Identifying Critical Nodes in Multimodal Transport Networks Based on Resilience Theory

Abstract

Identifying critical nodes in a multimodal transport network is fundamental to enhancing the resilience of transport systems against natural or man-made disasters. Although research on identifying critical nodes in complex networks has been deeply accumulated, its practical application in transport systems is still limited and requires further exploration. To bridge this research gap, an improved integrated method for identifying critical nodes in multimodal transport networks is proposed based on resilience theory. This method comprehensively incorporates disaster consequences and recovery time into resilience assessment, and an Improved Binary Particle Swarm Optimization (IBPSO) algorithm is developed to solve this problem efficiently. Then, a real case in China is conducted to verify the effectiveness of the proposed method. The results show that the proposed method can rapidly identify critical nodes in multimodal transport networks from the resilience perspective. These critical nodes, which are mainly concentrated at choke points with long repair durations, are often overlooked in traditional methods. Compared with the five baseline methods, the proposed method improves identification efficiency by more than 20%, verifying its better performance. The research findings can provide a scientific foundation for decisions on the effective protection and rapid recovery of critical nodes in the network.

1. Introduction

Multimodal transport networks are essential lifeline engineering that provide transportation services between cities [1]. According to the official data provided by the National Bureau of Statistics of China [2], the total passenger traffic volume has already exceeded 15.7 billion trips in 2023, which consists of 3.85 billion by rail, 11.01 billion by road, 257.71 million by waterway, and 619.58 million by air, respectively. There is no doubt that the multimodal transport network plays an important role in economic and social development in China, especially in terms of roads and railways. However, due to the complexity and interdependence of multimodal transport networks, this system presents considerable vulnerability.

As reported by the Mineta Transportation Institute [3], terrorist attacks on multimodal transport networks continue to occur every year. From 2004 to 2021, a total of 3836 such attacks have been recorded worldwide. Within these recorded events, critical nodes have a significant impact on transport efficiency and the reliability of the network. The failure of critical nodes in any one mode of multimodal transport networks will affect the other modes [4]. A disruptive event at a critical transportation hub will lead to widespread failure [5]. Therefore, the reliable operation of multimodal transport networks confronts serious challenges due to the uncertainty and suddenness of emergencies. It is urgent and necessary to identify critical nodes in multimodal transport networks to carry out effective protection.

Multimodal transport networks refer to the connection of two or more modes of transport networks to form a comprehensive transport system. It is essential to identify critical nodes in such networks to maintain their stability during emergencies. In previous research, it is common to identify critical nodes of a single transportation mode, such as a road network [6] and a railway network [7]. However, owing to the interdependence of multimodal transport networks, the critical nodes identified in a single transport network may not meet the demands of practical applications. Thus, it is worthwhile to explore a new critical node identification method for multimodal transport networks. Furthermore, identifying critical nodes in multimodal transport networks is a complex problem, but modeling with complex network theory can simplify this process [8,9,10]. Currently, research on critical node identification for multimodal transport networks has achieved considerable progress [1,4,11]. Typical methods for evaluating critical node importance can be divided into two main categories: the first category relies on network topology features, using centrality metrics to evaluate the structural importance of nodes. These metrics include degree centrality (DC) [12,13], betweenness centrality (BC) [14], closeness centrality (CC) [15] and structural holes (SH) [4], etc. The core idea is that centrality is positively correlated with node importance. However, such methods primarily focus on static topological characteristics, while functional characteristics remain insufficiently explored. Therefore, it is hard to accurately identify critical nodes of multimodal transport networks in the complex real world [4]. Integrating functional characteristics of multimodal transport networks is expected to narrow the gap between assessment results and practical operational demands. The second category measures the decline of network performance after a node is removed to indirectly evaluate node importance. Network operational metrics are used as the Performance Response Function (PRF) to measure network performance. Common metrics include network efficiency [16,17], connectivity [18] and time [19], etc. Previous research on critical node identification in complex networks has mainly concentrated on mitigating the cascading consequences of disruptive events [20], while the recovery process has received limited attention.

In recent years, the concept of resilience has gained increasing attention among scholars in the field of disaster prevention and mitigation [21,22,23,24,25]. Resilience characterizes a system’s capacity to absorb external disruptions and rapidly recover from operational interruptions [26]. Key factors include disaster consequences and recovery time. Extensive research has been devoted to resilience assessment in infrastructure. Bruneau [27] quantified resilience as the area between the degraded performance curve of an infrastructure system and its pre-disruption performance curve. As the research further develops, Reed et al. [28] refined the basic model and proposed a new metric, defining resilience as the area under the system’s performance curve. Subsequent studies extended Bruneau’s framework to address multi-stage, multi-pattern, and multi-scale performance variations in real-world transport networks, leading to more refined representations such as the “resilience trapezoid” [29]. Although these studies do not directly address critical node identification in multimodal transport networks, they provide a solid theoretical foundation for the present research.

Moreover, current critical node identification methods mainly focus on the global importance of nodes, without considering specific travel demands or predefined origin-destination (OD) pair requirements. In contrast, practical engineering scenarios such as multimodal transport networks often require targeted protection of network connectivity for specific OD pairs. Consequently, there is an urgent need for a critical node identification method suitable for these specific scenarios.

To bridge this research gap, this study proposes a formulation suitable for scenarios with predefined OD pairs, which can provide targeted decision support for network resilience optimization and emergency management. A PRF metric is designed to quantify the transport efficiency of the network. Then, an improved integrated method for identifying critical nodes in multimodal transport networks is proposed based on resilience theory, which comprehensively incorporates disaster consequences and recovery time into resilience assessment. To solve this problem efficiently, an IBPSO algorithm is developed. Subsequently, the effectiveness and advantages of the proposed method are verified with an example of a real transportation system in China. The critical nodes in the multimodal transport network are identified and compared with the five baseline methods. Finally, a systematic evaluation is conducted to quantitatively analyze the resilience recovery curves under different attack strategies. The research findings can provide a scientific foundation for decisions on the effective protection and rapid recovery of critical nodes to enhance the resilience of multimodal transport networks under extreme conditions.

