Resilience concepts in integrated urban transport: a comprehensive review on multi-mode framework

3.1.1 Attribute-based metrics.

As mentioned in Section 2, the classical framework to describe transportation resilience consists of four attributes: robustness, redundancy, resourcefulness and rapidity. In addition to that, much research extends the framework with other attributes. Attribute-based metrics directly quantify some attributes on the performance only at special periods.

In the pre-perturbation (disruption) phase, Hua and Ong (2017) used total unaffected passenger flow to describe the extent of robustness. Beiler et al. (2013) calculated redundancy as the alternative routes between critical ODs and mentioned a travel time index as “travel time in peak hours under normal conditions,” while Soltani-Sobh et al. (2016) proposed the operational cost and expected failure cost as indicators to distinguish the influence caused by a specific perturbation. Besides, Yoo and Yeo (2016) introduced adaptive capacity metrics by calculating the number or area in which adjacent nodes can replace an attacked node. Murray-Tuite (2006a, 2006b) quantified adaptability by the atypical uses of the transportation system, safety by the number of traffic incidents and mobility by the networkwide efficiency.

In the post-perturbation (recovery) phase, the resilience triangle in Figure 1 indicates T₃-T₂ as the restoration time, P(T₁) as the maximum performance, P(T₂) as the robustness of the system and [P(T₃) − P(T₂)]/(T₃-T₂) as the rapidity of recovery, which are the outcome of robustness, redundancy and resourcefulness (Tierney and Bruneau, 2007). Wang et al. (2015) represented rapidity as the speed of the system returning to the normal state and recovery as the time required to alleviate the systematic disarray. Furthermore, D'Lima and Medda (2015) emphasized recoverability on the cost and speed of returning to a new equilibrium.

3.1.2 Performance-based resilience metrics.

Unlike attribute-based metrics, performance-based metrics aim to assess the performance of the transportation system during the whole affected period. Zhou et al. (2019) identified and sort out three of the most widely used performance-based metrics in previous literature. The first one is the time-based degradation of system performance proposed by Twumasi-Boakye and Sobanjo (2018). The second is the time-dependent ratio of recovery on quantifying performance loss by Liao et al. (2018). The third is the expected fraction of demand satisfied in post-perturbation phase using the recovery costs algorithm introduced by Chen and Miller-Hooks (2012).

The resilience index is first defined by Reed et al. (2009) equation (1), and it is based on the concept of the resilience triangle. According to Figure 1, where R(T) is the resilience index at time point T, T₁ is the time point when the perturbation begins. Similarly, as displayed in equation (2), Q(T) is the performance percentage of the specified transportation system, t is the timestamp when perturbation first occurs and t + α is the timestamp when the system's performance completely recovers. Their theory described resilience by the metric of performance loss from the full system performance during the whole disruptive event. It is not surprising that the value of Q(T) is equal to the area of the resilience triangle in Figure 1.

Then this algorithm of system performance was modified and improved by Bocchini and Frangopol (2012) and Adjetey-Bahun et al. (2016). As shown in equation (3), the metric of performance loss in equation (1) is divided by the time interval α between perturbation first occurs and system's performance completely recovers. In the modified algorithm, resilience is measured by the metric of average performance loss over the specified time interval α.

Liao et al. (2018) introduced five system states in the whole perturbative event in Figure 4: original stable state, system disruption state, disrupted state, recovery state and stable recovered state. The five states are the important watershed for the time-dependent metric of transportation resilience. The metric is the ratio of recovery at the time t_i to function loss at other specified time points. As shown in equation (4), R_F (t_i | p_j) is the resilience metric at the time t_i under the resilience perturbation p_j, while F (t_i | p_j) is the function of system performance at the time point t_i with the perturbation p_j. t₀ is the function of the system at original stable state. F (t_m | p_j) is the minimum function. In this algorithm, transportation resilience is treated as a more dynamic concept and is more consistent with the ability to bounce back.

Chen and Miller-Hooks (2012) proposed another widely used metric which is usually used in the resilience in road and metro networks. The expected fraction of demand satisfied in the post-perturbation phase using the recovery costs is calculated in equation (5), where d_w is the maximum demand satisfied with the OD pair ω at the post-perturbation phase and D_w is the demand satisfied with the same OD pair at the pre-perturbation phase.

