A Review of Resilience Assessment and Recovery Strategies of Urban Rail Transit Networks

Abstract

Rail transit is an important means of ensuring the sustainable development of urban transportation, but disturbance events caused by natural disasters, human factors, and other influences can lead to disruptions in rail transit operations. To cope with the impact of disturbance events on urban rail transit networks, and to explore the changes in rail transit network performance and recovery strategies under the influence of disturbance events from a resilience perspective, this paper overviews the existing research on resilience assessment and recovery strategies for urban rail transit networks. Firstly, the characteristics of the urban rail transit network and the model construction method are analyzed. Secondly, on the basis of combing the connotation development of system resilience, urban rail transit network resilience is defined, while the existing resilience metrics and assessment indexes are classified and summarized. Finally, the failure scenarios and recovery strategies of urban rail transportation network are deeply studied and discussed. The research results show that urban rail transit network resilience has been widely concerned by scholars, and certain results have been achieved in three aspects of resilience connotation, resilience assessment and recovery strategy. Nevertheless, further research is needed on these aspects. We propose future research directions that involve exploring modeling methods aligned with actual network topologies, developing unified indexes for resilience assessment and focusing on resilience assessment and recovery strategies under uncertain disturbance events. The research results can provide an important reference for the resilient operation and sustainable construction of urban rail transit.

1. Introduction

With a dense population and concentrated buildings, a city is a complex and internally interdependent system. The interconnection of various infrastructure enables the city to operate normally [1]. The urban rail transit, with its characteristics of large volume and fast speed, has become an important tool for people to travel, effectively alleviating the pressure of urban ground traffic. At present, the rail transit in major cities is developing rapidly towards network operation. As of 31 December 2022 (according to the 2022 Annual Statistics and Analysis Report of Urban Rail Transit released by the China Urban Rail Transit Association), a total of 55 cities in China have opened 308 urban rail transit operation lines, and the total length of the operating lines is 10,287.45 km. Of these, 26 cities has four or more operating lines and three or more transfer stations, accounting for 47.27% of the total number of cities operating in urban rail transit [2]. This shows that a considerable part of the cities with rail transit are tending to network operation. The operating mileage of rail transit in the Chinese network of operating cities is shown in Figure 1. In the context of networked operation, the interaction between stations and lines increases. Once the system is disturbed by natural disasters, mechanical equipment failures, or human factors, it will quickly spread to other parts of the network, affecting the operation of a single line and even paralyzing the entire railway network [3,4]. For example, on 20 July 2021, the extreme rainstorm in Zhengzhou caused severe urban waterlogging, which destroyed the water retaining wall and poured into the subway tunnel, resulting in the death of 14 people [5]. On 11 November 2022, the power supply equipment of Shanghai Rail Transit Line 11 failed, causing a train to immediately short circuit at the Malu section. This incident resulted in two trains stopping, trapping more than 1300 passengers and delaying them for over 60 min [6].

The two characteristics of disturbance events are the suddenness and the uncertainty of the degree of disturbance, which make improving the ability of the urban rail transit network to deal with disturbance events a focus for scholars. The “Resilience City” in the 21st century is a kind of safety management concept based on the formation of an ecosystem theory, which is designed to adapt to external emergencies and be symbiotic with the target [7]. Scholars have gradually paid attention to the impact of disturbance events on the performance of the traffic network and the performance loss in the recovery process.

This paper begins with the construction of an urban rail transit network model based on complex network theory and summarizes the shortcomings of the existing research. Then, it analyzes the resilience assessment, failure scenarios and recovery strategies of the urban rail transit network. Finally, this paper provides a new perspective for the development of the optimal recovery strategies of the urban rail transit network considering resilience.

The rest of this paper proceeds as follows. Section 2 briefly introduces the characteristics and construction methods of urban rail transit network models, which lays the groundwork for later analysis. In Section 3, the assessment methods and indexes are summarized based on the development and connotation of resilience. The research of failure scenarios and recovery strategies for urban rail transit networks is analyzed in Section 4. This paper ends with discussion in Section 5.

2. Construction of the Urban Rail Transit Network Model Based on Complex Network Theory

As a mature and effective network research theory, complex network has been widely used in many fields. As a complex network, the urban rail transit network abstracts the stations as nodes and the lines as the connecting edges, and its characteristics and model construction methods are the basis for the development of urban rail transit network research.

