Robustness Modeling and Optimization of Multi-Layer Storage and Supply Networks for Equipment Support

Abstract

To investigate the vulnerability of equipment transportation systems under cascading failures, this study models the system as a two-layer coupled network consisting of a storage–supply layer and a transportation layer within a multilayer complex network framework. The model captures both intra-layer structural properties and inter-layer coupling dependencies. A unified multilayer node-importance metric is proposed, integrating layer-level structural contribution with intra-layer centrality to achieve cross-layer comparable and interpretable node ranking. Based on this, a load-capacity–state coupled cascading failure model is developed, incorporating node capacity constraints, dynamic state transitions, and importance-aware load redistribution. Benchmarking experiments against degree, betweenness, multiplex PageRank, and TOPSIS-based methods show that the proposed metric leads to the fastest robustness degradation and the smallest area under the robustness curve, indicating superior capability in identifying cascade-critical nodes. The results demonstrate that targeted attacks significantly degrade network performance, especially when coupled nodes fail. The findings suggest that prioritizing protection of critical nodes and optimizing load redistribution can effectively enhance system resilience.

1. Introduction

The multi-level storage and supply system is an important foundation for ensuring the efficient supply of equipment and materials and the emergency response capability. It plays a key role in national defense support, emergency rescue, and strategic material allocation. With the continuous expansion of the scale and increasing complexity of the storage and supply system, a multi-level and coupled storage and transportation network structure has been formed among the central reserve, regional reserves, and terminal supply nodes. When a certain level of reserve center or transportation channel is affected by natural disasters, equipment failures, or human sabotage and other sudden events, the material processing and supply capacity of the node will decline, and the upstream and downstream nodes connected to it may also be affected in a chain reaction, thereby triggering a cascading failure of the entire storage and supply network. This process leads to delays in material allocation and even the collapse of the system, seriously threatening the robustness and sustainable operation of the equipment support system. Therefore, it is necessary to conduct a systematic analysis of the structural characteristics, key nodes, and inter-layer dependencies of the multi-level storage and supply system from the perspective of complex networks, establish a robustness quantification and recovery strategy model, and provide scientific support for improving the anti-destruction and recovery capabilities of the storage and supply system.

The multi-level storage and supply system is composed of multiple levels of storage nodes, transportation routes and supply channels, and can be essentially regarded as a multi-layer complex network. Early research on complex networks mainly focused on the topological characteristics of the network. Watts and Barabási et al. revealed the small-world and scale-free network properties of complex networks [1], laying the theoretical foundation for subsequent research on network robustness and dynamics. Currently, the research hotspots in complex networks are concentrated on the cascading failure and vulnerability analysis of single-layer networks [2,3]. For instance, Liu et al. studied the resilience of urban rail transit networks through an improved edge weight function [4], and the results indicated that the system’s robustness significantly declined under selective attacks. Feng Fenling et al. proposed a node-importance metric calculation method based on grey relational degree and random forest for multimodal transport networks [5]. Wang Nuo et al. revealed the evolution characteristics of the vulnerability of the global container shipping network under intentional attacks through network pressure testing methods [6,7,8]. However, most of these studies are based on single-layer or weakly coupled network models and fail to fully reflect the complexity of inter-layer interdependence, node heterogeneity and load transfer mechanisms in multi-level systems.

With the development of multi-layer network theory, scholars have begun to focus on the coupling structure between networks and the cross-layer node interaction mechanism [9]. Aleta et al. analyzed the modeling methods and dynamic characteristics of multi-layer networks [10], and Halu et al. proposed a multi-layer PageRank (PR) algorithm considering cross-layer interaction to describe the centrality of nodes in the overall network [11]. In China, Wu Zongning et al. systematically reviewed the research progress in the modeling, topological properties, and robustness of multi-layer networks, and pointed out the application prospects of multi-layer networks in transportation, energy, and supply chain systems [12]. Shen Li et al. constructed a metro-bus composite network model for cities, and the results showed that the composite network had better survivability under various attack modes than single-layer networks, but their model did not consider the dynamic changes in node capacity and load [13]. Feng Fenling et al. proposed a method for calculating the node-importance metric in multi-layer networks by combining the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method and the grey relational degree method in the study of the China–Europe Railway Express transportation network, and verified the impact of inter-layer interaction on the overall robustness of the network [14,15,16]. Wang Xinglong et al. further analyzed the accessibility robustness of the aviation multi-layer dynamic network and found that the survivability of the network presented nonlinear changes in different time periods [17].

To sum up, scholars at home and abroad have achieved rich results in the research on the robustness of complex traffic and logistics networks. Such as Quizhpe et al. reviewed how microgrid planning must jointly consider structural redundancy, reliability, and recovery-oriented design. Although the application domain differs from equipment support systems, this perspective is conceptually relevant to the present study [18]. However, there is still a lack of research on the multi-layer complex network modeling, node-importance metric calculation, and dynamic evolution mechanism of robustness for multi-level storage and supply systems. Existing achievements mostly focus on single-layer topological structures and lack systematic characterization of the functional differences, coupling relationships, and load transfer characteristics of nodes at different levels in the storage and supply system. In addition, the research on the recovery mechanism after network damage is relatively weak, and a systematic recovery strategy framework based on the node-importance metric and hierarchical coordination has not yet been formed. Therefore, this paper, based on the theory of multi-layer complex networks, constructs a topological model of multi-level storage and supply networks, analyzes the structural characteristics and the distribution law of the node-importance metric between layers; on this basis, designs a cascading failure model considering node capacity, load distribution, and state evolution, revealing the key factors affecting the dynamic changes in the robustness (anti-destruction capability) of storage and supply networks; and further proposes a network recovery strategy based on the priority recovery of important nodes. The research results can provide theoretical references and decision-making basis for the structural optimization, system resilience (recovery capability) improvement, and operation safety guarantee of the equipment storage and supply system.

This paper proposes the following innovative points from an integrated “structure-function-evolution” perspective:

Before presenting the proposed metric, it is necessary to clarify its position relative to existing multilayer centrality methods. The proposed node-importance metric is not intended as a universal replacement for degree, betweenness, multiplex PageRank, or TOPSIS-style multilayer ranking methods. Rather, it is designed for equipment support networks characterized by functional asymmetry between layers, strong cross-layer dependence, and the need to explicitly couple node ranking with cascading load redistribution. Degree and betweenness mainly capture local connectivity or shortest-path control, multiplex PageRank emphasizes recursive prestige across layers, and TOPSIS-style methods rely on exogenously selected indicators and normalization procedures. In contrast, the proposed metric integrates layer-level structural contribution and intra-layer node influence into a unified cross-layer comparable score, which is directly usable in the subsequent cascading-failure redistribution mechanism.