Furthermore, Omer et al. (2012) used the travel time as a critical metric for transportation system performance. Similar to Twumasi-Boakye's algorithm, the dynamic trends of resilience over time are represented in equation (6), where t_{ij (pre-perturbation)} represents the travel time during the pre-perturbation phase between node i and j, while t_{ij (post-perturbation)} indicates the travel time during the pre-perturbation phase between node i and j. Also, in the rail transit system, days of lost service (LSDs) are used to measure its resilience (Chan and Schofer, 2016). LSD can be generalized to study the resilience of public transportation systems facing severe perturbations such as extreme weather events (Zhou et al., 2019). Equation (7) shows how LSD is calculated, where RVM_P represents revenue vehicle miles per day at disrupted state, RVM_o is that at the original stable state, t₀ is the time point when perturbation first occurs and t is the time when the system retrieves its normal service level.

In addition to the metrics that measure time costs above, Vugrin et al. (2014) tried to specify the amount of resource expenditures required for recovery using two metrics. In equation (8), the recovery effort in total (E) is calculated as the sum of resource expenditures on recovery task j on the route i in mode k at the time point t. b_ijkt is a binary variable judging if this certain recovery task is successfully completed. Then, the impact of perturbation on system performance (SI) is represented by the total cost because of insufficient flow capacity in equation (9), where C_i is the cost on route i and Ci0 is the cost at the pre-perturbation phase. α_OD is a coefficient of extra penalty costs; D_OD is the portion of travel demand exceeding current capacity at the OD pair l-m. Finally, the resilience could be represented as the sum of E and SI in equation (10), where β is a weighting coefficient.

Compared to the attributed-based metrics, performance-based metrics are more reasonable for describing the transportation resilience because they have fully covered the performance of the transportation system since the perturbation first occurred. Some of the metrics also cover time cost and resource expenditure and the ability to sustain during the disruptive event, which all make them more robust and comprehensive.

3.1.3 Topology-based metrics.

Topology-based metrics focus more on the transportation network structure than its internal dynamic nature. They are built up with the basic graph-based properties such as nodes and traffic flow links of the transportation network. Thus, they are also called “centrality metrics.” Schintler et al. (2007) and Berche et al. (2009) proposed two widely used topology-based network parameters: average shortest paths and size of giant components. Both reflect the capacity that the transportation network keeps connected during the perturbation. According to Schintler et al. (2007), average shortest paths assess the strength of transportation network connection. The size of giant components reflects the percentage of links and nodes that remain functional during the perturbation. Besides, Aydin et al. (2018) introduced the concept of network betweenness centrality and efficiency. Osei-Asamoah and Lownes proposed network efficiency further as a metric in equation (11), where L_ij is the length of shortest path of node i−j.

In addition, Ip and Wang (2011) introduced a metric using the weighted number of reliable passageways to calculate the reliability of a certain node and the network resilience. In equation (12), ω_i and W_j are, respectively, the population weight and self-exhausted weight of the node i−j and P_k (i, j) is calculated as the reliability of a passageway of the node i-j. Hartmann (2014) proposed the algorithm of network backup capacity (C_b) of node i−j in equation (13), where ΔB_ij is the increase of edge-betweenness after the removal of the edge of with largest edge-betweenness. Testa et al. (2015) enriched the theory on betweenness centrality and further calculated network redundancy in equation (14), where NIP is the number of independent paths between node i to condition of the complete graph, I (i, j) is the number of independent paths between node i-j and σi2 is the neighborhood of the vertices in the neighborhood of i. In conclusion, these metrics are defined in different ways and forms, but their essence concentrates mostly on comparing the topological details of the transportation system with the corresponding complete graph.

Sun et al. (2020) explored the cities' resilience in the global air transportation network to single airport perturbations. They used two types of passenger-focused topological metrics to quantify the impact on the resilience of cities' air transportation: The unaffected passengers and the reroutable passengers on aggregated (airport) links. A large fraction of unaffected passengers indicates that the airport perturbation has a small effect on the travelers, while a large fraction of reroutable passengers indicates that the airport disruption has a smaller effect on the travelers (Sun et al., 2020).

3.2.1 Performance optimization model.

The performance optimization model aims to promise the best results in the system's performance under multiple constraints. According to Liao et al. (2018), mathematical modeling and optimization methods in the performance optimization model are efficient and promising tools for measuring resilience. They could be either dynamic or static and can cover different modes. However, the suggested methods have limitations because the recovery measures' effectiveness is highly based on human judgment. The threats need to be identified using a yet-to-be-developed disaster database (Liao et al., 2018).