2.1. Characteristics of the Urban Rail Transit Network

A network is a combination of many basic individuals with certain characteristics and functions, and there are certain interrelationships between internal compositions. There are four basic models of complex networks: regular network [8], random network [9], small-world network [10] and scale-free network [11]. Among them, the proposal of the small-world network and scale-free network reveals the formation law and evolution mechanism of many real networks [12], which can relatively accurately describe the characteristics of real networks. At present, the existing research has proved that the urban rail transit network has small-world and scale-free characteristics. Latora et al. [13], Deng et al. [14] and Meng et al. [15], respectively, confirmed that the Boston, Nanjing and Shenzhen city urban rail transit network have small-world and scale-free characteristics, and the complex network theory was applied to the study of the urban rail transit network.

2.2. Construction Method of the Urban Rail Transit Network Model

The construction of an urban rail transit network model is a process which abstracts the urban rail transit network and changes it into a simplified complex network model. It provides a basic model basis for analyzing the topological structure characteristics and various characteristics of the urban rail transit network. Sienkiewicz et al. and Von Ferber et al. conducted a study on the construction method of a public transportation network model [16,17,18,19] The four common construction methods are: Space L [20,21], Space P [22,23], Space B and Space C. The construction method is shown in

Table 1. Construction method of the topological structure of the urban rail transit network.

Construction Method	Overview of the Method	Sketch Map
Space L	Urban rail transit stations are regarded as network nodes. If two stations are adjacent on a single rail transit line, there are connected edges between them.
Space P	Urban rail transit stations are regarded as network nodes. If the stations have a direct transportation line, there is a connection between them.
Space B	The stations and lines in the urban rail transit network are regarded as network nodes and the nodes abstracted as a line have connected edges with any station node on the line.
Space C	The lines in the urban rail transit network are regarded as the network nodes. If different lines can be reached by transfer, the abstract nodes of these lines are connected with each other.

. The Space L focuses on the actual topology structure of the urban rail transit network, but it cannot reflect the transfer properties. Space P focuses on demonstrating the network transfer properties, but it cannot reflect the actual network structure, and its model construction is complex with a large calculation volume. Space B and Space C abstract the lines as the network nodes, and they have certain differences with the actual topological structure. Compared with other methods, Space L can recreate the real structure of the rail transit network. Therefore, many studies use Space L to build the urban rail transit network model, few of them use Space P, and Space B or Space C are rarely used.

In addition, based on complex network theory, scholars construct the undirected network model [21,24,25]. However, the unweighted undirected network ignores the characteristics of unequal distance or the overlap of line in the real world, as well as the effect of travel time, route and direction which caused by the passenger flow. Therefore, it is difficult for the undirected unweighted network to truly and accurately describe the rail transit network with the characteristics of spatial and temporal change. Based on this, some scholars construct network models from different angles, which are mainly divided into two methods [26], similarity weight and dissimilarity weight. Similarity weight is generally used to describe traffic in a network, which is positively correlated with density. In contrast, dissimilarity weight is generally used to describe the connection cost of network nodes correlated with distance. According to the complexity and topological structure characteristics of urban rail transit network, the similarity weight method is more suitable for the close relationship between nodes in the real network. Therefore, many scholars choose the similarity weight method to build the model. For example, Meng et al. [15] constructed a weighted network model by using the passenger flow of the interval section as the side weight.

Combined with the characteristics of the rail transit network, some scholars have improved the network model construction method based on the above methods. Ni et al. [27] considered the different connection weights of different stations and transfer stations, introduced virtual transfer stations and sides to build a rail transit network, and split the transfer stations into multiple interconnected nodes for research. Feng et al. [28] only considered transfer and terminal stations, and merged the same path into one path to establish a network. Chen et al. [29] analyzed the relationship between the lines where the stations are located with Space Syntax and constructed an urban rail network model by considering the traffic characteristics of the urban rail network that runs on specific lines. The above model construction methods of rail transit networks are all based on single-layer networks, and all nodes in the network exist in the same layer network. But for some special network structures, such as when two adjacent stations are transfer stations on the two lines, the modeling approach of a single-layer network will fail. Therefore, Yin et al. [30] considered the independence of different lines and the heterogeneity between lines and transfer sides, and proposed the application of the multi-layer network to construct the rail transit network model. In that way, each line is abstracted as a layer of the network, stations of lines are abstracted as nodes of a single-layer network, the transfer stations exist in the multi-layer network and the transfer stations of different layers are connected with each other, forming the transfer edge.