Accordingly, this paper makes four main contributions. First, a domain-specific two-layer coupled network model is developed for the equipment support system, explicitly representing the storage–supply layer, the transportation layer, and their inter-layer dependencies. Second, a unified multilayer node-importance metric is proposed by combining layer-level relative structural importance with intra-layer node centrality, so that nodes from functionally different layers can be compared on a common scale. Third, a capacity–state–importance cascading-failure model is established to describe gradual degradation, overload propagation, and importance-guided load redistribution under both random and targeted attacks. Fourth, simulation-based robustness analysis is conducted to derive resilience-oriented planning implications, including priority protection of high-importance nodes, redundancy allocation, and optimization of cross-layer load redistribution.

It should also be emphasized that the proposed metric is particularly suitable when the two layers play clearly different functional roles, when coupled nodes simultaneously undertake storage and transportation responsibilities, and when node importance must be explicitly embedded into the load redistribution rule of the cascading model. For highly homogeneous or weakly coupled networks, the advantage of the proposed metric over simpler centrality measures may be less pronounced.

This paper is structured as follows: Section 1 presents the modeling framework and coupling characteristic metrics for the two-layer storage–supply and transportation network; Section 2 proposes a multi-layer node importance evaluation method; Section 3 establishes a cascading failure model linking capacity, status, and importance, along with simulation settings; Section 4 reports and discusses the impact of random/malicious disturbances, capacity and fault-tolerance parameters, and load redistribution strategies on robustness; Section 5 summarizes the findings and proposes resilience-oriented engineering recommendations and future research directions.

2. Double-Layer Equipment Transportation Network Model and Topological Characteristics

2.1. Construction of a Double-Layer Equipment Transportation Network Model

This study adopts the assumption of an undirected and unweighted network due to the following characteristics of military equipment transportation processes. First, logistical reversibility: when forward transportation routes are disrupted, emergency equipment can be redirected backward along the original routes; therefore, directionality is not a primary concern in network robustness analysis. Second, homogeneity of transport capacity: military logistics channels are standardized, and each convoy unit operates under uniform load limits, thereby minimizing capacity heterogeneity. Third, classification constraints: for reasons of national security, detailed data on transport capacity and costs are classified. Accordingly, the networks constructed in this study are modeled as undirected and unweighted.

By using the network coupling theory, the double-layer equipment transportation network is defined as A triple $\left(G,E,A\right)$. Among them, $G_s=\left(V_s,E_s,A_s\right)$ represents the LayerStorage–Supply Layer, and $G_t=\left(V_t,E_t,A_t\right)$ represents the Transportation Layer.

In the model, $V_s$ and $V_t$ respectively represent the node sets of each network layer, $E_s$ and $E_t$ represent the edge sets within the layer, and $A_s$ and $A_t$ represent the adjacency matrices within the layer. If nodes i and j have edges in that layer, then $A_{ij}=1$; otherwise, it is 0.

The inter-layer relationship of the network is represented by the inter-layer coupling matrix $C=\left\{c_{ij}^{st}\right\}$, where the matrix element $c_{ij}^{st}=1$ indicates that there is a material allocation or transportation connection between the storage and supply layer node $i$ and the transportation layer node $j$, and vice versa is 0.

When building a two-layer equipment transportation network, the coupling relationship between layers is established based on the principle of “node sharing and material flow direction”, that is: if a certain node city has both a reserve warehouse and a transportation hub at the same time, or if this transportation node directly undertakes the transfer task of equipment and materials, it is regarded as the existence of a coupling relationship between layers [19].

Therefore, the double-layer equipment transportation network can be abstracted as a coupled system between the storage and supply network and the transportation network, as shown in Figure 1. The deep blue network layer represents the storage and supply network layer, which is mainly composed of the central warehouse, regional warehouses and forward warehouses. The light orange network layer represents the transportation network layer, which includes major transportation nodes such as railways, highways and air transport.

In recent years, modeling and analyzing the properties of multilayer networks has become a hot topic in complex network research. Some scholars utilize two-dimensional supra-adjacency matrices to flatten multilayer networks, enabling the unified representation of interactions across different layers.

This paper adopts this approach, utilizing a hyper-adjacency matrix as the mathematical model for a two-layer equipment transportation network. Since the research object is an undirected, unweighted multi-layer network, the hyper-adjacency matrix exhibits symmetry and can be expressed as:

$$A=\left[\begin{array}{cc}{A}_s& C\\ C^T& A_t\end{array}\right]$$

(1)

Based on collected data from the equipment storage, supply, and transportation system, the storage and supply network layer comprises 86 storage nodes and 124 material flow paths; the transportation network layer includes 102 transportation nodes and 148 transportation routes. Among these, 12 nodes simultaneously perform both storage/supply and transportation functions, forming inter-layer coupled nodes. Statistically, the entire two-layer network comprises 176 independent nodes and 402 edges.

Using the Space-L method to number and abstract storage/supply nodes and transportation nodes, the established two-layer equipment transportation network model is shown in Figure 1.

2.2. Data Abstraction and Reproducibility Protocol

The multilayer network designed for equipment support comprises two core functional layers. Their respective node definitions and aggregation logic are outlined as follows: Storage–Supply Layer: Nodes in this layer correspond to physical storage and supply facilities within the support system, including strategic depots, regional reserve centers, forward warehouses, and integrated support bases. Crucially, each node represents an aggregated business entity rather than a single geographical coordinate, providing an initial layer of protection for sensitive information.

Transportation Layer: This layer identifies key hubs in the material circulation process, such as railway freight stations, highway junctions, airports, and port logistics centers. Collectively, these nodes form the backbone transportation infrastructure facilitating the cross-regional flow of equipment and supplies.

(1) Aggregation Rules

To prevent the leakage of sensitive data through excessive granularity while simultaneously streamlining the network for efficient analysis, multiple facilities with similar roles and close proximity are merged into single nodes. This aggregation adheres to two rigorous criteria:

Functional Similarity: Facilities must perform identical or comparable roles within the supply chain to ensure functional consistency post-aggregation.

Spatial Proximity: Facilities are grouped only if they reside within the same administrative region or operational zone, thereby maintaining geographical coherence.

(2) Intra-layer Edge Construction Rules

To accurately reflect the inherent structural characteristics of the system, edges within each layer are defined by long-term, stable relationships rather than transient operational data. The specific rules are as follows:

Storage–Supply Layer: An undirected edge is established between two nodes if a stable, planned allocation relationship—such as routine supply, reserve transfer, or hierarchical distribution—exists between them. These edges characterize normalized material interactions.