The dynamic traffic assignment (DTA) is a widespread and effective optimization model. Sommer et al. (2016) introduced only an optimization-based adaptive traffic signal system on vehicular modes. Its optimal objects are increasing the vehicular performance of the transportation system, controlling flow to provide a more stable traffic network and reducing congestions and vehicular stops. (Sommer et al., 2016). Geroliminis and Sun (2011) used a macroscopic fundamental diagram in the model to optimize the linking space-mean road traffic flow, density and speed existing in a large urban area. Their models both used the average vehicular flow on a network as a function of the number of vehicles with independent trip OD pairs and route choices inside a selected road system to evaluate its robustness and redundancy level.

For the multi-mode DTA, Nogal et al. (2016) introduced the dynamic system optimal equilibrium to a restricted DTA time-travel cost optimization model on all modes. An optimization model for multi-mode traffic is introduced by Ye and Ukkusuri (2015) for the recovery of the transportation system network calculating the most optimal sequence for recovery nodes, links and routes to optimize resilience in the maximum recovery rate and minimum recovery expenditures, while the effect of their model is highly depended on the post-perturbation resource priority management and systematic budget allocation. Feixiong et al. (2012) introduced the Supernetwork model to optimize travelers' choice of all modes. This model considers the real-world activity and trip chain and embedded travelers' preferences for transport modes and locations in the model. All the multi-mode models have their functions of interaction between modes to ensure optimization accuracy.

Though optimization models for multi-mode traffic are helpful in the recovery strategies at the post-perturbation phase, they also require a considerable number of computational resources, accurate mathematical formulation and strategic policy-supported goals to ensure their effectiveness (Serdar et al., 2022). The limitations of their model are the dependence on the network impedance value that is selected and quantified based on a series of questions in the user travel survey. This nature makes the model difficult to be extended to a large scale and significantly relies on the accuracy of the survey conducted (Nogal et al., 2016).

3.2.2 Traffic simulation model.

The traffic simulation model examines the system's performance under both hypothetical and actual scenarios with precedented parameters. It is beneficial for detecting the critical functional components, systematic vulnerabilities and any other unexpected weak points. However, Nkenyereye et al. (2019) pointed out that its substantial processing capacity required the magnitude of what can be assessed, making the method difficult to scale up to the magnitude of urban transportation.

In most research, simulation is suitable for assessing unique characteristics of different modes of the transportation network. It is explicitly implemented to assess the performance metrics instead of tracking the changes of topological and other changes of the transportation network.

For road system simulation, Murray-Tuite (2006a, 2006b) the traffic assignment-simulation methodology DYNASMART-P generated system user equilibrium traffic assignments for a test network are sound to assess adaptability, mobility, safety and rapidity. Duy et al. (2019) combined the hydrological data with operational data of local road networks into the traffic simulation model to assess the resilience in a coastal city's transportation system under the impact of a flood disaster. Ganin et al. (2019) conducted a simulation model evaluating the selected intelligent traffic signals in the road system under cyber-attacks, and the resilience of the system under different scenarios is simulated. They formulated the data input such as transportation network density and average delay into the model of traffic signals effectiveness. The simulation result showed how much loss that the locking of signals could cause to the resilience than the normal infrastructure failure. (Ganin et al., 2019). The simulation outcome of this mode reflects the real process accurately, and the resilience change of the road system is examined under precedented metrics.

For railway system simulation, Osei-Asamoah and Lownes (2014) used a traffic simulation model to assess the resilience of railway transit in the urban transportation system by comparing the model outputs under normal situations and that with perturbations. The input performance metrics selected are travel time delay and ridership by hours (passengers entering each station in each hour). They assumed that these passengers were all to take the shortest way to the destination. The outcome of this mode is a little subjective based on its assumptions; the comparison between scenarios has a certain level of deviation.

The method is suitable for predicting future scenarios and detecting inconspicuous resilience problems. But its main limitation is resource-intensive and difficult to scale to accommodate urban or larger magnitude networks, while a more precise calibration transportation performance is also required.