In recent years, with the development of complex network theory, super-network models can not only analyze the structure and performance of single-layer networks, but also consider the interactions between single-layer networks and other types of networks due to their more complex characteristics such as multilayers, multilevel and multiple attributes. The network model based on a super-network is made by layering the superposition and connecting the sub-networks of different layers by adding virtual commutation curves, and this type of network can portray the intricate and multiple correlations between several different types of networks. The network model is now gradually being applied in the transport field. Yamada et al. [31] proposed a discrete network design problem based on the super-network optimization of a freight network. Yu et al. [32] constructed a super-network model of Nanjing urban rail transit and compared it with the traditional Space L and Space P models. Tilg et al. [33] employed a super-network model to improve aggregated traffic models based on the macroscopic fundamental diagram (MFD), and conducted a case study for the realistic network of Sioux Falls.

In conclusion, the existing research achievements on the characteristics and model construction of the urban rail transit network are relatively rich. With the revelation of small-world and scale-free characteristics, scholars have conducted a lot of research on the urban transportation network based on the theory of complex network. In terms of network model construction, it has developed from a simple modeling stage to being combined with the characteristics of a rail transit network. The Space L and the Space P method commonly used in the existing model-building methods cannot reflect the actual topological structure and transfer nature of the network. According to the characteristics of the rail transit network, this method can solve the problem where the Space L cannot reflect the nature of the transfer to a certain extent, which introduced the virtual transfer station and the virtual transfer side. While the rail transport network has a multi-layer nature, it is difficult to reflect the heterogeneity and interrelationship of different lines in the model construction of a single-layer network; applying the multi-layer network or super-network to the model construction of the rail transport network is more compatible with the actual characteristics of the network.

3. Assessment of the Resilience of Urban Rail Transit Networks

As a complex and huge system, the urban rail transit network has the risk of network interruptions caused by disturbance events in the operation process, which requires the rail transit network to have strong resistance and recovery in the face of disturbance events. Therefore, scholars have introduced the concept of system resilience and applied it to the realm of urban rail transit to analyze how system performance evolves throughout the entire disturbance event.

3.1. Development and Connotation of Resilience Theory

“Resilience”, originating from “resilio” (Indo-European-Italian), means restoration to the original/original state. Resilience, first used in the field of mechanics, refers to the ability of a material to deform and store the recovery potential energy due to force without complete fracture or complete deformation. In 1973, Holling [34] first introduced resilience into the field of ecology to describe the persistence of ecosystems and their ability to absorb all kinds of changes and disturbances. Subsequently, Pimm [35] proposed a different view that resilience is the rate at which the system returns to its original equilibrium state after being disturbed. Both resilience views focus on the structure and performance maintenance of the system; the difference is that the former focuses on the strength of the disturbance that the system can withstand, while the latter focuses on the comprehensive ability of recovery, resistance, persistence and change in the system after interference [36].

Over the past 50 years, the theory of resilience has found extensive application across diverse research domains, spanning from mechanics and ecology to social, economic, environmental changes and disaster science [37]. In 2006, Murray-tuite [38] first proposed that the resilience of a transportation system refers to the ability to maintain or quickly restore its original functions after interruption or disaster and divided it into ten dimensions: redundancy, diversity, resilience, adaptability, safety, efficiency, autonomy, collaboration, accessibility and reconstitution. The UK Department for Transport [39] defines transport resilience in relation to transport capacity: it denotes the ability of a transport network to endure the impacts of severe weather conditions, to function effectively under such conditions and to swiftly restore operations thereafter. In urban rail transit, there is no accurate and uniform definition. Zhang et al. [40] defined urban rail transit network resilience as the level of network connectivity after a node interruption and the ability to restore connectivity to an acceptable level after recovery strategies. Bešinović et al. [41] defined rail transport system resilience as the ability of the rail system to provide effective services under normal conditions and to resist, absorb, adapt and quickly recover from interruptions or disasters. Therefore, this paper defines urban rail transit network resilience as the ability of the network to resist, absorb and adapt to the disturbance event and quickly recover from the disturbance to an acceptable level through the emergency phase and then to the initial network performance level.