Transportation Layer: Undirected edges connect nodes linked by primary transport corridors, such as railway lines, trunk highways, or fixed air routes. These edges prioritize the backbone structural connectivity of the infrastructure over real-time traffic intensity.

(3) Inter-layer Coupling Rules

Inter-layer coupling is grounded in the actual operational logic of the support system, deliberately eschewing abstract assumptions to ensure authenticity. An inter-layer edge connects a storage–supply node i and a transportation node j if at least one of the following conditions is met:

Co-located Dual-functionality: The storage facility and transport hub are situated in the same city or operational zone and jointly participate in material scheduling and flow.

Direct Transshipment Responsibility: The transportation node is tasked with the loading, unloading, or handling of materials for the corresponding storage node, signifying a direct operational link.

Functional Dependency: Core logistical phases, such as inbound and outbound movements for a storage node, rely on the transport hub as a primary gateway.

Coupling patterns extend beyond simple one-to-one mappings. A single storage–supply node may connect to multiple transportation nodes to reflect the diversity of shipping channels. Conversely, a transport hub can serve several storage nodes simultaneously, showcasing its radial service capacity.

(4) Data Anonymization and Reproducibility

To guarantee the reproducibility and verifiability of research results while strictly safeguarding sensitive data, we employ the following anonymization strategies and provide a standardized, reproducible dataset:

Anonymized Node Coding: All nodes are encoded with abstract identifiers—specifically, $S_1\text{–}{S}_{86}$ for storage nodes and $T_1\text{–}{T}_{102}$ for transportation nodes—to prevent identity disclosure.

Sensitive Information Removal: Geographic coordinates, specific facility names, and actual operational parameters are entirely purged from the dataset.

Topology Preservation: Only the topological relationships between nodes are retained, ensuring that subsequent network analysis remains viable.

For scenarios where data sharing is restricted, this paper describes a synthetic network generation procedure. The resulting synthetic networks faithfully preserve the key statistical properties of the original system, including degree distribution, node clustering coefficient, assortativity, and coupling ratio.

(5) Construction Workflow

The comprehensive data processing and construction workflow for the equipment support multilayer network is illustrated in Figure 2. It follows five core steps: Raw Operational Structure, Data Anonymization, Node Aggregation, Layer Allocation, Edge Construction (Intra- and Inter-layer), Final Supra-adjacency Matrix ($A$, see Equation (1)).

By employing standardized abstraction and anonymization, this scheme ensures model reproducibility without compromising sensitive operational details. Ultimately, it achieves an effective balance between the security imperatives of equipment support systems and the rigorous demands of scientific inquiry.

2.3. Network Topology Eigenvalue Analysis

2.3.1. Distributional Characterization of Structural Heterogeneity

To comprehensively characterize the structural heterogeneity of the multilayer equipment transportation network, this study extends the analysis beyond node degree and examines the distributional properties of three key metrics: node degree, betweenness centrality, and node load.

While degree distribution is commonly used to identify scale-free properties, it does not fully capture the functional role of nodes in flow redistribution and cascading failures. Therefore, betweenness centrality is introduced to reflect the control of shortest-path flows, and node load (including initial load and peak redistributed load during cascading processes) is analyzed to characterize dynamic stress concentration.

For each metric, we perform a systematic distributional comparison using three candidate models: power-law, log-normal, and exponential distributions. The fitting is conducted using maximum likelihood estimation (MLE), and the goodness-of-fit is evaluated using the Kolmogorov–Smirnov (KS) statistic. In addition, likelihood ratio tests and information criteria (AIC/BIC) are employed to compare competing models.

The complementary cumulative distribution functions (CCDFs) of the three metrics are shown in Figure 3.

The results indicate that: The degree distribution follows a power-law behavior with exponent ${\gamma }_d\approx 2.7$, consistent with typical scale-free network.; The betweenness centrality distribution exhibits a heavier tail, with exponent ${\gamma }_b\approx 2.2$, indicating stronger concentration of flow control in a small number of nodes. The load distribution shows a mixed heavy-tail behavior, where the power-law model is competitive with the log-normal model, suggesting that dynamic load redistribution introduces additional variability beyond static topology.

Across all metrics, the power-law model is consistently favored over the exponential distribution based on KS statistics and likelihood ratio tests, confirming the presence of strong heterogeneity in both structural and functional dimensions.

These results demonstrate that the network exhibits not only topological scale-free network properties but also functional heavy-tailed behavior in flow and load dynamics, which is critical for understanding cascading failure propagation.

2.3.2. Node Clustering Coefficient

The node clustering coefficient $C_i$ describes the degree of connection tightness between node neighbors [20]. In a two-layer network, the aggregation coefficient of node $v_i$ is defined as:

$$C_i={\displaystyle \frac{2e_i}{k_i\left(k_i-1\right)}}$$

(2)

Here, $e_i$ represents the actual number of edges that exist between the neighboring nodes of node $v_i.$ For cross-layer nodes, considering the joint adjacency relationship between the storage and supply layer and the transportation layer, the node clustering coefficient is expanded to:

$${\displaystyle C_i^{multi}}={\displaystyle \frac{{\sum }_{\alpha ,\beta }{\displaystyle e_i^{\left(\alpha ,\beta \right)}}}{{\sum }_{\alpha ,\beta }{\displaystyle k_i^{\left(\alpha \right)}}{\displaystyle k_i^{\left(\beta \right)}}}}$$

(3)

The calculation results are shown in Figure 4. The node clustering coefficient of the multi-layer network is mainly distributed between 0.15 and 0.75, with an average value of 0.46, indicating that the network as a whole has a strong local agglomeration, that is, there are relatively dense local connections between the equipment storage and supply and transportation nodes. Some highly concentrated nodes correspond to the close supporting relationship between regional warehousing centers and military production bases, reflecting that the system has a high degree of redundancy and synergy within the local area.

2.3.3. Node PageRank Value

The PageRank algorithm is widely used for measuring the node-importance metric in single-layer networks. In a two-layer equipment transportation network, the PageRank value of a node is not only related to the quantity and quality of its neighbors, but also affected by the relative importance of the network layer [21].

Let the adjacency matrices of the storage and supply layer and the transportation layer be $A_s$ and $A_t$ respectively, and the PageRank values of their nodes be $P{R}_i^s$ and $P{R}_i^t$ respectively. Then, the PageRank within the layer can be calculated by the classical algorithm:

$$P{R}_i^{\alpha }={\displaystyle \frac{1-d}{N}}+d\sum _{j\in M_i}{\displaystyle \frac{P{R}_j^{\alpha }}{k_j^{\alpha }}}$$

(4)

Here, $d$ is the damping factor (taken as 0.85), and $M_i$ is the set of nodes pointing to node $i$.