The rest of this paper is structured as follows: In Section 2, the methodology for identifying critical nodes in multimodal transport networks is explained in detail, including the network topology model, resilience assessment model, critical node identification model, IBPSO algorithm, and simulation process. In Section 3, the proposed method is applied to a real multimodal transport network in a certain region of China, and the results are compared with five baseline methods. In Section 4, the scope of our formulation and its limitations are discussed. Finally, the conclusions and future research directions are presented in the last section.

2. Methodology

2.1. Network Modeling

Multimodal transport networks contain multiple transportation modes that serve to transfer cargo from one location to another. These transport networks can be modeled as abstract networks. Assume that each transportation mode is a sub-transportation network. The road, railway, aviation, and waterway transportation modes are used as an example to demonstrate the construction process of a multimodal transport network model, which can be expressed as $\mathcal{M}=\{R,T,A,W\}$. For each mode of transportation, it can be abstracted as graphs G_κ = (V_κ, E_κ, W_κ) and $\kappa \in \mathcal{M}$, where V_κ denotes the node set of transportation mode κ. These nodes can be cities, transportation hubs, critical bridges, airports and ports. E_κ denotes the edge set of transportation mode κ, and W_κ denotes the weight of the edges. The edges denote the direct connections between nodes and are weighted by travel time, which is defined as the actual distance divided by the operating speed of each mode κ.

Among different transport networks, an inter-layer connection layer is constructed. Where $E_{uw}^{yz}$ denotes the inter-layer edge set, which can be expressed as:

$$E_{uw}^{yz}={\stackrel{\wedge }{e}}_{uw}^{y\to z}=(u^y,w^z),y,z\in \mathcal{M}=\{R,T,A,W\}$$

(1)

where y and z represent two different transport modes belonging to the set $\mathcal{M}$. u^y and w^z denote nodes u in layer y and w in layer z, respectively.

For illustrative purposes, four individual networks are mapped onto a single plane based on complex networks to form multimodal transport networks, as shown in Figure 1. Different colors represent distinct transport network layers, while dashed lines in different colors mark the positions of critical nodes from different transport modes in the multimodal transport network.

2.2. Model for Resilience Assessment

The quantitative resilience assessment methods can be classified into two categories: the PRF method [22,23,24,25,26,27,28,29,30] and the index system method [31]. Among them, the PRF method has become the primary approach in research on quantifying the resilience of multimodal transport networks. It can capture the dynamic performance of the network over time during disruptions and visually present the functional recovery processes. In this study, the PRF method is used to dynamically assess the resilience of multimodal transport networks, and it quantifies the recovery capacity after external disruptions using the area under the PRF curve per unit time interval. As shown in Figure 2a, the typical post-disaster PRF curve of a multimodal transport network fully reflects the multi-phase response characteristics of the network under a disruptive event. Once the critical nodes fail, the network performance suffers a sharp decline in functionality, and then gradually recovers to the steady-state level via adaptive adjustments. The recovery of network performance has significant time-varying characteristics. Therefore, the index system method is difficult to accurately depict its dynamic characteristics. It is necessary to establish a time-varying resilience assessment method based on the PRF of the network.

Multimodal transport networks usually use transport efficiency as a key metric. This metric serves as an indicator of performance response. Assume that several critical nodes simultaneously suffer disruptive shocks at time t₀. The consequences of the disaster directly lead to interruptions in the multimodal transport network. The transport efficiency drops to its minimum level. Once repair resources are provided, the repair team carries out synchronized work on the damaged nodes in the initial phase. This strategy greatly accelerates the initial recovery process of the network. Assume that the multimodal transport network contains N critical OD pairs, and the transport efficiency of the n-th OD pair at the k-th recovery stage is denoted as $f_{o{d}_n}^k$. Considering that the shortest path post-disaster remains unchanged until the damaged node is restored, the transport efficiency during the interval t₀~t₁ remains constant and is denoted as $f_{o{d}_n}^1$. After time t₁, the damaged node with the minimum repair duration is restored first, and a new shortest path is generated. The transport efficiency rapidly improves to $f_{o{d}_n}^2$. Subsequently, the next stage of repair starts. The restoration of subsequent nodes by emergency repairs enhances the transport efficiency to $f_{o{d}_n}^3$. Similarly, as the recovery time increases, the damaged nodes resume normal operation in an orderly manner. The network performance finally returns to the pre-disaster normal level $f_{o{d}_n}^{init}$ at the moment t_k. The transport efficiency during the periods $t_0~t_1,t_1~t_2,\cdot \cdot \cdot ,t_{k-1}~t_k$ is $f_{o{d}_n}^1,f_{o{d}_n}^2,···,f_{o{d}_n}^k$, respectively, where $f_{o{d}_n}^1\le f_{o{d}_n}^2···\le f_{o{d}_n}^k$. Considering that t_k may vary under different attack strategies, it is difficult to conduct a resilience assessment uniformly at the same time. Therefore, when conducting resilience calculations, the time period from t₀ to t_max is taken into account, where t_max denotes the maximum repair duration of critical nodes in the multimodal transport network under a specific disruptive scenario.

Based on resilience theory, the resilience level R of a network can be quantitatively characterized using the area enclosed by the PRF curve and the time axis t within the post-disaster recovery time (t₀~t_max). The area of this region reflects the combined effect of disaster consequences and recovery time, and it can comprehensively characterize the resilience of the network. The resilience value R of the multimodal transport network is defined as the sum of the resilience values of all OD pairs.

$$R={\displaystyle \sum _{n=1}^N{R}_n}$$

(2)

where R_n denotes the resilience values of the n-th OD pair.

Assume that the multimodal transport network undergoes a recovery process consisting of K sequential stages. As shown in Figure 2a, for the n-th OD pair, R_n is defined as the sum of the areas enclosed by the PRF curve and the time axis t under every recovery stage, which can be calculated by Equation (3).

$$\begin{array}{ll}{R}_n& =S_{R1}+S_{R2}+\cdot \cdot \cdot +S_{RK}+S_{R\max }={\displaystyle \sum _{k=1}^K{S}_{Rk}}+S_{R\max }\\ & =f_{o{d}_n}^1(t_1-t_0)+f_{o{d}_n}^2(t_2-t_1)+\cdot \cdot \cdot +f_{o{d}_n}^K(t_K-t_{K-1})+f_{o{d}_n}^{init}(t_{\max }-t_K)\\ & ={\displaystyle \sum _{k=1}^K[f_{o{d}_n}^k}\cdot (t_k-t_{k-1})]+f_{o{d}_n}^{init}(t_{\max }-t_K)\end{array}$$

(3)

where S_Rk denotes the area enclosed by the transport efficiency curve and the time axis during the k-th recovery stage. S_Rmax denotes the area enclosed by the transport efficiency curve during stable operation and the time axis during the periods $t_k~t_{\max }$.