According to the definition of resilience, the network performance changes under perturbation events are shown in Figure 2. The abscissa in the figure represents the time $t$, the ordinate represents the network performance response function $Q(t)$. The resilience process can be divided into five stages: at the initial $t_0~t_{\mathrm{d}}$, network performance remains in a steady state $Q(t_0)$, at this time, the rail transit network is in a normal operation state and reflects the reliability of the network; the disturbance event occurred at $t_{\mathrm{d}}$, causing failure of network node or connection fault and network performance has declined rapidly, which embodies the vulnerability of the network until $t_{\mathrm{a}}$, network performance drops to a minimum $Q(t_{\mathrm{a}})$; $t_{\mathrm{a}}~t_{\mathrm{r}}$ is the response stage, the operation and management department of urban rail transit shall formulate efficient and reasonable recovery strategies according to the network damage situation and the existing resources situation; $t_{\mathrm{r}}~t_{\mathrm{s}}$ is the emergency recovery stage of the network, with the guidance of recovery strategies and assistance from external resources, network performance returns to an acceptable level; until $t_{\mathrm{e}}$, the network fully returns to the performance level before the perturbation event, considering the disturbance events to influence the elimination, the rail transit network has returned to normal operation.

In conclusion, reliability and vulnerability are aspects of system performance that gradually emerge throughout the full life cycle of a disaster, influenced by disturbance events [36], which can be regarded as a characterization of the resilience of the system at different moments. In contrast, resilience can not only reflect the resistance, absorption and response ability of the system in the whole operation cycle, but can also reflect the recovery and adaptability of the system. It can describe the system performance more comprehensively and analyze the changes in the performance of the urban rail transit network under disturbance events.

3.2. Resilience Measurement Method of the Urban Rail Transit Network

Resilience measurement is a critical aspect in the study of urban rail transit network resilience. Accurately measuring network resilience provides a theoretical basis for transit operation and management authorities to develop rational and effective recovery strategies. Measurement methods are categorized into qualitative assessment and quantitative assessment approaches.

The qualitative analysis method generally analyses the factors that may affect the system resilience through a questionnaire survey, fuzzy analytic hierarchy process or expert scoring method, and establishes a resilience assessment index framework to assess the system resilience. Mostafavi [42], through the investigation, established a set of qualitative analysis indicators from the four aspects of the current situation, driving factors, obstacles and improvement strategies; and then, the expert scoring method was used to construct the resilience assessment index system of the transportation network. Vugrin [43] comprehensively considering the absorption, adaptation and recovery of the system, proposed a qualitative analysis of the system resilience evaluation framework. The quantitative assessment method originated in seismic engineering research, primarily relying on data to establish mathematical models for quantifying system resilience. Depending on the data source, these methods can be categorized into empirical analysis and simulation analysis. Empirical analysis is for the perturbation events that have already occurred. For example, Chan et al. [44] and Mudigonda et al. [45] assessed the resilience of transportation infrastructure to disruptions based on operational data during Hurricane Sandy. Diab et al. [46] examined the relationship between outdoor subway track segments, weather conditions, frequency and duration of service disruptions using 2013 data from the Toronto subway system. Xia et al. [47], based on the perspective of resilience, analyzed a total of 1911 urban rail transit operation accidents from 2007 to 2018. However, due to the complexity and suddenness of disturbance events in urban rail transit networks, obtaining real-time data during such events can be challenging. Therefore, current research often employs simulation analysis methods. This involves simulating attacks on network components and observing the entire process of response and recovery. Based on this, resilience performance curves or other measurements characterize the resilience of urban rail transit networks [48]. The quantitative assessment methods of existing resilience are summarized in

Table 2. Existing quantitative assessment method of resilience.