After considering the cross-layer influence, the comprehensive PageRank value of the node in the entire two-layer network can be expressed as:

$$P{R}_i^{total}={\omega }_{s}P{R}_i^s+{\omega }_{t}P{R}_i^t+\delta \sum _j{C}_{ij}P{R}_j^{cross}$$

(5)

Among them, ${\omega }_s$ and ${\omega }_t$ are the weight coefficients of the storage and supply layer and the transportation layer respectively, and $\delta$ is the inter-layer coupling adjustment parameter.

The calculation results are shown in Figure 5. Among the top 10 nodes in terms of PR value, nodes such as the Central Strategic Reserve Repository, regional comprehensive support centers, and national-level railway freight hubs account for 70%, indicating that they play a crucial role in aggregating material and information flows in the network.

2.3.4. Network Layer Coupling Characteristics

To analyze the degree of association between the storage and supply layer and the transportation layer, Spearman’s correlation coefficient (Spearman’s $\rho$) is introduced, calculated as follows:

$$\rho =1-{\displaystyle \frac{6{\displaystyle {\sum }_{i=1}^n}{d}_i^2}{n\left(n^2-1\right)}}$$

(6)

Here, $d_i$ represents the rank difference in the core values of the same node in different layers of the network, and the value range of ρ is $\left[-1,1\right]$. When $\rho >0$, the network shows assortative coupling; that is, high-connection nodes tend to connect to high-connection nodes. When $\rho <0$, the network shows a mismatch coupling [21].

After calculation, the $\rho$ of the double-layer equipment transportation network is −0.117, indicating that there is a mild mismatch coupling feature between the storage and supply layer and the transportation layer; that is, the reserve nodes with high connectivity tend to be connected to the transportation nodes with smaller connectivity values through the transportation network. This mismatch feature reflects the supply mode of “central warehouse—distributed transportation hub” in the system, which is in line with the hierarchical and division of labor structure of the actual equipment storage and supply system.

2.4. Calculation Method for the Node-Importance Metric in Multi-Level Equipment Transportation Networks

Unlike single-layer transportation networks, in the calculation of node importance in multi-layer equipment transportation networks, not only the structural position and connection characteristics of nodes within each layer need to be considered, but also the relative importance and coupling influence between network layers should be comprehensively reflected. Therefore, this paper conducts research from two aspects: (1) Calculating the relative importance of each network layer to reflect the contribution of the storage and supply layer and the transportation layer to the overall network connectivity; (2) Calculate the intra-layer node-importance metric within each layer to reflect their core status in the local topological structure; Finally, the global node-importance metric is obtained through the fusion model.

2.4.1. Positioning Against Existing Multilayer Centrality Methods

Before defining the proposed metric, we clarify its methodological position relative to representative node-ranking approaches used in complex and multilayer networks. In single-layer settings, degree and betweenness are widely adopted because they quantify local connectivity and shortest-path control, respectively. In multilayer settings, multiplex PageRank extends recursive prestige propagation across layers, while TOPSIS-style multilayer methods combine multiple indicators through weighted decision analysis. These methods are informative, but they are not equally suitable for the present problem.

The equipment support network studied in this paper exhibits three characteristics that motivate a different design. First, the storage–supply layer and the transportation layer are functionally asymmetric rather than interchangeable. Second, coupled nodes may simultaneously play storage, transfer, and cross-layer bridging roles, so their criticality cannot be adequately described by a purely intra-layer ranking. Third, the node-importance score is not used only for static ranking, but also serves as an explicit control variable in the subsequent cascading-failure load redistribution rule. Therefore, the proposed metric is designed as a cross-layer comparable importance score that jointly reflects layer-level structural contribution and intra-layer node influence.

For clarity,

Table 1. Comparison between representative multilayer centrality methods and the proposed node-importance metric.

Method	Functional Asymmetry	Cross-Layer Comparability	Coupled Nodes Contribution	Cascading LOAD Applicability
Degree	No	No	No	No
Betweenness	Partially	No	No	No
Multiplex PageRank	Partially	Limited	Partially	No
TOPSIS-based	Indicator-dependent	Yes	Yes	No
Proposed Metric	Yes	Yes	Yes	Yes

summarizes the differences between representative baseline methods and the proposed metric. The proposed approach should not be interpreted as uniformly superior in all multilayer networks. Its main advantage is expected in functionally asymmetric and strongly coupled systems where node importance must be directly linked to cascading-failure control. In contrast, for highly homogeneous or weakly coupled networks, simpler measures may provide comparable rankings at lower modeling cost.

2.4.2. Unified Definition and Calculation Steps

Based on the above positioning, the proposed metric is defined as a hierarchical fusion of layer-level structural importance and intra-layer node centrality. The calculation proceeds in three steps.

Step 1: Layer Weight Calculation.

We first quantify the relative structural contribution of each layer to the global connectivity of the coupled network. Let E denote the set of all intra-layer edges in the multilayer network, and let $E_s$ and $E_t$ denote the edge sets of the storage–supply layer and the transportation layer, respectively [22]. The contribution of layer l is measured by the normalized sum of edge betweenness centrality over all edges belonging to that layer. Accordingly, the layer weight $w_l$ is defined as

$$w_l={\displaystyle \frac{{\sum }_{e\in E_l} b_e}{{\sum }_{e\in E} b_e}},$$

(7)

where $b_e$ denotes the betweenness centrality of edge e in the global coupled topology. This definition ensures that a layer undertaking more shortest-path transmission tasks in the coupled network receives a larger structural weight.

Step 2: Intra-layer Node Centrality.

Within each layer, node-importance metric is measured using PageRank centrality, which captures the prestige of a node in terms of both the quantity and quality of its adjacent nodes. For a node i belonging to layer l, its intra-layer centrality is denoted by $P{R}_i^{\left(l\right)},$ and is obtained from the standard PageRank iteration on the corresponding layer-specific adjacency matrix. This step reflects the local structural influence of the node within its own functional subnet.

Step 3: Global Importance Fusion.

The final global importance of node i in layer l is obtained by fusing the layer weight and the intra-layer node centrality. Specifically, the global importance score is defined as

$$I_i^{\left(l\right)}=w_l\cdot P{R}_i^{\left(l\right)}.$$

(8)

For a coupled node that appears in both layers, its total importance is defined as the sum of its layer-specific contributions, i.e.,

$$I_i=I_i^{\left(s\right)}+I_i^{\left(t\right)}$$

(9)

Through this design, the proposed metric simultaneously captures three aspects: the relative structural role of each layer in the global network, the local influence of nodes within each layer, and the dual contribution of coupled nodes across layers. Therefore, the resulting score is cross-layer comparable and can be directly used as an input variable in the subsequent cascading-failure redistribution mechanism.