3.2.3 Topological model.

Corresponding to the topological metrics mentioned before, topological models focus on calculating the shortest paths and the size of giant components. Berche et al. (2009), Osei-Asamoah and Lownes (2014) and Ta et al. (2009) applied the topological model to calculate the shortest paths by the distribution of node degree. Dunn and Wilkinson (2016) and Yoo and Yeo (2016) calculated the size of giant components and network efficiency by detecting the proportion of nodes or links that is connected as a cluster under different perturbation scenarios. Unlike optimization models and simulation models, topological is simpler in form. It simplifies transportation systems into quantities of links (representing the direct routes between adjacent nodes) and nodes (representing intersections and OD points). It has wide applications in urban transportation, but it is best suited for water or air logistical transportation.

Calibrating the network's connectivity from nodes to links is the key for topological model to assess transportation resilience through topological metrics. For instance, Zhenwu et al. (2019) implemented the “percolation theory” to the network of Chinese cities through clusters of nodes to identify its vulnerable connections (Zhenwu et al., 2019). Cerqueti, Ferraro and Iovanella (2019) used weighted links in the model to investigate the failure propagation (Cerqueti et al., 2019). Testa et al. (2015) applied the connectivity to assess the resilience of a topological transportation network during the random disconnections of nodes and links. Then, an integrated program (SLOSH) was applied to detect the damaged links and nodes in the network and measure the resulting resilience under a storm surge (Testa et al., 2015). Only when the connectivity calibrating is precise can the model be efficient in assessing the topological metrics of the network.

Though the topological model is relatively simple to develop and apply, it has certain limitations compared to optimization and simulation models. It cannot account for any accidents in the links, especially for accidents that could severely affect the network's capacity. It also ignores the different modes of traffic as well as their available alternatives in the nodes and links.

In addition, Zanin et al. (2018) urged that there are four major common pitfalls while applying the network theory to measure the topology of transportation systems. 1) The scale-freeness, which is the most important topological properties, requires both large enough networks and the application of suitable statistical tests to avoid obtaining biased results. 2) Beyond that, another pitfall that stems from topological metrics is highly associated with the number of nodes and links in the selected network. Thus, comparing different networks using same set of such metrics will lead to unreliable results. 3) Inappropriate choices of node sequence will result in unfounded conclusions comparing to the actual condition of real-world networks. Thus, there is a trade-off between the assessment of high-quality attack sequence and run time of the model, because the quality and run time both relate to the number of network nodes. The computation of variant betweenness needs to be appropriate in the large network. 4) Comparisons between random and targeted attacks have to be performed with care when specific metrics do not represent the node importance very well for the network (Zanin et al., 2018).

3.2.4 Data-driven model.

In recent years, because of data collection and storage development, data-driven methods have become a more efficient way to assess transportation resilience. Unlike the three mentioned methodologies, the data-driven method focuses on the direct data collected to calculate and assess the change of the system's properties under different scenarios. However, it ignores the operation mechanism of the transportation system during the entire process. The essence of the data-driven model is choosing proper and representative data such as ridership, service frequency and vehicle miles traveled of different modes. Statistical methods are sometimes used to pre-process the data before being used as performance indicators (Zhou et al., 2019).

One essential requirement for a data-driven model to assess transportation resilience is tracking the operational status and the performance of the system responding to perturbations. Donovan and Work (2017) built up their data-driven model to assess resilience by GPS data records from taxi sensors. The data they collected was travel distance and time of taxies. They used specific rules to remove some outliers and then normalized the trips by calculating the travel pace based on the travel distance and time changes throughout the week. Then the statistical Mahalanobis distance was applied to detect how the resilience is affected by special events (Donovan and Work, 2017). The tracking of system status is not only critical for assessing resilience but also one core requirement of cyber-physic transportation system (CPS).

Another critical requirement of the models to investigate the outcomes of perturbations is analyzing several types of data representing transportation systems. Diab and Shalaby (2019) applied the statistical analysis and filtering of metro line service records to assess resilience in terms of the “Lost Days of Service” because of the impact of perturbations (Diab and Shalaby, 2019). They collected time series data of metro series and retrieved data of metro networks and travelers' behavior. Roy et al. (2019) used data from the social media platform in their data-driven model to assess how resilience was affected by earthquakes and storms. The diversity of data allows the model to comprehensively represent the system trends, where data from performance and topological aspects are helpful.