Assessment Measure			Sketch Map	Formula
quantitative assessment	Based on the resilience performance curve	Resilience triangle		$R={\displaystyle {\int }_{t_{\mathrm{d}}}^{t_{\mathrm{s}}}\left[100\%-Q(t)\right]}dt$
		Performance integral ratio		$R={\displaystyle \frac{{\displaystyle {\int }_{t_{\mathrm{d}}}^{t_{\mathrm{s}}}Q(t)dt}}{{\displaystyle {\int }_{t_{\mathrm{d}}}^{t_{\mathrm{s}}}Q(t_0)dt}}}$
		Decrease ratio		$R={\displaystyle \frac{Q{(t_{\mathrm{d}})}_{\max }-Q(t_{\mathrm{d}})}{Q{(t_{\mathrm{d}})}_{\max }}}$
	Based on the resilience performance characterization	Ability to adapt, absorb, and recover	—	$R=R_{\mathrm{s}}·R_{\mathrm{x}}·R_{\mathrm{h}}$
		Ability to recovery and absorb	—	$R=\alpha ·R_x+\beta ·R_h$

$R$ is resilience. $Q(t)$ is network performance. $Q(t_0)$ is network initial performance. $Q(t_{\mathrm{d}})$ is the actual degradation in network performance after interference. $Q{(t_{\mathrm{d}})}_{\max }$ is the maximum possible decrease in network performance after disturbance. $R_{\mathrm{s}}$ is network adaptability. $R_{\mathrm{x}}$ is network absorption capacity. $R_{\mathrm{h}}$ is network recovery ability. $t$ is time. $t_{\mathrm{d}}$ is disturbance start time. $t_{\mathrm{s}}$ is performance full recovery time. $\alpha$, $\beta$ is the weight of $R_{\mathrm{x}}$ and $R_{\mathrm{h}}$, respectively.

.

3.2.1. Based on the Resilience Performance Curves

Bruneau [49] proposed that the resilience triangle is the most commonly used resilience index measurement method to measure the performance of the system and the performance of the integral measurement of the network of resilience and its universal advantages, but this method assumes that the normal performance level of the system is 100%, and the system performance after the disturbance event instantly drops to the lowest [50]. Reed [51], on the basis of the resilience triangle, put forward the method of performance integral ratio, which is the network performance from the time of disturbance to full performance recovery, and used network performance at the normal state as the standard to measure network resilience. This can better reflect the performance loss and normal performance level in the recovery process, but cannot reflect the relationship of the network interference system resilience over time. Rose [52] defined resilience as the ability of the entity or system to maintain system performance when affected by disturbance, with network interference being measured as the ratio of the difference between the maximum possible decrease in system performance and the actual decrease in system performance after the network has been subjected to a disturbance to the maximum possible decrease in system performance; however, the difficulty of this method is in estimating the maximum decline in system performance.

3.2.2. Based on the Resilience Performance Characterization

Based on some or all of the capabilities of a system, the measure suggests that resilience includes resistance, absorption, recovery and adaptation abilities. Absorption refers to the ability of the system to actively absorb the disturbance impact and minimize the negative impact, adaptability is the ability of the system to adjust itself and cope with the disturbance impact and recovery refers to the ability of the system to quickly achieve effective recovery under the influence of disturbance events [53]. Francis et al. [54] represented resilience as the product of system adaptation, absorption and recovery abilities during a disturbance event. Cheng et al. [55] measured the resilience level of an infrastructure network as a weighted sum of absorption and recovery abilities.

From the analysis summarized above, the qualitative assessment method has a certain subjectivity; therefore, most studies used the quantitative assessment method to measure resilience. Two commonly used methods are based on resilience performance curves. The resilience triangle method can accurately express the resilience loss value of the urban rail transit network under disturbance events, and the performance integral ratio rule focuses on the relationship between network performance loss and initial network performance; while based on resilience capacity characterization and combining qualitative and quantitative methods, the adaptation, absorption and recover ability of resilience are also subject to certain subjective influences. Therefore, measuring resilience based on resilience performance curve is a relatively good measure at present.

3.3. Assessment Index of Urban Rail Transit Network Resilience

System resilience in urban rail transit operations encompasses the network’s abilities to resist, absorb, recover, and adapt throughout its operational lifecycle. When quantitatively assessing system resilience based on resilience performance curves, a primary concern is how to measure the changes in network performance. Current research primarily defines network performance response functions by selecting indicators from two perspectives: network topology and passenger travel service.