2.4.3. Sensitivity Analysis and Numerical Example

To illustrate how layer weights affect node ranking, we present a simplified numerical example involving three hypothetical nodes: Node A, a storage depot located only in the storage–supply layer; Node B, a transport hub located only in the transportation layer; and Node C, a coupled node that simultaneously appears in both layers. Their ranking outcomes under different layer-weight settings are shown in

Table 2. Sensitivity of node-importance metric to layer weights.

Node Type	Intra-Layer Score	${\mathit{w}}_{\mathit{s}}=0.8,{\mathit{w}}_{\mathit{t}}=0.2$	${\mathit{w}}_{\mathit{s}}=0.5,{\mathit{w}}_{\mathit{t}}=0.5$	${\mathit{w}}_{\mathit{s}}=0.2,{\mathit{w}}_{\mathit{t}}=0.8$
Node A (Storage)	0.034	0.0272 (Rank 1)	0.0170 (Rank 2)	0.0068 (Rank 3)
Node B (Transport)	0.018	0.0036 (Rank 3)	0.0090 (Rank 3)	0.0144 (Rank 2)
Node C (Coupled)	0.0200/0.0200	0.0200 (Rank 2)	0.0200 (Rank 1)	0.0200 (Rank 1)

.

The example shows that the ranking of single-layer nodes is sensitive to the relative structural importance of the layer in which they are located. When the storage–supply layer dominates the overall network, Node A receives the highest global importance. When the transportation layer becomes more dominant, Node B moves upward in the ranking. By contrast, the coupled node remains comparatively stable because its importance is jointly supported by both layers. This property reflects the intended behavior of the proposed metric: it preserves cross-layer comparability while recognizing the structural advantage of nodes that simultaneously contribute to multiple functional layers [23].

To further illustrate the empirical meaning of the layer weights in the studied network, the calculated results show that the storage–supply layer has a larger normalized edge-betweenness contribution than the transportation layer. This indicates that, in the present equipment support network, the storage–supply layer plays a relatively more central structural role in maintaining global connectivity. Consequently, storage–supply hubs and cross-layer coupled nodes tend to rank prominently in the final importance list, which is consistent with the hierarchical characteristics of the actual equipment support system.

3. Simulation Modeling of Vulnerability in Double-Layer Equipment Transportation Network

3.1. Node Status Analysis

In the equipment storage, supply and transportation system, the operational load level of nodes directly determines whether they can maintain normal operations and is one of the important indicators for measuring system reliability. In traditional research, nodes are often simplified into two states: “normal” and “completely failed”. When the load of a node exceeds its capacity, it is removed from the network. However, in the actual equipment transportation system, the functional decline of nodes is often gradual and recoverable. Therefore, a simple binary division cannot accurately reflect the dynamic characteristics of the system.

To capture the progressive functional degradation of nodes, we introduce a state variable ${\sigma }_i\left(t\right)\in 0,1,2$, which is distinct from the continuous variable representing the physical load, ${\mathrm{L}}_{\mathrm{i}}\left(\mathrm{t}\right)\in {\mathbb{R}}^{+}$. The state transitions of nodes are governed by the relationship between load and capacity; the detailed classification of node states, their physical interpretations, and the corresponding load redistribution rules are summarized in

Table 3. Node state definition and load redistribution rules.

$\mathbf{State} {\mathsf{\sigma }}_{\mathbf{i}}\left(\mathit{t}\right)$	Condition	Physical Meaning	Load Redistribution Rule
0 (Normal)	$0<L_i\left(t\right)\le Q_i$	Fully operational	No redistribution
1 (Overloaded)	$Q_i<L_i\left(t\right)\le S_i$	Performance degradation, partial failure	Redistribute excess load ${\mathsf{\Delta }}_{\mathrm{i}}=L_i\left(t\right)-Q_i$
2 (Failed)	$L_i\left(t\right)>S_i$	Complete failure, removed from the network	Redistribute entire load $L_i\left(t\right)$

.

The state transition dynamics are formally defined as:

$${\sigma }_i\left(t+1\right)=\left\{\begin{array}{cc}0,& \mathrm{i}\mathrm{f} L_i\left(t\right)\le Q_i\hfill \\ 1,& \mathrm{i}\mathrm{f} Q_i<L_i\left(t\right)\le S_i\hfill \\ 2,& \mathrm{i}\mathrm{f} L_i\left(t\right)>S_i\hfill \end{array}\right.$$

(10)

During the simulation process, when a node is in an overloaded state, only the portion of the load exceeding the safe capacity is transferred to its adjacent nodes. When a node enters a complete failure state, its entire load is redistributed to other nodes according to the rules to simulate the cascading failure propagation process.

To examine whether the conclusions depend on the simplifying assumptions, three sensitivity scenarios were further considered: weighted links, partially directed flows, and heterogeneous node capacities.

3.2. Construction of Load-Capacity Cascading Failure Model

To characterize the dynamic evolution of the equipment support network under disturbances, this study constructs a load-capacity–state coupled model. The model integrates initial node load, capacity constraints, and an importance-based load redistribution mechanism to simulate the propagation of failures within the network.

3.2.1. Initial Load of Nodes

In the equipment transportation network, the load of a node represents its capacity to handle material flow or perform support tasks. Due to limitations in acquiring real-time operational flow data, this study adopts node degree as a proxy for the initial load, reflecting the node’s structural exposure and potential traffic aggregation capability. The initial load of node $i$, denoted as $L_i\left(0\right)$, is defined as:

$$L_i\left(0\right)=d_i$$

(11)

where $d_i$ denotes the degree of node $i$.

The validity of this definition is empirically supported. Based on statistical analysis of operational data from 12 non-classified regional warehouses, node degree shows a significant positive correlation with average monthly material throughput (Pearson $r=0.82, p<0.01$), confirming its effectiveness as a structural proxy for node load.

3.2.2. Node Capacity and State Model

Based on engineering practice and design standards, two capacity thresholds are defined for each node:

Safe capacity:

$$C_i^s=\left(1+\alpha \right)L_i\left(0\right)$$

(12)

Limit capacity:

$$C_i^m=\left(1+\beta \right)L_i\left(0\right)$$

(13)

where $\alpha$ is the safety tolerance coefficient and $\beta$ is the limit capacity coefficient, satisfying $\beta >\alpha >0$.