Accordingly, the attack benefit S of the multimodal transport network is defined as the sum of the shaded area of all OD pairs, which can be expressed as:

$$S={\displaystyle \sum _{n=1}^N{S}_n}$$

(4)

where S_n denotes the attack benefits of the n-th OD pair.

As shown in Figure 2a, for the n-th OD pair, S_n is defined as the sum of the areas between the post-disaster PRF curve and the initial normal efficiency level $f_{o{d}_n}^{init}$ across all recovery stages, which can be calculated by Equation (5).

$$\begin{array}{l}{S}_n=S_{S1}+S_{S2}+\cdot \cdot \cdot +S_{SK}={\displaystyle \sum _{k=1}^K{S}_{Sk}}\\ =(1-f_{o{d}_n}^1/f_{o{d}_n}^{init})\cdot (t_1-t_0)+(1-f_{o{d}_n}^2/f_{o{d}_n}^{init})\cdot (t_2-t_1)+\cdot \cdot \cdot +(1-f_{o{d}_n}^K/f_{o{d}_n}^{init})\cdot (t_K-t_{K-1})\\ ={\displaystyle \sum _{k=1}^K(1-f_{o{d}_n}^k/f_{o{d}_n}^{init})}\cdot (t_k-t_{k-1})\end{array}$$

(5)

where S_Sk denotes the area enclosed by the post-disaster PRF curve and the initial normal efficiency level $f_{o{d}_n}^{init}$ during the k-th recovery stage. $f_{o{d}_n}^{init}$ is usually taken as a constant.

The initial transport efficiency $f_{o{d}_n}^{init}$ can be obtained via Equation (6) once the initial topological network is determined.

$$f_{o{d}_n}^{init}={\displaystyle \frac{1}{t_{o{d}_n}^{min}}}$$

(6)

where $t_{o{d}_n}^{min}$ denotes the minimum travel time of the shortest path from origin node o to destination node d for the n-th OD pair, which is obtained by Dijkstra’s algorithm.

Similarly, the transport efficiency $f_{o{d}_n}^k$ can be obtained by Equation (7).

$$f_{o{d}_n}^k={\displaystyle \frac{1}{t_{o{d}_n}^k}}$$

(7)

where $t_{o{d}_n}^k$ is the minimum travel time during the k-th recovery stage of the multimodal transport network from origin node o to destination node d for the n-th OD pair. Based on the Dijkstra algorithm, the minimum travel time under different recovery stages is calculated, and the transport efficiency $f_{o{d}_n}^k$ can be obtained using Equation (7). Meanwhile, the resilience values R_n and the attack benefits S_n can be evaluated using Equation (3) and Equation (5), respectively.

As shown in Figure 2b, the area enclosed by the initial normal efficiency level $f_{o{d}_n}^{init}$ and the time axis t over the interval $t_0~t_{\max }$ consists of two components: R_n and S_n. Therefore, the mathematical relationship between the resilience values R_n and the attack benefits S_n for the n-th OD pair is given by Equation (8).

$$f_{o{d}_n}^{init}(t_{\max }-t_0)=S_n+R_n$$

(8)

2.3. Model for Identifying Critical Nodes

The proposed method identifies critical nodes not only by considering whether node failures can disconnect the network (structural importance), but also by evaluating the cumulative performance loss during the dynamic recovery process, where nodes are restored sequentially according to different repair times (functional importance). The objective is to find the attack strategy that minimizes the resilience value R or equivalently maximizes the attack benefit S.

For a multimodal transport network graph G_κ = (V_κ, E_κ, W_κ) with a specific origin node o ∈ V_κ, destination node d ∈ V_κ, and attack node set A ⊆ V_κ, we aim to find the minimum attack node set A_min ⊆ A, such that, after removing all nodes in A_min and their incident edges, there are no connected paths between o and d.

$$A_{\min }=\arg {\min }_{A\in X}\left|A\right|$$

(9)

where X denotes the set of all attack node sets satisfying the above constraints, which is defined as:

$$X=\{A\subseteq V_{\kappa }\mid no path o\to d in G_{\kappa }^{\prime }\}$$

(10)

$G_{\kappa }^{\prime }=(V_{\kappa }^{\prime },E_{\kappa }^{\prime })$ represents the remaining graph after removing the attack node set A and its adjacent edges from the initial network G_κ, where V_κ’ and E_κ’ denote the sets of nodes and edges in V_κ and E_κ that do not include nodes in A. For example, as shown in Figure 3, if the set of all attack node sets is $X=\{(T_1,R_1),(T_3,R_3),(R_1,T_2),(R_3,T_2),(T_1,R_2,R_3),(R_1,R_2,T_3)\}$, it is easy to find $A_{\min }=\{(T_1,R_1),(T_3,R_3),(R_1,T_2),(R_3,T_2)\}$. After any subset of the nodes in A_min fails, there will be no connecting path between o and d, and the number of elements in A_min is the minimum.

The attacker’s optimal strategy is to select the attack plan $\mathcal{P}$ that minimizes network resilience among all plans. This plan must satisfy the constraint of the minimum node set A_min. Then, all nodes in plan $\mathcal{P}$ are the critical nodes of the multimodal transport network. The mathematical expression is as follows:

$$\mathcal{P}=\{\mathcal{P}\in A_{\min }|\underset{\mathcal{P}\in {\mathcal{A}}_{\min }}{\min }R(G_{\kappa })\}$$

(11)

where R(G_κ) represents the resilience value of the multimodal transport network G_κ.