3.3.1. Network Topology

The characteristics of the network topology structure include network connectivity, transfer ability, average path length and diameter, which can reflect various abilities of the network structure from various angles. Among them, network connectivity is the premise and basis for ensuring the normal operation of the rail transit network and the normal transportation of passenger flow. In the statistical parameters of complex networks, static and dynamic indicators can be used to analyze the connectivity. The static indicators such as node degree, degree distribution and connectivity can only describe the general characteristics of the network, while the dynamic indicators such as the global network efficiency and maximum connected subgraph ratio can reflect the overall change trend in the network under disturbance. Therefore, most studies [56,57,58,59] have selected network average efficiency as the index for assessing network connectivity and analyzing changes in the performance of urban rail transit networks under the influence of disruptive events. Some scholars evaluate network resilience by integrating multiple parameters. For example, Qiao et al. [60] calculated network efficiency, average shortest distance and network diameter individually. Zhang et al. [59] chose three topological structure characteristics of network degree, network accessibility and network average efficiency as the resilience assessment index. Ding et al. [61] proposed to measure urban rail network resilience based on the sum of RTTN node degree values and the sum of the shortest distance between all node pairs. Although the selection of parameters based on the network topology structure can better describe the actual network structure, this method ignores the important position of passenger travel in the operation and management of urban rail transit and cannot reflect the relationship between the network topology structure and passenger travel service.

3.3.2. Passenger Travel Service

In the view that the resilience assessment index based on the network topology structure cannot reflect the actual operation of the urban rail transit network, many scholars measure the network performance from the perspective of passenger travel service. Lv et al. [62] considered the influence of the line passenger flow and network service efficiency, proposed a service resilience index based on traffic loading and constructed a network resilience assessment model based on service efficiency. Tang et al. [63] took the network passenger flow as the index of urban rail transit network travel service. Knoester et al. [64] proposed a composite performance indicator, which was calculated as the weighted sum of traffic punctuality and traffic intensity. Ma et al. [65] constructed an urban rail transit resilience evaluation model, in which all the indicators were from the perspective of passenger travel service, including travel utility accessibility, the passenger flow loss rate, the travel time loss rate and the travel recovery efficiency. Itani et al. [66] used passenger travel delays and the number of bridged buses after disruptions to evaluate the resilience of Toronto’s transit network.

Some scholars comprehensively consider the network topology structure and passenger travel service to define the network performance response function. For example, Lu et al. [58] built new indicators to measure network performance in three aspects: comprehensive reference number, travel time and passenger flow. D’Lima [67] used the speed of recovery of passenger numbers after emergencies as an indicator of resilience. Zhang [68] considered the flow of passengers and used the node strength to evaluate the importance of nodes in the network. Yin et al. [69] proposed two types of quantitative resilience loss indicators based on the whole network accessibility of urban rail transit and the proportion of unaffected passengers in the whole network. Li et al. [70] started from these two aspects, reflecting the network topology with the node degree value and the number of reasonable paths, and proposed new indexes for network resilience evaluation by considering the differences in passenger trips and passenger flows. In addition, there are two main types of multi-indicator research: composite indicators [71,72,73] and weighted indicators. The former is mainly combined by analyzing the physical relationship between indicators; the latter is weighted according to importance, etc., and it is difficult to avoid the problem of subjectivity.

In conclusion, the qualitative analysis of resilience measures exhibits considerable subjectivity, whereas quantitative assessment methods enhance scientific rigor and comprehensiveness. Current research defines network performance response functions through quantitative analysis and utilizes resilience performance curves, resilience triangles and performance integral ratio methods to quantitatively assess the resilience of urban rail transit networks. However, scholars differ in their selection of indices for network performance response functions, primarily focusing on network topology structure and passenger travel services. This lack of a unified standard has resulted in some deviation in the analysis results among different scholars.

4. Failure Scenarios and Recovery Strategies of the Urban Rail Transit Network

The diverse causes of failures in urban rail transit networks necessitate a clear delineation of failure scenarios to facilitate the development of efficient and rational recovery strategies. This approach aims to reduce accident losses and enable stations to maintain the capability of implementing train operation plans, thereby ensuring the network can provide normal operational services. Such efforts are crucial for enhancing the operational efficiency and security assurance of urban rail transit network systems. The urban rail transit network disturbance scenario and recovery strategy framework are shown in Figure 3.