The efficiency and accuracy of the data-driven model depend on the data source availability, data filtering, data processing and data analytics. Any misrepresenting network properties, such as topography, capacity and actual physical damage, might lead to entirely opposite outcomes, misleading future performance predictions.

4.1.1 The road traffic scenario.

According to the schematic of performance of a resilient system (Wan et al., 2018), road system resistance to disturbance mainly goes through two stages: pre-perturbance and post-perturbance. In the pre-perturbance stage, the disturbance resistance strategies aim to maintain the system running in the original state and to keep the system capacity and demand unchanged according to the plan.

1. Demand management

   Macro demand management plays a vital role in improving the system's resilience in advance, and it is one of the most effective ways. Specific measures include releasing timely information (Brudny and Krawiec, 2013), encouraging the choice of different modes of transportation (Liu et al., 2017) and increasing relevant incentive policies (Chen et al., 2020). This strategy can effectively prevent the impact caused by transportation facilities collapse, traffic congestion and other disturbances.

   Variable message sign (VMS) is a main platform to release guidance information. Yi et al. proposed a method for location optimization of urban VMS (Yi et al., 2019). They also gave an evaluation model for existing VMS and the recommended frontal distance and the height of characters. Bus bridging or taxi bridging has been widely used to connect stations affected by disruptions which help passengers resume their journeys. A two-stage model was developed to optimize the bridging plan and its allocation to buses, with the goal of balancing operational priorities between minimizing bus bridging time and reducing passenger delays (Gu et al., 2018). In addition, park and ride service is a part of comprehensive demand management. It has been proved that despite the complexity of finding parking spaces in the city center, park and ride services have made progress in alleviating traffic congestion (Memon et al., 2014).

2. Variable speed limits control

   Variable speed limits (VSLs) have a positive impact on improving highway traffic safety and efficiency, especially for dealing with severe weather, traffic congestion and other disturbance types. VSL control adopts advanced traffic flow and traffic environment detection technology and automatically adjusts the current speed limit value based on the preset control strategy. It reflects the changing road traffic environment characteristics in real-time and releases the information to road users in real-time.

   Based on the traffic flow theory, some scholars have studied the potential impact of VSLs on the traffic operation efficiency of the expressway, including using this technology to eliminate the shock wave (Popov et al., 2008) in case of congestion in the bottleneck section, so as to maintain the traffic operation in the working area in a more uniform and stable state (Kwon et al., 2007) and reduce the delay of isolated confluence area and confluence area (Carlson et al., 2010).

   The academic circles have not reached a unified understanding of the impact of VSL control on traffic capacity. The study of Dutch highway data showed that VSLs could not improve the capacity. In some recent studies, Karimi et al. (2004) believed that the VSLs control is only to replace the part of the flow occupancy curve lower than the key occupancy rate with a straight line segment, and its slope represents the speed after the implementation of VSL. Its theory is that there is no improvement in traffic capacity. Papageorgiou (Carlson et al., 2010) observed the traffic flow of a two-way six-lane road for 27 days, obtained the traffic flow characteristics under different speed limits and drew two conclusions: First, when the upstream occupancy rate of the mainline is lower than the key occupancy rate, the implementation of VSL control will reduce the average speed and increase the density, which can reduce the traffic flow to the bottleneck area, so as to delay or even eliminate the decline of traffic capacity in the downstream bottleneck area. Second, too low speed limit will reduce the capacity, and too high-speed limit will not affect the capacity.

3. Infrastructure monitoring and maintenance

   After long-term use, transportation infrastructure is constantly worn, resulting in deficient performance and diverse types of distress or damages (Hadjidemetriou et al., 2020). Infrastructure maintenance management has become an effective means to alleviate transportation facilities collapse and other related disturbances. A World Bank study revealed that every 10% increase in road roughness led to a 5–20% reduction in road life (Janoff et al., 1985; Smith et al., 1997) and an additional 0.5–4.1% increase in fuel consumption (McLean and Foley, 1998). Therefore, early comprehensive performance inspection of road, bridges, culverts and tunnels together with the disposal of specific damages can bring sizable economic benefits as well as ensure facility operation quality and serviceability.