2.4. Improved Binary Particle Swarm Optimization Algorithm

Particle Swarm Optimization (PSO) is a classical stochastic optimization technique based on a population [32]. Continuous particle swarm optimization assumes that there are m particles in the D-dimensional search space ${ℝ}^D$. The i-th particle of the swarm is parameterized by a spatial coordinate vector $\overrightarrow{x_i}=(x_{i1},x_{i2},\dots ,x_{iD})$ and a corresponding velocity vector $\overrightarrow{v_i}=(v_{i1},v_{i2},\dots ,v_{iD})$. Let $P_i^{\mathrm{best}}=(p_{i1},p_{i2},\dots ,p_{iD})$ denote the best visited position by the i-th particle, and $P^{\mathrm{gbest}}=(p_{g1},p_{g2},\dots ,p_{gD})$ represents the best position found by the entire swarm so far. Then the position and velocity of the particle can be updated according to the following equations:

$${\nu }_{ij}^{r+1}=\lambda {\nu }_{ij}^r+c_1{\eta }_1(p_i^{best}-x_{ij}^r)+c_2{\eta }_2(p^{gbest}-x_{ij}^r)$$

(12)

$$x_{ij}^{r+1}=x_{ij}^r+v_{ij}^{r+1}$$

(13)

where i = (1, 2, …, m), j = (1, 2, …, D), c₁ is the weight of attraction to the previous best location of the current particle. c₂ is the weight of attraction to the previous best location of the whole particle swarm. r is the current iteration number. η₁ and η₂ are independent uniform random numbers in [0, 1]. In this equation, λ represents the inertia weight, which indicates the influence of the particle’s previous velocity on its current velocity and can be obtained using the following equation.

$$\lambda ={\lambda }_{\max }-{\displaystyle \frac{r}{r_{\max }}}({\lambda }_{\max }-{\lambda }_{\min })$$

(14)

To solve the discrete decision-making problems in reality, Kennedy and Eberhart proposed a discrete binary version of PSO for binary problems. The main difference between the binary Particle Swarm Optimization (BPSO) algorithm and the continuous version is that the velocity of the particles is determined based on the probability that a bit will change to one. According to this definition, the velocity must be limited within the range of [0, 1]. Therefore, a sigmoid function can be used to map all real-valued numbers of velocity to the range [0, 1].

$$sig(v)={\displaystyle \frac{1}{1+e^{-v}}}$$

(15)

The new particle position can be obtained using the following equation.

$$x_{ij}^{r+1}=\left\{\begin{array}{ll}1& \mathrm{if} U(0,1)<sig\left(v_{ij}^{r+1}\right)\\ 0& \hfill \mathrm{otherwise}\hfill \end{array}\right.$$

(16)

The function U(0,1) is a uniform random number generator. Each time this function is called, it will generate a new value within the range [0, 1].

However, the original sigmoid function is unable to meet the accuracy requirements for identifying critical nodes in multimodal transport networks within a limited time. Therefore, a new function is introduced using the following equation.

$$sig(v)={\displaystyle \frac{1}{1+e^{-(v-\mu )}}}$$

(17)

where μ is the selection bias threshold, which is used to adjust the strictness of the algorithm in determining the importance of nodes.

The fitness function for evaluating each candidate attack plan $\mathcal{P}$ in the IBPSO algorithm is defined as:

$$fitness(\mathcal{P})=\left|\mathcal{P}\right|•\alpha +R(\mathcal{P})$$

(18)

where α denotes the penalty coefficient, and $R(\mathcal{P})$ represents the resilience of the network after the execution of the attack plan $\mathcal{P}$.

2.5. Simulation Process

To identify critical nodes in a multimodal transport network, a simulation process based on resilience theory is constructed as shown in Figure 4, with the following specific steps:

Step 1: Build a multimodal transport network. The road, railway, aviation, and waterway modes of transportation can be abstracted as graphs G_κ = (V_κ, E_κ, W_κ), $\kappa \in \mathcal{M}$. Each node is associated with a set of attributes, including its type, location, and repair duration. Repair duration can be estimated based on the engineering repair workload and the repair capacity of teams under specific disaster intensities. Each edge is weighted by the travel time.

Step 2: Calculate the initial transport efficiency $f_{o{d}_n}^{init}$ of the network. Based on the constructed multimodal transport network, the initial transport efficiency from the origin node o to the destination node d can be calculated to evaluate the baseline performance of the network.

Step 3: Calculate the number of nodes in the minimum cut set of the network. Based on the maximum flow minimum cut theorem, the minimum cut set of the network can be calculated to evaluate the minimum number of nodes, $N_{od}^{\min }$, which is required to cut off the multimodal transport network.

Step 4: Determine the parameters of the IBPSO model. Based on the number of nodes in the minimum cut set, the selection bias threshold μ of the algorithm can be determined. Each particle in the IBPSO algorithm represents an attack plan $\mathcal{P}$. The initial attack plan is generated based on particle initialization.

Step 5: Update the attack plan based on the situation of the particles. Each particle is updated to form an attack plan for the network. Based on the attack plan, the corresponding nodes will be removed from the topological network to disrupt its structural connectivity at the same time.

Step 6: Calculate the post-disaster transport efficiency $f_{o{d}_n}^k$ of the network. When the network is damaged, the repair work starts simultaneously. Even though repair durations vary due to different characteristics of the damaged nodes, the transport efficiency still shows an orderly recovery trend as the damaged nodes are gradually restored. Therefore, the post-disaster transport efficiency $f_{o{d}_n}^k$ of the damaged network is calculated to quantify its operational performance under disruption. If there are no connected paths between the origin node and the destination node in the remaining network, the travel time is defined as infinite, and $f_{o{d}_n}^k$ = 0. Meanwhile, the transport efficiency can be calculated across different time periods.

Step 7: Calculate the resilience of the network. The resilience of the network can be calculated using Equations (2) and (3).

Step 8: Evaluate the fitness function and update P_i^best, P^gbest. The fitness function of the network is calculated according to the attack plan and the recovery process. P_i^best and P^gbest are updated since the calculation begins.

Step 9: Output the optimal critical nodes of the multimodal transport network. In a multimodal transport network, the optimal attack plan can be calculated using the IBPSO algorithm, and the plan with the lowest fitness value is selected. All nodes in this plan are the critical nodes of the multimodal transport network.

3. Experiments and Results

To verify the effectiveness of the proposed method, a typical multimodal transport network was selected as an example. The research results were compared with the five baseline methods for identifying critical nodes in complex networks. Meanwhile, the network resilience was compared under the failure of critical nodes identified by the six methods.