4.1. Failure Scenarios of the Urban Rail Transit Network

Following disturbance events leading to station or interval failures in urban rail transit systems, developing a recovery strategy to swiftly restore the network to an acceptable level is the core focus of resilience research. Existing studies often simulate network failures using deliberate or random attack methods, focusing on scenarios characterized by low-frequency but high-impact disturbances such as severe weather events, major infrastructure failures or deliberate attacks. It can be roughly divided into station failure and interval failure. Taking station failure as the research object, Zhang et al. [74] simulated two failure scenarios: the failure of three stations on a single line, and large-scale power outages resulting in the failure of 20 stations across the region. Chen et al. [75] modeled and analyzed three failure scenarios: the failure of a single primary station or edge, the failure of a single multi-line interchange station and the failure of multiple stations or edges. Zheng et al. [76] designed three scenarios based on the scope and severity of disruption: randomly selecting ten stations, failure of ten randomly selected stations, failure within a radius based on flood-prone areas and deliberate attacks targeting the top ten most critical stations.

In the above study, the station or interval is considered independently, that is, the failure of a station or interval does not affect the station or interval of the adjacent section. Ignoring the operation management department to reduce the influence of the disturbance event, it usually adjusts the train crossing route and selects the nearest station with return conditions on both sides of the damaged position. In urban rail network vulnerability studies, a few studies [77] have considered the impact of this factor on network performance, while it has not yet attracted attention in resilience studies. Most of the scenarios about disturbance event set by scholars according to the research content has strong subjectivity. When disturbance events occur, the failure of a station or interval often triggers a cascade effect. Station failures result in significant passenger congestion, prompting passengers to choose alternative travel routes, either actively or passively redistributing to other stations or lines. This redistribution can overload other stations or lines, resulting in additional congestion and failures. The potential for cascading failures is currently being investigated in some of the studies on rail transport. Adams et al. [78] constructed a multi-hazard risk model for UK railways with cascading damage pathways based on historical and current data. Guo et al. [79] considered the cascading effects of failure and recovery and revealed the impact of different failure spreads and recoveries with realistic examples. However, it is rarely represented in resilience studies at present [80].

4.2. Urban Rail Transit Network Recovery Strategy

The key to the study of fault recovery strategy is how to reasonably arrange the recovery sequence of the failure station or interval in the network. The existing recovery strategy mainly includes exhaustive recovery, random recovery and priority recovery. Exhaustive recovery involves listing all possible recovery strategies one by one to select the best strategy. For instance, Li et al. [70], Zhang et al. [40] and Saadat et al. [81] used the exhaustive method to analyze optimal recovery strategies for rail transit networks in Beijing, Shanghai and Washington after faults. However, this method is evidently unsuitable for complex scenarios with numerous alternative strategies. Random recovery randomly determines the recovery order of failure nodes or connected edges. This method has great uncertainty, low recovery efficiency, long time, and is not applicable under limited resources. Priority recovery is determined by the subjective preferences of decision makers, who establish the recovery sequence based on specific criteria such as node degree or importance. For example, Yadav et al. [80] chose network centrality (degree centrality, betweenness centrality, eigenvector centrality) as the index of the preferred recovery strategy, where these nodes are added back to the network in decreasing order of the chosen node centrality metric during recovery. Zhang et al. [74] chose node degree as the indicator of priority recovery strategy, and in contrast to a random recovery strategy, the priority recovery strategy is more helpful to improve the performance of the network throughout the recovery stage. However, while superior to random recovery, this method may miss the optimal scheme when faced with numerous alternatives, often resulting in suboptimal outcomes.

The recovery process of rail transit network is dynamically changing, and the above static recovery strategy may not achieve the optimal recovery effect, and it is difficult to be applied to the practical engineering field. As a result, scholars are increasingly focusing on recovery process and model-based recovery strategies. Martello et al. [82] designed three performance curves considering failure magnitude: linear recovery for minor disruptions, trapezoidal performance recovery for major disruptions and assuming system shutdown within the first 20% of the recovery period for extreme disruptions. Lv et al. [62] proposed an optimization model with the goal of maximizing the service resilience of the rail transit network based on the genetic algorithm to obtain the optimal recovery order of failure stations. Xu et al. [75] used the network average efficiency to construct an urban rail transit resilience model and found that the selection of an optimal recovery sequence for the failed components can effectively minimize resilience losses. Yin et al. [69] noted that the network passenger flow dynamically changes during the repair process. They established a recovery model aiming to quantify resilience loss with travel efficiency and overall network recovery as goals. The results indicate that the model-based recovery strategy effectively restores network functionality after damage.