   Maintenance units regularly patrol transportation infrastructure in jurisdictions to determine performance in time and consider disposal plans. However, on the one hand, traditional labor-based inspections are unsuitable for application in a wide range of transportation network considering their low efficiency and operational complexity. The timeliness of the inspection data is unable to meet the coverage rate. On the other hand, the increase in total facilities puts forward higher requirements for maintenance (Hormozabad et al., 2021). It costs more efforts for the maintenance management department to effectively track and predict the decay trend of conditions. Although the compound growth rate of infrastructure maintenance investment has increased, it is difficult to ensure the rational distribution of huge maintenance funds.

4.1.2 The subway traffic scenario.

As a large-volume transportation method in the city, subway traffic has high autonomy and operational complexity, which leads to its vulnerability (Bešinović, 2020). Under various emergencies such as natural disasters, equipment failures and man-made damages, it is prone to cascade failures and requires more time to recover. Therefore, the enhancing strategies of subway traffic resilience have been widely studied. Considering the resilience characteristics of subway traffic, strategies for pre-perturbance focuses on improving network vulnerability to unexpected changes. The overall resilience is improved by enhancing local resistance. Therefore, strategies for pre-perturbance can be categorized as influencing factor analysis, vulnerable elements identification and reinforcement and network structure optimization.

1. Influencing factor analysis

   Identifying the key influencing factors for subway traffic to resist perturbation is of great significance to ensure the normal operation of the subway and can help managers to take corresponding protective measures. Previous studies have explored the influencing factors of resilience from the perspectives of comprehensive evaluation, scenario simulation analysis and control measures analysis.

   Influencing factor analysis research are concerned with scenario specific. Researchers delved into weather-related disruptions and disasters in subway traffic, such as fire (Rie and Ryu, 2020; Giachetti et al., 2017), earthquakes (Fan et al., 2021), hurricanes (Zhu, Yuan, et al., 2016; Zhu, Yuan, et al., 2017) and weather conditions (Chan and Schofer, 2016; Diab and Shalaby, 2020; Wang et al., 2020). Quan et al. (2011) used scenario simulation to focus on the impact of rainstorms with different return periods on the Shanghai Metro. Lyu (2018) conducted a flood risk assessment of Guangzhou's subway system, considering 12 factors such as rainfall, altitude, slope and subway line density. Aoki et al. (2016) analyzed the impact of different flood control equipment on the flood control capacity of subway stations. They concluded that the subway station entrance is the station most affected by heavy rain.

   Usually, these studies estimated the frequency and duration of service interruptions. Moreover, the temporal and spatial distribution of disruptions and recovery processes under various perturbances are uncovered to provide planners with useful system design information. The influencing factor analysis strategy is based on historical data or simulation methods, which can accurately describe the resilience changes of system interruptions and provide a reference for future resilience regulation. However, there are problems of incomplete consideration factors and unrealistic model estimation. The reliability is supposed to be improved through more comprehensive data.

2. Vulnerable elements identification and reinforcement

   Vulnerable elements identification aims to find out the critical elements in the subway traffic system, which could lead to severe impacts when being failed. It usually requires evaluating the functional loss of the system based on assuming the failure of elements. The greater the loss, the higher the importance of the evaluated element. The corresponding reinforcement measures are applied to those vulnerable elements.

   Vulnerable elements identification is conducted through topology analysis. Deng et al. (2015) proposed a new framework based on network theory and The Failure Mode, Effects and Criticality Analysis method to analyze network efficiency by network theory and risk matrix in The Failure Mode, Effects and Criticality Analysis method. King et al. (2020) used a combination of quantitative methods founded in Graph Theory to quantify the resilience of the transit network and identified the critical stations in Toronto's subway network. Jing et al. (2020) developed a new method to determine the critical stations in metro systems based on route redundancy. The results indicated that the critical stations are not necessarily transferred stations or large node degree stations. Regarding the vulnerability of network components, Bababeik et al. (2018) proposed a bi-objective location and allocation model for relief trains in the rail network, determining the optimal location and allocation of relief trains to enhance the resilience level of the rail network.

   Vulnerability analysis strategies identify critical subway stations by modeling the service network. It could assist in a cost-effective resource allocation for strategically protecting the subway network and informed decision-making for emergency evacuation planning. Note that because of the heterogeneity of the network and different importance evaluation criteria, analysis models should be robust against the failure and removal of randomly chosen network elements.

3. Network structure optimization

   Optimizing the topology of the subway traffic network is one of the effective ways to improve network resilience and mitigate network damage. Based on different perspectives, researchers evaluated the network structure from the aspects of network connectivity, service level, operational reliability and so on, which aimed to enhance the resilience through structure optimization.