3.1. Experimental Network

To illustrate the effectiveness of the proposed method, the multimodal transport network comprising roads and railways in a specific region of China was adopted as a case study. To ensure data security, the network was appropriately simplified without affecting the calculation results. The topological network was depicted in Figure 5. This multimodal transport network included 100 nodes, including 118 roads and 70 bridges. The railway bridges were labeled as B1–B12, the highway bridges as Q1–Q58, and the urban nodes as C2–C6. To reflect variations in repair durations among different bridges, a color gradient from light to dark was used to indicate repair times for critical nodes subjected to extreme damage under deliberate attacks. To comprehensively verify the effectiveness of the proposed method, critical node identification analysis was conducted under a single origin-destination (SOD) pair scenario (o₁–d₁) and multiple origin-destination (MOD)pair scenarios (o₁–d₁ and o₂–d₂), respectively.

3.2. Analysis of Critical Nodes

The analysis of critical nodes consists of two parts: a SOD pair scenario and a MOD pair scenario. Based on the proposed IBPSO algorithm, the critical nodes in the multimodal transport network are calculated. Key parameter settings of the two algorithms are presented in

Table 1. Key parameter settings of two algorithms in case study.

Parameters	λ	c1	c2	r	m	α	μ
IBPSO	0.6	1.8	1.8	800	80	100	5
BPSO	0.6	1.8	1.8	800	80	100	-

.

To verify the effectiveness of the proposed algorithm, the fitness function curves of the two algorithms are shown in Figure 6. The traditional BPSO algorithm demonstrates a slow convergence rate, making it challenging to identify critical nodes rapidly and accurately within a limited number of iterations. When the iteration number reaches 800, the fitness function value of the traditional BPSO algorithm is 2108.18, indicating that 19 nodes are still regarded as critical nodes. In contrast, the IBPSO algorithm achieves a fitness function value of 396.49, successfully identifying only three critical nodes, which verifies the effectiveness and advantages of the proposed improvement strategy. Furthermore, to illustrate that the newly introduced parameter significantly improves convergence performance and calculation stability in complex network scenarios, we further analyze the fitness convergence of the IBPSO algorithm under different values of the parameter μ to verify the rationality and necessity of the algorithmic improvement, as shown in Figure 7.

To assess the stability of the improved algorithm, box plots of the optimal fitness values obtained from 30 independent runs are presented in Figure 8. Compared with the standard BPSO algorithm, the IBPSO algorithm exhibits a narrower box range and shorter whiskers, indicating that the optimal fitness results of IBPSO have much smaller fluctuations during 30 independent runs. The IBPSO algorithm shows a significantly lower median fitness value. All the above results demonstrate that the IBPSO algorithm has outstanding operational stability and better optimization performance than the traditional BPSO algorithm. Furthermore, the computational complexity of the improved BPSO remains O (m × D × r_max), which is consistent with the standard BPSO. The introduction of the bias parameter in the sigmoid function only adds a little extra computation and does not increase the computational complexity level of the algorithm.

In the SOD pair scenario, the primary objective of critical node identification is to protect the critical nodes in the network to ensure the stable operation of a specific single transport channel. This case study utilizes the o₁–d₁ pair as an example. As shown in Figure 4, based on the simulation process, the PRF curves and resilience value R of the multimodal transport network under different attack scenarios are calculated. The results demonstrate that nodes Q14, B10, and Q28 are identified as the critical nodes in the multimodal transport network by the proposed method. All three bridges are situated at key strategic chokepoints, and their repair durations are relatively long. Obviously, the simultaneous failure of these three critical nodes would completely disrupt the transport route from the origin node o₁ to the destination node d₁. Therefore, targeted protection should be carried out for these critical nodes to strengthen the resilience of the network. Importantly, if only nodes Q14 and Q28 are damaged, the optimal route from o₁ to d₁ can still be established via the railway network, indicating that damage to the road bridges alone will not lead to a decline in the network transport efficiency.

In MOD pair scenarios, the primary objective of critical node identification is to protect the critical nodes in the network to ensure the stable operation of multiple specific transport channels. In this case study, two typical OD pairs, o₁–d₁ and o₂–d₂, are selected for the case analysis. However, consistent with the SOD pair scenario, nodes Q14, B10, and Q28 are identified as critical nodes of the multimodal transport network by the proposed algorithm. The simultaneous failure of the critical nodes would completely disrupt the transport route of the two typical OD pairs. Therefore, the proposed method can effectively identify critical nodes for both SOD pair and MOD pair scenarios.

3.3. Comparison of Critical Nodes with Other Methods

To verify the advantages of the critical node identification method based on resilience theory, a comparison is conducted between it and traditional measures including DC, BC, CC and SH. Moreover, this study also systematically compares the proposed resilience-based method (PRM) with the conventional resilience-based method (CRM) that adopts network efficiency as the evaluation indicator. Critical nodes are defined as those whose failure would lead to a significant decrease in network efficiency. The critical nodes identified by the five methods are summarized as follows.

3.3.1. Degree-Based Method

When identifying critical nodes in complex networks, one of the classic approaches is to identify critical nodes based on degree centrality [33]. Node degree offers a simple, efficient, and intuitive metric for critical node identification. It demonstrates significant advantages in topological networks, particularly in large-scale networks and rapid response scenarios. It provides a common standard for evaluating node importance across complex networks and directly reflects the number of connections of nodes within the network. The more connections a node has, the greater its local influence within the network. Therefore, in this case, the top 27 critical nodes sorted by node degree are shown in Figure 9. The maximum node degree in the network is 4 and these highly connected nodes primarily include C2, C3, C4, C5, C6, G11 and G15.

3.3.2. Betweenness-Based Method

Betweenness centrality evaluates the structural importance of a node by quantifying its intermediary control capacity in the global logistics transport process, and it reflects the number of shortest path connections of a node in the network [34]. Furthermore, it quantifies both the ability and the importance of a specific node acting as a “bridge” in the network.

Betweenness centrality serves as a global metric that offers a more comprehensive assessment of node importance within networks, particularly suitable for scenarios where the overall structure and connectivity of the network need to be considered. In contrast to node degree, which merely counts the number of direct connections a node possesses, betweenness centrality emphasizes a node’s position and role within the entire network.