The timing of recovering failed stations or intervals is crucial for formulating recovery strategies. However, in practical engineering applications, the recovery efforts of urban rail transit networks are influenced by various uncertain factors, including the severity of damage at failure stations, availability of recovery resources, the number of repair teams and the duration required for recovery [83,84,85]. However, these aspects have not yet received sufficient attention in rail transit recovery studies. Existing research tends to oversimplify the repair process, often assuming uniform repair times and resource requirements across different stations or intervals [69,74]. For example, in the study of the optimal recovery strategy, Lv et al. [62] assumed that the repair of the transfer station takes 2 days and the non-transfer station takes 1 day. Zhang et al. [86] assumed that the interval recovery time is proportional to the length of the interval. Considering the budget limit, Zhang et al. [59] assumed that the repair time follows the normal distribution to consider the uncertainty, constructed the recovery model based on the system resilience and solved the network recovery timing scheme.

In conclusion, scholars have recognized that the recovery process of urban rail transit networks is dynamic, evolving from a simple static recovery strategy to a model-based approach. Through simulation and other methods, it can better simulate the changes in the actual network and complete the recovery of network damage. However, in simulating disturbance event failure scenarios, scholars have generally overlooked the interdependence of stations or intervals and have paid insufficient attention to adjustments in train scheduling across different stations and intervals. This oversight disregards how adjustments in train scheduling impact the timing and sequence of network recovery. At the same time, the existing studies rarely take into account the uncertain factors of the recovery of urban rail transit network, such as the uncertainty of recovery time and recovery resources, which leads to the ideal of the research results and difficulties in practical engineering application.

5. Discussion

To ensure people’s travel safety and promote the development of urban rail transit, improving the resistance and recovery of the urban rail transit network to handle disturbance events is essential and important. In recent years, the resilience of the urban rail transit network has been widely concerned by scholars, and some achievements have been made in network model construction, resilience assessment and recovery strategy. However, the current dynamic process of network model construction, failure scenario simulation and recovery is still inconsistent with reality. With the rapid development of urban rail transit towards a networked direction, any disturbance event can quickly propagate throughout the entire network. Therefore, research on resilience assessment and recovery strategies for urban rail transit networks is crucial. The major findings from this research are summarized as follows:

• The construction of an urban rail network model based on complex network theory serves as the foundation for conducting research on rail resilience. The existing Space L and Space P methods cannot simultaneously reflect the actual network topological structure and its transfer properties. Therefore, subsequent research should consider the heterogeneity among different lines and transfer properties to build a model that more accurately represents these characteristics. This will enhance the practical application value of the research results.
• The selection of appropriate assessment indexes and metrics is crucial for evaluating urban rail transit resilience. Most existing research on resilience in urban rail transit networks has utilized methods such as the resilience triangle and performance integral ratio for quantitative analysis. However, there is no unified standard among scholars for selecting performance response function indices, leading to deviations in analysis results. In subsequent research, it would be prudent to begin with the definition of urban rail transit network resilience and then select a comprehensive set of indicators that can simultaneously reflect both the network topology structure and passenger travel services.
• The recovery strategy should be developed based on the actual operational characteristics of urban rail transport. Currently, many studies on the recovery strategy of urban rail transit networks treat damaged stations or sections independently when determining the extent of failures. Few studies consider how urban rail transit management adjusts train operations across different stations and sections during disturbance events. This oversight neglects the interconnectedness of stations and intervals within the same operational segment of the rail transit system, resulting in a gap between study outcomes and real-world scenarios.
• The restoration of urban rail transit networks during disturbance events depends not only on recovery resources and the number of repair teams but also on the extent of damage to stations or intervals of the rail network and the repair capabilities of the teams. Most existing studies overlook the influence of these factors on recovery outcomes, with a few treating them as fixed values. Therefore, the focus of future research will be on how to incorporate the uncertainty of disruptive events into the study of repair strategies.

The urban rail transit network, as an important infrastructure for urban transportation, promotes environmental and economic sustainability with its energy-saving, high-capacity and non-polluting characteristics. Enhancing the resilience of urban rail transit systems is paramount for ensuring the delivery of safe and efficient services. We believe that a thorough literature review on resilience and recovery in urban rail transit can serve as a reference for the safe operations. This helps improve the resilience of urban rail systems and promotes sustainable urban transport development.