According to the real-time load $L_i\left(t\right)$, the node state $s_i\left(t\right)$ is classified into three stages:

$$s_i\left(t\right)=\left\{\begin{array}{cc}0,& L_i\left(t\right)\le C_i^s\left(\mathrm{normal}\right)\\ 1,& C_i^s<L_i\left(t\right)\le C_i^m\left(\mathrm{overloaded}\right)\\ 2,& L_i\left(t\right)>C_i^m\left(\mathrm{failed}\right)\end{array}\right.$$

(14)

Accordingly, the amount of load that needs to be redistributed from node $i$, denoted as $\mathsf{\Delta }{L}_i\left(t\right)$, is given by:

$$\mathsf{\Delta }{L}_i\left(t\right)=\left\{\begin{array}{cc}0,& s_i\left(t\right)=0\\ L_i\left(t\right)-C_i^s,& s_i\left(t\right)=1\\ L_i\left(t\right),& s_i\left(t\right)=2\end{array}\right.$$

(15)

The parameter settings in this study are based on the GJB 4355-2002 General Requirements for Military Material Storage Warehouses. Specifically, $\mathsf{\alpha }=0.4$ is adopted to comply with standard warehouse design margins, while $\mathsf{\beta }=1.2$ is used to simulate the maximum load-bearing capacity under emergency conditions.

3.2.3. Load Redistribution Mechanism

When a node enters an overloaded or failed state, its excess load is redistributed to its set of active neighboring nodes, denoted as ${\mathcal{N}}_i$. To prevent cascading collapse of critical hub nodes, an importance-based negative feedback allocation strategy is introduced.

The weight assigned to neighbor node $j$, denoted as ${\omega }_{ij}\left(t\right)$, is defined as:

$${\omega }_{ij}\left(t\right)={\displaystyle \frac{\left(H_j+ϵ{)}^{-1}\right.}{{\sum }_{k\in {\mathcal{N}}_i}(H_k+ϵ{)}^{-1}}}$$

(16)

where $H_j$ represents the global importance score of node $j$, and $ϵ$ is a small positive constant to ensure numerical stability.

The load of node $j$ at time step $t+1$ is then updated as:

$$L_j\left(t+1\right)=L_j\left(t\right)+{\omega }_{ij}\left(t\right)\cdot \mathsf{\Delta }{L}_i\left(t\right)$$

(17)

This mechanism ensures that nodes with higher importance receive a smaller proportion of redistributed load, thereby reducing the risk of overload-induced cascading failures at critical nodes.

3.2.4. Cascading Failure Simulation Algorithm

The cascading failure process is simulated using the following Algorithm 1.

Algorithm 1. Cascading Failure Simulation

Input: Network topology $G\left(V,E\right)$, initial load $L\left(0\right)$, threshold parameters $\alpha ,\beta$, and initial attack set $A$. Output: Final set of failed nodes $F$, and robustness metrics $G$ and $\eta$. Procedure: Initialization: Set $F=\mathrm{\varnothing }$, and initialize the time step $t=0$. 2. Perturbation: Remove the nodes in the attack set $A$, and redistribute their initial loads to neighboring nodes according to the load redistribution rule defined in Equation (18). 3. Iterative Evolution: While new overloaded or failed nodes are generated, perform the following steps: Update the real-time load $L_i\left(t\right)$ of affected nodes; Update the node state $s_i\left(t\right)$ according to Equation (4); If $s_i\left(t\right)=2$, add node $i$ to the failure set $F$, and redistribute its load to neighboring nodes; Update the time step: $t=t+1$. 4. Termination: The simulation stops when no new failed nodes appear in the network. 5. Performance Evaluation: The robustness of the network is evaluated using the following metrics: Largest connected component ratio: $$G={\displaystyle \frac{N_{\mathrm{largest}}}{N_0}}$$ Global network efficiency: $$\eta ={\displaystyle \frac{1}{N\left(N-1\right)}}{\displaystyle \sum _{i\ne j}}{\displaystyle \frac{1}{d_{ij}}}$$

This algorithm captures the progressive propagation of cascading failures and enables quantitative evaluation of network robustness under different disturbance scenarios.

3.3. Quantitative Indicators of Network Vulnerability

The vulnerability of a multi-layer equipment transportation network reflects its ability to maintain normal operation after being disrupted or attacked. This paper adopts two indicators, namely the largest connected component ratio and global efficiency, to conduct quantitative analysis from the aspects of structural integrity and operational performance.

3.3.1. Maximum Network Connectivity Ratio

When a system is disturbed or a node fails, the network may be decomposed into several connected subgraphs. Among them, the connected subgraph with the largest number of nodes is called the maximum connected subgraph, and its scale reflects the overall connectivity of the network.

Define the largest connected component ratio $G$ of the network as:

$$G={\displaystyle \frac{N_e}{N_0}}$$

(18)

Among them, $N_0$ represents the number of nodes in the system’s initial maximum connected subgraph, and $N_e$ represents the number of nodes in the maximum connected subgraph after the attack. The larger the $G$ value is, the better the structural integrity of the system is maintained after it is damaged.

3.3.2. Global Network Efficiency

The global efficiency of the network measures the convenience of material transmission between nodes and is defined as the average of the reciprocals of the shortest path distances between all node pairs:

$$\eta ={\displaystyle \frac{1}{N(N-1)}}{\displaystyle \sum _{i=1}^N} {\displaystyle \sum _{j=1}^N} {\eta }_{ij}$$

(19)

Here, ${\eta }_{ij}={\displaystyle \frac{1}{d_{ij}}}$, where $d_{ij}$ is the shortest path distance between nodes $i$ and $j$, and $N$ is the total number of nodes in the network.

When there are a large number of failed nodes or connection interruptions in the network, the average shortest path increases and the network efficiency decreases, reflecting an enhanced system vulnerability.

3.4. Simulation Experiment Process Design

To study the cascading behavior of the double-layer equipment transportation network under different failure conditions, the following simulation process is designed:

(1)

Network construction stage: Establish a two-layer equipment transportation network model composed of the storage and supply layer and the transportation layer, forming an undirected and unweighted network;

(2)

Parameter initialization stage: Set the initial load $L_i^0$ of the node, capacity parameters $\alpha$ and $\beta$, and calculate the safe capacity and limit capacity of the node;

(3)

Attack strategy stage: In the simulation, two methods, namely random attack and targeted attack (based on node importance), are respectively adopted to trigger node failure;

(4)

Cascading evolution stage: According to the load redistribution rule (Equation (17)), the load and status of nodes are dynamically updated, and the process of node status changes in the network is recorded.

(5)

Termination condition: The simulation terminates when no new failed nodes are generated in the network.

(6)

Result analysis stage: Calculate the largest connected component ratio G and global efficiency η for each stage, and plot the vulnerability change curve. The simulation process is shown in Figure 6.