A node with higher betweenness centrality has greater across-region significance within the network and can more effectively control coordination between different domains. Consequently, the top 30 critical nodes are identified, ranked by descending betweenness centrality values, as shown in Figure 10. Notably, nodes with higher betweenness include T5, T4, B8, B7, C4, C5, G11, T6 and C6. Once the critical nodes with high betweenness in the multimodal transport network are damaged, the connectivity of the network will be significantly reduced.

3.3.3. Conventional Resilience-Based Method

Conventional resilience-based methods assess node importance by evaluating the reduction in network efficiency following the removal of a node [16]. When a critical node in the multimodal transport network fails, the global connectivity of the system will be weakened, directly leading to a decrease in network efficiency. The greater the network efficiency loss caused by one node failure, the more critical this node is to the multimodal transport network.

As a typical standard network performance metric, network efficiency effectively reflects the overall connectivity of the multimodal transport network. This study identifies the top 30 critical nodes according to the descending order of network efficiency loss following node deletion, as shown in Figure 11. These critical nodes include C4, C6, T4, T3, C3, Q19, T2, Q47, T6 and C3, which are mainly located in the central region of the network. The failure of these critical nodes will diminish the overall network efficiency, but it fails to fully disconnect the paths of OD pairs. Furthermore, this resilience-based method does not take into account the impact of recovery time on network resilience.

3.3.4. Closeness-Based Method

Closeness centrality evaluates the importance of nodes by calculating the average shortest path distance from one node to all other nodes across the entire network. Nodes with higher closeness centrality hold shorter path distances to other network units, which means they can achieve faster freight transmission throughout the multimodal transport network.

As a representative global topological indicator, closeness centrality fully depicts how conveniently a node can reach all other locations across the entire multimodal transport network. This study identifies the top 30 critical nodes in descending order of closeness centrality values, as shown in Figure 12. These critical nodes mainly consist of T4, C4, B6, B7, B8, B5, T5, Q47 and T6, which are mainly concentrated in the central part of the network. The failure of these critical nodes will make the average travel distance between all network nodes rise markedly, but such node failures cannot fully disconnect the fixed paths between OD pairs. Meanwhile, this indicator only reflects static geometric advantages of node positions and fails to consider the dynamic operational resilience and post-failure recovery characteristics of actual multimodal transport networks.

3.3.5. Structural Holes-Based Method

Structural hole theory evaluates the importance of nodes by measuring gaps in the connection between separate node groups in the network. Nodes that occupy key structural holes act as the only bridge between isolated parts of the system. This means they occupy key positions in the connection between different regions across the entire multimodal transport network.

As a typical topological indicator, structural hole metrics reflect the unique controlling ability of nodes for connections among different network regions. Even if only node G16 holds prominent structural hole characteristics in the network, the structural hole algorithm can calculate the corresponding index value for all nodes. Each node is quantified separately based on its connection relationships with adjacent nodes. In this way, all nodes can be sorted in descending order, which helps distinguish nodes with different importance degrees. The results show that the critical nodes include G16, T4, C4, B6, B7, B8, B5, T5, Q47 and T6. Among them, G16 is located at the junction of two independent transport subnetworks.

3.3.6. Visualization of Critical Nodes

To better compare the critical nodes identified by different methods, Figure 13 presents the results of all six approaches using distinct color markers for intuitive comparison.

In this figure, cyan, green, magenta, blue, yellow, and red nodes denote critical nodes identified by CRM, SH, CC, BC, DC, and PRM, respectively. The six methods exhibit both consistency and differences in identification results: The critical nodes identified by the six methods are mainly located at the nodes along the railway lines and cities. Mixed critical nodes refer to locations that are classified as critical by more than one method. However, apart from these nodes, different identification methods have their own unique characteristics. The PRM focuses specifically on choke points with long repair durations. These nodes are fewer in number but directly determine the overall connectivity and recovery capacity of the network, making them the most critical for post-disaster resilience. For example, nodes marked in red, which are identified by the PRM, are all located on the main transport corridor serving both the SOD pair scenario and the MOD pair scenarios. When these nodes fail, the transport connection between the OD pair will completely break down.

3.4. Comparison of Deliberate Attacks on Critical Nodes

To compare the effectiveness of the six critical node identification methods, this study removes the top-ranked nodes identified by each method from the multimodal transport network, respectively. We quantify the decline in transport efficiency according to the increase in the number of failed nodes. This trend is shown in Figure 14.

As illustrated in Figure 14a, deliberate attacks on critical nodes identified by the PRM lead to the fastest decline in transport efficiency with only a small number of failed nodes, demonstrating the advantages of the PRM over other methods.

In the initial state, the optimal transport paths for the multimodal transport network are generated with the objective of minimizing total travel time based on its transport characteristics. When disasters damage critical nodes along the optimal paths of the network, the transport efficiency declines rapidly. This phenomenon occurs because the disruption of critical nodes forces operators to find new optimal paths within the remaining network. However, due to the inherent characteristics of the multimodal transport network, such rerouting certainly results in prolonged travel times and reduced transport efficiency. The decline in transport efficiency depends on the robustness of the infrastructure surrounding the failed critical node. When the adjacent network exhibits high robustness, alternative routes can be quickly identified once a node fails, effectively reducing the impact on transport efficiency of the multimodal transport network. Conversely, if the local robustness is insufficient, the newly generated optimal paths require extensive detours, resulting in significantly reduced transport efficiency.

Under the resilience-based attack strategy, in the SOD pair scenario, as the number of failed nodes increases from 0 to 3, network transport efficiency drops rapidly to 0, and no accessible connected paths remain in the multimodal transport network. The results show that only 3 critical nodes are needed to disrupt the network, far fewer than the 9 nodes required by the other methods. For the MOD pair scenarios, as shown in Figure 14b, three critical nodes are still sufficient to disconnect the o₂-d₂ pair. These nodes remain consistent with those obtained in the SOD pair scenario, but the decline in transport efficiency is more abrupt than that observed for the o₁-d₁ pair scenario. Under the objective of disabling the multimodal transport network with the minimum number of node failures, multiple plans can achieve network disruption. However, the PRM can solve the optimal attack plan, which maximizes the comprehensive impact on both recovery time and transport efficiency of the network.

