modes based on the calculated impact of urban flooding. It prioritizes recovery sequences after disruptions for optimal post-disaster repair and proposes targeted connectivity enhancing strategies. Based on actual issues and research conclusions, corresponding improvement strategies are proposed to reference the UMS’s safe operation management in adverse weather.

(1)

Focus on monitoring flood-prone areas in the UMS network.

In Section 3, the bluespot identification algorithm was utilized to simulate flooding conditions across the city under various rainstorm risk levels, with the Nanjing UMS serving as a case study. The urban flooding layer was superimposed onto the UMS network layer to identify the vulnerable areas of the UMS during different scenarios of rainstorm-induced waterlogging. Among the Nanjing UMS stations, Xinghuo Road, Taifeng Road, Daming Road, and Dongliu stations were the most frequently affected by inundation. Historical rainfall data confirmed that the simulation results of these inundation events were consistent with actual occurrences, as these stations have indeed experienced multiple flooding incidents.

Enhancement Strategy: Focus on monitoring flood-prone areas within the network during heavy rainfall. This monitoring should include, but not be limited to, water levels at flood-prone stations and adjacent roads, changes in passenger traffic at these stations, emergency flood prevention plans at stations, reserves of emergency supplies, the effectiveness of flood defense structures, such as floodgates and waterstop belts, and the resilience of the power system. Collaborate with the meteorological department to conduct real-time monitoring of rainfall and station operations, and analyze and present the data using information technology. The deployment of sensors and cameras, along with the development of information platforms, requires a comprehensive action plan.

(2)

Protect critical lines and maintain maximum network connectivity and operational capacity. A complex network can be seen not only as a topology diagram of discrete points and connections but as a direct composition of lines. When the entire network requires shutdown for adjustment, the impact of different line shutdowns on the overall performance of the UMS varies significantly. The number of effective OD pairs in the Nanjing Metro network is 33,438, and when Line 1 malfunctions, the network performance drops by more than 80%. The impact of Line 1’s shutdown on the overall network operation is almost devastating.

Enhancement strategy: Critical lines are those that significantly impact network performance during the operation of the UMS. In actual operation management, attention should not only be focused on flood-prone locations due to waterlogging disturbances but on the real-time tracking of the safety status of key lines. The consequences of critical line failures are even greater than those of failures at flood-prone locations.

(3)

Determine the recovery sequence of stations and the entire line based on the network resilience value. During flooding events, the UMS network may experience multiple inundations, and the backflow of rainwater can damage the network’s structure and equipment. In the event of an operational disruption, the initial priority is to swiftly evacuate any trapped passengers. Subsequently, resources should be mobilized to repair the damaged lines and equipment within the stations. Variations in resource allocation and the order of site repairs will lead to differing levels of network resilience during the recovery process. A higher flood resilience value, which is characterized by the network’s connectivity, indicates a stronger system response to disasters and a more efficient and rapid return to normal operations.

Enhancement strategy: The formation of resilience follows a temporal evolution that includes absorption, resistance, repair, and adaptation. An external indicator of waterlogging resilience in the UMS operations is the network structure’s connectivity. During network repairs, nodes with higher degree values should be prioritized, and the network’s transfer functions should be promptly restored. If multiple nodes share the same degree value, they should be ranked based on passenger flow and station importance. Subsequently, attention should be directed to addressing the deep waterlogging in key urban road areas, as the severity of waterlogging on roads directly impacts the damage to stations. When the number of damaged points is limited, the restoration sequence can be determined by the network resilience calculations. The connectivity and efficiency of the network consistently guide the selection of the line structure repair sequence.

Due to the technical limitations of the existing algorithms and software platforms, this study has several constraints. For simulating urban flooding scenarios, this paper employs ArcGIS as the modeling platform and utilizes an improved passive inundation algorithm to identify each bluespot within the city. However, numerous assumptions were necessary for this study. The water balance equation does not perfectly align with actual runoff changes and the urban pipeline network’s drainage capacity is estimated using a rainfall intensity formula, which deviates from reality. The actual conditions of the pipeline network are extremely complex, making it challenging to fully comprehend. Additionally, the urban DEM data uses 30-m precision, which affects the calculation results for the bluespots. In future research, data sources will be updated and refined to enhance the accuracy of the calculations.