3.5. Multi-Dimensional Robustness Evaluation Metrics

To avoid potential bias arising from single-indicator evaluation and to provide a comprehensive assessment of network robustness, this study adopts a multi-dimensional evaluation framework. Specifically, four complementary metrics are employed to characterize system performance from structural, functional, and dynamic perspectives: largest connected component ratio, global efficiency, service-level proxy, and recovery time. In addition, an aggregated robustness measure based on the area under the robustness curve (AURC) is introduced for comparative analysis.

(1) Largest Connected Component Ratio

The largest connected component ratio $G\left(t\right)$ is used to quantify the structural integrity of the network under disturbance:

$$G\left(t\right)={\displaystyle \frac{N_{\mathrm{largest}}\left(t\right)}{N_0}}$$

(20)

where $N_{\mathrm{largest}}\left(t\right)$ denotes the number of nodes in the largest connected subgraph at time step $t$, and $N_0$ is the total number of nodes in the original network.

This metric reflects the extent to which the network remains connected after cascading failures.

(2) Global Efficiency

The global efficiency $E_{glob}\left(t\right)$ measures the overall efficiency of information or material transmission:

$$E_{glob}\left(t\right)={\displaystyle \frac{1}{N\left(N-1\right)}}\sum _{i\ne j}{\displaystyle \frac{1}{d_{ij}\left(t\right)}}$$

(21)

where $d_{ij}\left(t\right)$ is the shortest path length between nodes $i$ and $j$ at time step $t$.

Compared with connectivity-based metrics, global efficiency captures the degradation of transport performance due to increased path lengths and fragmentation.

(3) Service-Level Proxy

To evaluate the operational capability of the system, a service-level proxy $SL\left(t\right)$ is introduced:

$$SL\left(t\right)={\displaystyle \frac{{\sum }_{i\in V}\mathbb{1}(s_i\left(t\right)\ne 2)\cdot d_i^{dem}}{{\sum }_{i\in V}{d}_i^{dem}}}$$

(22)

where $s_i\left(t\right)$ denotes the state of node $i$ at time $t$ (with $s_i=2$ indicating failure), and $d_i^{dem}$ represents the demand associated with node $i$.

In the absence of real demand data, node-type-based weights are used as proxies. This metric reflects the proportion of demand that can still be satisfied under disruption.

(4) Recovery Time

Recovery time $T_{rec}$ is defined as the minimum number of time steps required for the system to recover to 90% of its initial performance level:

$$T_{rec}=\mathrm{m}\mathrm{i}\mathrm{n}\{t\mid E_{glob}(t)\ge 0.9E_{glob}(0\left)\right\} \#$$

(23)

This metric captures the resilience of the system in terms of recovery speed after cascading failures.

(5) Area Under Robustness Curve (AURC)

To provide an overall assessment of robustness throughout the attack process, the area under the robustness curve is defined as:

$$AURC={\int }_0^T{E}_{glob}\left(t\right) dt$$

(24)

In practice, this integral is approximated using numerical integration. A larger AURC indicates better overall robustness.

4. Simulation Verification and Analysis

4.1. Benchmarking the Proposed Node-Importance Metric

To isolate the incremental value of the proposed node-importance metric, a benchmarking experiment was conducted against four representative ranking methods: degree centrality, betweenness centrality, multiplex PageRank, and TOPSIS-style multilayer importance. For each method, nodes were ranked from high to low importance and then removed sequentially under a targeted-attack setting. The attack budgets were set to 5%, 10%, 15%, and 20% of the total nodes, respectively.

Table 4. Robustness indicators ($G$ and $\eta$) under different attack strategies and budgets.

Ranking Method	$\frac{\mathsf{\eta }}{{\mathsf{\eta }}_0}$ (5%)	$\frac{\mathsf{\eta }}{{\mathsf{\eta }}_0}$ (10%)	$\frac{\mathsf{\eta }}{{\mathsf{\eta }}_0}$ (15%)	$\frac{\mathsf{\eta }}{{\mathsf{\eta }}_0}$ (20%)	$\frac{\mathit{G}}{{\mathit{G}}_0}$ (5%)	$\frac{\mathit{G}}{{\mathit{G}}_0}$ (10%)	$\frac{\mathit{G}}{{\mathit{G}}_0}$ (15%)	$\frac{\mathit{G}}{{\mathit{G}}_0}$ (20%)	AUC (η)	AUC (G)
Degree Centrality	0.79	0.63	0.47	0.34	0.92	0.84	0.76	0.69	0.64	0.841
Betweenness Centrality	0.76	0.58	0.42	0.3	0.9	0.82	0.73	0.67	0.603	0.821
Multiplex PageRank	0.73	0.54	0.38	0.26	0.89	0.79	0.7	0.64	0.57	0.8
TOPSIS-based Method	0.69	0.49	0.33	0.22	0.87	0.76	0.68	0.62	0.53	0.78
Proposed Metric	0.64	0.43	0.28	0.18	0.84	0.72	0.64	0.6	0.485	0.75

summarizes the resulting largest connected component ratio ($G$), global efficiency $(\eta$), and the corresponding Area Under the Curve (AUC) values for each method. To ensure a fair comparison, all ranking methods were tested on the same coupled network instance under identical settings ($\alpha = 0.4, \beta = 1.2$).

To ensure a fair comparison, all ranking methods were tested on the same coupled network instance under identical cascading-failure settings. The safety tolerance coefficient ($\alpha = 0.4$) and limit coefficient ($\beta =1.2$), respectively, and the same load redistribution rule was adopted throughout the experiment. Therefore, the only difference among the compared methods lies in the attack sequence generated by different node-ranking strategies.

Following the robustness evaluation framework used in this paper, two indicators were employed: the largest connected component ratio and the global efficiency. A more effective node-importance metric should identify the truly critical nodes earlier; therefore, when used to generate the targeted-attack sequence, it is expected to cause a faster degradation of network robustness. To provide an aggregate comparison across attack budgets, the area under the robustness curve (AUC) was further calculated for each method. In this benchmark, a smaller AUC indicates stronger capability in identifying nodes critical to cascading failure propagation. The results are shown in Figure 7.

In addition to the above benchmarking experiment, the targeted attack generated by the proposed metric is further compared with random attacks to analyze the vulnerability evolution of the network under different disturbance modes.

4.2. The Impact of Attack Methods on Network Vulnerability

During the operation of the equipment storage and supply system, the failure causes of nodes can be classified into two categories: random failure and deliberate damage. Random failure corresponds to functional abnormalities of warehouse nodes or transportation hubs caused by non-human factors such as natural disasters, equipment aging, and communication disruptions. Targeted attacks, on the other hand, correspond to planned sabotage targeting key nodes such as human interference, cyber attacks, and transportation disruptions.