   Measure coefficients are taken as objective functions, and the construction cost is the constraint function to establish the network structure optimization model. When evaluating the reliability of network connectivity, most researchers choose the relative size of the largest connected subgraph (Wang, 2008), the connectivity probability of a node pair (Liu et al., 2020) or the network efficiency (Zeng et al., 2021) as the measurement parameters. The network service level can be measured by the increase rate of passenger travel cost (Hua and Ong, 2017) and network transmission capacity saturation (Liu et al., 2020). Operational reliability usually refers to the reliability of network travel time, which can be measured by the average shortest travel time of the network and the rate of change of network nodes to the shortest travel time (Shi et al., 2019).

   Network topology optimization is to increase the number of backup nodes or lines to improve connectivity or increase the capacity of key nodes or lines to adapt to the distribution of passenger traffic. However, considering that the geographical location of subway network nodes often depends on passenger flow demand, this method's application is relatively limited.

4.2.1 The road traffic scenario.

In the post-perturbance, after system performance of road traffic declines, recovery strategies involve rebuilding system accessibility and restoring its functionality as quickly as possible.

1. Variable lane management

   Variable lane control has become an effective means to alleviate traffic congestion, road construction, epidemic and other disturbances in various countries. The variable lane control scheme is applied to tidal traffic flow on urban roads. In this scenario, the traffic flow in one direction is large or even congested. In contrast, the traffic flow in the other direction is small, resulting in idle road resources, waste of road resources and low traffic capacity. The network-based variable lane control method is analyzed theoretically in the existing research on variable lane (Pan et al., 2022). By establishing a bilevel programming model (Meng and Khoo, 2008) based on the optimal system performance, the corresponding solution method is designed to calculate the attributes of variable lane control nodes (Wang et al., 2013).

   The main objects of variable lane management include road channelization, information control scheme and social vehicles. However, most of them only consider the variable lane control for a certain section to adjust the traffic flow of some tidal sections and do not improve the system capacity from the overall point of view. At the same time, this kind of technology is relatively slow to solve. For the variable lane adjustment under accident conditions, the focus is on the traffic evacuation speed after the accident, so the application scenarios are limited.

2. Ramp control

   From the perspective of system control, ramp control can be divided into three types: timing control, single point dynamic control and dynamic coordination control. They have a positive impact on dealing with severe weather, traffic congestion, epidemic and other disturbance types.

   Timing control is the simplest ramp control method, which is a static control. The regulation rate of timed ramp control is one or more fixed values, which is determined based on analyzing the historical traffic information of the road section. Timed ramp control has the advantages of a simple algorithm, easy operation and convenient implementation, but it cannot correct the ramp regulation rate according to the real-time road traffic conditions.

   To make up for the defects of the timing control algorithm, many scholars have studied single-point dynamic control. In this dynamic control, the detector should be used to detect the traffic information on the road in real-time, and on this basis, the adjustment rate of the matching road should be adjusted in real-time for dynamic control. Among them, ALINEA control is the most widely used method. This control method was proposed by Greek scholars (Frejo and De Schutter, 2019). In this method, a critical value of mainline occupancy is given in advance. If the detected occupancy is different from the given value in each detection cycle, then the original adjustment rate is adjusted.

   Dynamic cooperative control is to regard several corresponding lanes on a fast road as a whole and determine the optimal regulation rate of each ramp through the road traffic information collected in real-time. The purpose of dynamic cooperative control is not to optimize the traffic operation of a ramp but to maximize the overall traffic benefit. Responding to traffic events or normal traffic phenomena can avoid or quickly eliminate frequent and occasional congestion. The main algorithms are ZONE (Geroliminis et al., 2011), BOTTLENEC and HERO (Papamichail et al., 2010).

   Ramp control is equivalent to the regulator of traffic demand of expressway and ground roads, mainline and branch line of the expressway. It has a significant effect when the demand pressure is slight, but when the traffic pressure increases, this method will cause a vicious circle of queue extension. Therefore, a more global consideration method is needed for overall control.