Under the degree-based node attack strategy, as the number of failed nodes with high-degree increases from 0 to 8, network transport efficiency maintains a stable level without obvious decline. The reason is that these high-degree nodes are not located on the optimal OD paths in the current network. Therefore, the failure of such nodes with a high degree has no significant influence on network transport efficiency. However, once the ninth high-degree node fails, the transport efficiency drops rapidly to 0. This phenomenon results from the cumulative impact of previous node failures, which significantly weakens the network’s ability to reconfigure its transport routes. Consequently, upon the failure of the ninth node, the remaining transport network can no longer generate new transport routes, leading to a sharp decline in transport efficiency. In fact, degree centrality only considers the direct connectivity of nodes while neglecting their positional significance and global structural importance within the network. Therefore, nodes with the same degree may have completely different impacts on the network transport efficiency.

Under the betweenness-based node attack strategy, as the number of failed nodes increases from 0 to 9, network transport efficiency decreases gradually in stages, falling from 100% to 56% and then dropping sharply to 0. When the third and fourth nodes fail, the transport efficiency remains unchanged. The reason is that these two nodes do not lie on the newly generated optimal path of the network after the failure of high-betweenness nodes such as T4 and T5. Hence, their removal has no direct effect on the overall transport efficiency of the network.

Under the conventional resilience-based attack strategy, the transport efficiency remains stable at 100% until two nodes fail. After that, the transport efficiency drops to 70%, 49%, 41%, respectively. It then gradually falls to 0 as the number of failed nodes reaches 10. This shows that this method can identify critical nodes, but its ability to cause a sharp efficiency decline is weaker than the resilience-based attack strategy.

Under the closeness-based node attack strategy, the transport efficiency drops rapidly to 70% upon the failure of the first node. It then stays at this level until the seventh node is removed. After that, the transport efficiency drops to 65%, even when 10 nodes have failed. This means that the closeness-based method only causes a partial efficiency drop. It cannot fully disrupt the transport network, even after multiple critical nodes fail.

Under the structural hole-based node attack strategy, the transport efficiency remains at 100% until three nodes fail. After the third node failure, the transport efficiency falls to 70%. After that, the transport efficiency drops to 48%. It then gradually drops to 42% when 10 nodes have failed. This method can identify some nodes with high structural importance, but these nodes do not control the main transport paths. As a result, the network can still maintain partial operation even after several structural hole nodes are removed.

In summary, the PRM needs only three critical nodes to paralyze the entire network, significantly fewer than the nine nodes required by both the degree-based and betweenness-based methods, and the 10 nodes needed by the CRM. Furthermore, even after targeting the top 10 nodes identified by the structural holes-based and closeness-based methods, the network continues to function. The markedly lower number of critical nodes required by the PRM confirms the advantages of our approach.

3.5. Comparison of Resilience Under Different Attack Strategies

The identification results reveal minor differences in the sets of critical nodes obtained from the six different methods. This is mainly because each method focuses on different topological or functional characteristics of the network. When the multimodal transport network faces deliberate attacks under extreme conditions, the six methods may be considered. To further compare the recovery curves of the network across the six attack scenarios, the PRF curve of the multimodal transport network is plotted under a sufficient recovery budget, as shown in Figure 15.

As illustrated in Figure 15a, the critical nodes identified by the PRM require the longest time to restore the transport system to normal operation after damage, demonstrating a significantly greater impact than other methods.

Under the proposed resilience-based attack strategy for the SOD pair scenario, the simultaneous failure of three critical bridge nodes (Q14, B10, and Q28) identified by the PRM leads to an abrupt collapse of the entire network. The network transport efficiency drops sharply to 0%, resulting in network collapse. Topological analysis shows that no connected paths exist between SOD pairs in the network. Subsequently, the multimodal transport network enters the recovery phase. The repair operations are carried out simultaneously at multiple locations. Among the damaged bridges, the Q14 bridge has the shortest repair duration and is the first to be fully restored. When it is fully restored at t = 105 days, network transport efficiency recovers to 85.82% of its baseline level. Subsequently, the Q28 node, with the second shortest repair duration, is also restored. However, network transport efficiency remains unchanged. The reason is that although the Q28 node has been restored, the optimal path of the network after recovery does not pass through this node. Afterward, the B10 bridge is restored at t = 165 days. With all damaged nodes fully restored, network transport efficiency is fully recovered to its initial level. For the MOD pair scenario, as shown in Figure 15b, with the Q14 bridge fully restored at t = 105 days, transport efficiency quickly recovers toward its baseline level. This demonstrates that the Q14 bridge plays an important role in the o₂-d₂ pair scenario.

Under the degree-based node attack strategy, the simultaneous failure of nine critical nodes identified by this method at t = 15 days results in an immediate drop of network transport efficiency to 0%. As emergency repair operations begin, the T2 node is fully restored at t = 45 days, and network transport efficiency rebounds rapidly to its baseline level.

Under the betweenness-based node attack strategy, at t = 15 days, the simultaneous failure of nine critical nodes identified by this method leads to a drop in network transport efficiency immediately to 0%. At t = 60 days, the first group of failed nodes with a repair duration of 45 days is restored. Correspondingly, network transport efficiency recovers to 85.82% of its baseline level. Subsequently, when the second group of failed nodes is fully restored at t = 135 days, the network returns to normal operation.

Under a conventional resilience-based node attack strategy, the simultaneous failure of the top 10 critical nodes identified by this method leads to an immediate drop in network transport efficiency to 0%. The transport efficiency remains at 0% for the following 30 days. After that, once all 10 critical nodes are fully restored simultaneously, the transport efficiency immediately recovers to its initial level.

Under the closeness-based node attack strategy, the simultaneous failure of the top 10 critical nodes identified by this method causes an immediate drop in the network transport efficiency to 65.23%. The transport efficiency remains at this level for the following 30 days. After that, once 8 critical nodes with a repair duration of 30 days are fully restored simultaneously, the efficiency immediately rises to 79.80%. Later, when the B6 bridge is restored at t = 105 days, the transport efficiency further increases to 85.82%. Finally, as all damaged nodes are fully restored at t = 135 days, the network transport efficiency recovers to its initial level.