4.2.2 The subway traffic scenario.

The independence and openness of the subway traffic make it highly dependable in operation. At the same time, it is vulnerable to disturbance and damage from various emergencies such as natural disasters, equipment failures and artificial damages. Compared with other modes of transportation, the subway traffic takes longer to recover (Chen et al., 2007). To prevent disturbances within the system from being further spread to road traffic, the immediate priority is passenger evacuation. While maintaining the service capacity of urban traffic leaves necessary space and time for the recovery of subway traffic. Given the above, strategies for post-perturbance can be bus-bridging and optimum recovery sequence decisions.

1. Bus-bridging

   Bus-bridging is an effective strategy in the contingency plan for subway emergencies and provides substantial support for passenger evacuation with its characteristics of rapidity, flexibility and accessibility. It can offer backup and continuation functions for the subway traffic during the reaction and recovery phase after an emergency.

   The European Commission (2004) regarded the coordination of services, network layout and emergency support between rail transit and public transport as the technical basis for implementing combined public transport. In the research report of TRB (2007), it was proposed that in the response and evacuation stages and the repair and service restoration stages after rail transit emergencies, the linkage support role of ground public transport should be imported to empower. Researchers studied the design of bus-bridging routes (Kuah and Perl, 1988; Martins and Pato, 1998; Deng et al., 2018), the scheduling model (Kepaptsoglou and Karlaftis, 2009; Jin et al., 2016; Itani and Shalaby, 2021), position of interchange stations (Tang et al., 2021; Derrible and Kennedy, 2010a, 2010b), emergency bus capacity and site selection (Teng and Xu., 2010; Gu et al., 2018) to introduce resilience enhancement in subway traffic.

   Current studies emphasized the research on the emergency connection of public transport in the case of subway operation failure, and the optimization objectives and constraints considered are different. Most of them focused on the emergency rescue of trapped passengers after the subway traffic operation was interrupted. The collaboration of multi-mode transportation might be a future research hotspot.

2. Optimum recovery sequence decisions

   The damage of different components in the subway network has distinct effects on the network. Moreover, the recovery of damaged components often requires different amounts, types of recovery resources and costs a certain amount of repair time and economic costs. Therefore, when multiple components in the network are damaged, the repair sequence of different components plays a crucial role in improving network resilience.

   Based on the optimal recovery objectives, the recovery sequence decision is to determine the optimal repair order of network failure units. Considering limited resources and costs, reasonable allocation and scheduling to optimize recovery resilience have received increasing concern. Zhang et al. (2018) determined the optimal recovery sequence and recovery time combined with the importance of nodes in the network. He also considered the performance cost after the subway interruption, including the loss of operating income and maintenance measures. Lu (2018) quantified the varying resilience of rail networks with time under different incidents, and the results showed that critical stations were identified differently depending on the duration time of different incidents and the characteristics of the failed stations. Jie-fei et al. (2020) and Qingchang et al. (2021) proposed resilience indexes to evaluate the performance of subway network resilience under different restoration schemes and recommend the scheme with the largest resilience index as the optimal scheme.

   Most studies usually build models based on the simplification of subway traffic networks and determine the optimal recovery sequence with the goal of minimum performance loss, shortest repair time and minimum recovery cost. But some practical factors, such as the physical length of tunnels between stations, the time and route of changing subway lines and subway lines with different volumes, are not included in the model.

3. Network self-organization adjustment

   The subway network consists of a large number of interacting nodes and links. Any adverse event that disrupts network component interactions and connectivity can significantly impact safety and efficiency. Therefore, it is important to maintain the connectivity of the line after the perturbation and reduce the impact of interruption caused by the interference without significantly reducing the network service capacity.

   Network self-organization is an effective method to increase regional resilience. For vulnerable nodes like transfer stations, adding an interloop could create redundancy to these vulnerable segments and improve network resilience by increasing network efficiency. Saadat et al. (2020) examined three alternatives to implanting loops in a network, each inserted based on subjective considerations such as location criticality, passenger flow and location connectivity. The results showed that network vulnerabilities can be significantly reduced by adding loops. Inserting a loop can reduce vulnerability by 24.6% on certain road segments. Derrible and Kennedy’s(2010a, 2010b) research showed that the number of looping paths available in a network significantly affected network resilience, which represented whether alternative routes could be used in an outage.

   As a post-perturbation recovery strategy, network self-organization adjustment can restore connectedness and reduce the total cost associated with a disruptive event. It would play a better control effect by combining bus-bridging and multi-mode transportation strategies.