Under the structural holes-based node attack strategy, the simultaneous failure of the top 10 critical nodes identified by this method causes an immediate drop in the network transport efficiency to 41.79%. The transport efficiency remains at this level for the following 30 days. After that, as all 10 critical nodes with a repair duration of 30 days are fully restored simultaneously, the transport efficiency immediately recovers to its initial level.

In conclusion, from the resilience perspective, the recovery time for the transport function of a multimodal transport network is longest when the system is paralyzed. Different attack strategies lead to different recovery curves. To facilitate a clearer comparison of this cumulative effect, resilience assessments can be conducted under various attack strategies against the network.

When the resilience is calculated over the period from 15 to 210 days, the resilience value R and attack benefit S under six different attack strategies can be obtained using Equations (2)–(5). The resilience value R and attack benefit S for MOD pair scenarios are also presented in the same figure. Among the six strategies, the resilience values of the network follow an ascending order for PRM, BC, CC, DC, CRM, and SH. The corresponding resilience values R are 96.49, 139.37, 162.14, 165, 165 and 177.54, respectively, while the corresponding attack benefits S are 98.51, 55.63, 32.86, 30, 30 and 17.46, respectively. For MOD pair scenarios, the overall resilience value can be calculated as the average value across all OD pairs. Therefore, the integrated resilience value R for the MOD pair scenarios in this case is 100.75. The proportional relationship between R and S is shown in Figure 16. PRM(SOD) and PRM(MOD) represent the PRM corresponding to the SOD and MOD pair scenarios, respectively.

In summary, the results of this case study show that the critical nodes identified by the PRM outperform those identified by BC, DC, CC, SH, and CRM, with attack benefits of 50.52%, 28.53%, 15.39%, 16.85%, 8.95% and 15.39%, respectively. The corresponding efficiency improvements are 21.99%, 35.13%, 33.66%, 41.56%, and 35.13%, respectively. The calculation method is presented in Appendix A. For MOD pair scenarios, the efficiency fluctuation is only 2.19%, which proves the applicability of the algorithm. These results effectively validate the advantages of the critical node identification method based on resilience theory.

4. Discussion

In this paper, the problem is formulated as finding a minimum node set that disconnects multiple given OD pairs. When it comes to an SOD pair, this problem is closely related to a minimum cut problem. However, this differs from general node importance ranking in complex networks, which evaluates the overall importance of nodes. General node importance ranking in complex networks aims to quantify the intrinsic influence of each node on the overall structure and function of the network. Its evaluation is usually based on the topological properties of the network (e.g., degree, betweenness, closeness) or dynamic characteristics, without relying on specific travel demands or connection requirements between any two nodes. It focuses on the global importance of nodes.

The scope of our formulation covers scenarios with predefined OD pairs, represented by multimodal transport networks, where travel requirements between specific OD pairs are predetermined. It is especially applicable to practical engineering contexts requiring targeted protection and control of network connectivity for designated OD pairs. A typical application is the identification of critical nodes for priority safeguarding, so as to maintain the connectivity of key OD pairs under extreme disturbances such as natural disasters and node failures. In such scenarios, the proposed formulation can provide targeted decision support for network resilience optimization and emergency management. However, our formulation relies on specified OD pairs, which limits its applicability to network vulnerability assessment in scenarios without predefined OD pairs. Moreover, the model is sensitive to the quantity and spatial distribution of OD pairs.

In addition, the simulation assumptions adopted in this study also introduce potential biases. The synchronous repair assumption adopted in the simulation accelerates the network recovery process, which may slightly overestimate the network’s resilience performance. Meanwhile, in this study, the fixed shortest path setting takes into account the dynamic route rerouting behavior under node failure. The optimal path is restricted to transmit only along the latest shortest paths, and alternative paths can be used to avoid failed nodes as much as possible. However, this design overestimates network resilience, as it assumes the network can always find effective alternative paths to bypass failed nodes and maintain normal transport efficiency, which simplifies the actual operational challenges. Furthermore, the simplified recovery process facilitates quantitative simulation and efficient comparison, while it cannot fully reflect the complex onsite maintenance schedule and resource constraints, which may also slightly overestimate the network’s resilience performance.

5. Conclusions

In this study, to address the problem of identifying critical nodes in multimodal transport networks, an improved integrated method is proposed based on resilience theory. This method considers disaster consequences and recovery time in network resilience and provides a more comprehensive assessment of the network performance under dynamic disruption scenarios. Meanwhile, to solve this problem efficiently, an IBPSO algorithm is developed. A typical multimodal transport network in China is taken as an example to demonstrate the parameter calibration procedure of the model. Afterward, the performance of the proposed method is compared with the other five methods. The major insights of our work are summarized as follows.

First, the proposed method is suitable for identifying critical nodes in multimodal transport networks from the resilience perspective, which extends the evaluation scope from the SOD pair scenario to the MOD pair scenarios. The main contribution of this study lies in the reasonable integration and targeted improvement of existing theories. Experimental results confirm that the IBPSO algorithm exhibits stronger optimization performance and higher solution accuracy for critical node identification. This study provides valuable insights into the application of resilience theory and complex network theory to transport systems.

Second, the results confirm that the proposed method outperforms the other five methods. Deliberate attacks on critical nodes identified by the proposed method result in the most rapid decline in network transport efficiency. The relevant validation results in the case study show efficiency improvements of 21.99%, 35.13%, 33.66%, 41.56%, and 35.13%, respectively. For MOD pair scenarios, the efficiency fluctuation is only 2.19%, which further verifies the applicability of the method.

Third, the model parameters can be calibrated to apply to different real multimodal transport networks. The critical nodes in the network are mainly concentrated at choke points with long repair durations, which are often overlooked in traditional methods. Targeted protection for these critical nodes can effectively enhance network resilience. The findings can provide a foundation for investigating pre-disaster protection and post-disaster recovery.

This research work still has limitations. This study mainly identified critical nodes from the perspective of the transportation network topology. In the future, it is necessary to integrate actual data such as transportation volume and economic metrics to further extend the method. In addition, relaxing the simplified simulation assumptions, including adopting more realistic recovery processes and enabling dynamic route rerouting during network disruptions, along with further validation on larger-scale multimodal transport networks, will significantly enhance the accuracy and practical realism of resilience evaluation, making the proposed method better meet the demands of practical engineering applications.