In simulation analysis, random attacks simulate the impact of unexpected events by randomly selecting nodes and rendering them invalid. Targeted attacks are based on the global importance ($H_i$) of nodes calculated in the previous text. Nodes are invalidated in order of importance from high to low to simulate purposeful targeted strikes.

Set the safety tolerance coefficient $\alpha =0.4$ and the limit coefficient $\beta =1.2$, and adopt the aforementioned load redistribution strategy based on the node-importance metric to conduct two types of attack simulations on the network respectively. The result is shown in Figure 8.

As can be seen from Figure 8, under the condition of targeted attack, the global efficiency of the network and the largest connected component ratio show a leapfrog downward trend as the number of node failures increases. When the number of failed nodes reaches 40, the global efficiency drops to 13.2% of the initial value, and the largest connected component ratio drops to 58%.

Under the condition of random attack, the decline in the index is relatively gentle. When the number of node failures reaches 70, the global efficiency still remains at 14.7%, and the largest connected component ratio is 49.3%. The reason lies in that in random attacks, the failed nodes are mostly low-importance nodes, and their failure will not have a significant impact on the overall circulation structure. However, in targeted attacks, which focus on destroying high-load hub nodes, local collapses will spread rapidly. The equipment transportation network has strong robustness under random failure, but shows high vulnerability under targeted attacks, verifying the rationality and sensitivity of the node-importance metric calculation model.

4.3. Analysis of Safety Tolerance Coefficient

The safety tolerance coefficient α reflects the margin size of the node’s design capacity relative to its initial load and is an important parameter for measuring the network redundancy capability.

To analyze the impact of the safety tolerance coefficient (α) on the robustness of the network, the attack method was set as a targeted attack, with the limit coefficient $\beta =1.2$. The changes in the global efficiency of the network and the largest connected component ratio under different α values were studied, as shown in Figure 9.

It can be known from the simulation results that as the number of attacked nodes increases, the overall performance index of the network shows a downward trend. When $\alpha <0.2$, the global efficiency and connectivity ratio of the network decline rapidly, and the vulnerability increases sharply.

When $0.2\le \alpha \le 0.4$, the downward trend slows down and the system has a certain resistance.

When $\alpha >0.4$, the network performance is basically stable, with only significant fluctuations under high-intensity attacks.

This indicates that moderately increasing the node safety tolerance coefficient (α) can effectively enhance the system’s anti-destruction capability, but excessive redundancy will cause resource waste and reduced efficiency.

Therefore, in the design of the equipment storage and supply system, appropriate redundant space should be reserved for core warehouses and backbone hubs in combination with the task load and operation level of nodes to balance reliability and economy.

4.4. Limit Coefficient Analysis

The limit coefficient β represents the ratio of the maximum load-bearing capacity of a node to its initial load, reflecting the load limit and stable boundary of the node under extreme conditions.

In the simulation, the attack mode is maintained as a targeted attack, and the safety tolerance coefficient (α) is set at α = 0.4. The changes in network robustness under different β values are studied, as shown in Figure 10.

The results show that as β increases, the overall invulnerability of the network is significantly enhanced. When β < 1.1, the global efficiency of the network and the largest connected component ratio decline rapidly, indicating that the limit capacity of nodes is insufficient to resist cascading failure propagation.

When β > 1.15, the vulnerability change slows down and the network performance remains stable.

It can be seen from this that enhancing the load-bearing capacity of key nodes can significantly reduce the risk of system cascading. In the actual equipment transportation system, priority should be given to strengthening the carrying capacity construction of central reserve depots, regional transfer hubs and trunk transportation nodes, so as to enhance the operational resilience of the system in the event of emergencies.

4.5. The Impact of Different Types of Nodes on Vulnerability

In a two-layer equipment transportation network, nodes can be classified into three categories: (1) Storage and supply nodes (central warehouse, regional warehouse, and forward warehouse); (2) Transportation nodes (railway, highway, aviation hubs, etc.); (3) Coupled nodes, that is, composite nodes that simultaneously undertake the functions of storage and supply as well as transportation.

To compare the impact of attacks on different types of nodes, α = 0.4 and β = 1.2 were set. Random and targeted attack strategies were adopted respectively to conduct failure simulations on various types of nodes. The results are shown in Figure 11.

The results show that the network performance declines most sharply after the failure of the coupled nodes, and both the global efficiency of the network and the connectivity ratio decrease rapidly. The failure of a single storage and supply node or transportation node has a relatively small overall impact.

This is because coupled nodes not only undertake the tasks of material reserve and distribution, but also are responsible for the connection and information exchange of cross-layer circulation, serving as the key hub of the inter-layer dependency structure. Therefore, in system planning and operation and maintenance, priority should be given to ensuring the safety and stability of coupled nodes, such as building redundant channels, backup nodes or cross-layer alternative lines and other measures.

4.6. The Impact of Load Redistribution Strategy on Network Vulnerability

To verify the rationality of the “load redistribution strategy based on node importance, status and capacity” proposed in this paper, it is compared and analyzed with three traditional strategies: the Uniform Allocation (UA); Reverse allocation method of neighbor nodes Residual Capacity-based Allocation (RC).

The simulation conditions were targeted attack, with α = 0.4 and β = 1.2. The results are shown in Figure 12.

As can be seen from the figure, the network vulnerability decreases the fastest under the Uniform Allocation (UA). The reason is that the difference in node capacity is not considered, which is prone to cause some nodes to be overloaded and trigger a secondary crash.

The robustness of neighbor reverse allocation and remaining capacity allocation has been slightly improved, but there is still a defect of ignoring the node-importance metric.

In contrast, the comprehensive weighting strategy proposed in this paper exhibits the minimum performance loss and the slowest degradation rate throughout the simulation process.

This indicates that in the propagation of cascading failures, if the node-importance metric, remaining capacity and real-time status can be considered simultaneously, the load can be reasonably allocated and the overall invulnerability of the system can be significantly enhanced.

In conclusion, the load redistribution mechanism is an important link affecting the robustness of the network. A reasonable redistribution strategy can effectively suppress the spread of failures and enhance the continuous operation capacity of the equipment transportation system in extreme environments.

4.7. Sensitivity Analysis Under Relaxed Modeling Assumptions

To further validate the robustness and generalizability of the proposed multilayer node-importance metric and cascading failure model, additional sensitivity experiments are conducted by relaxing several key modeling assumptions. Specifically, this study examines the impact of (i) weighted edges, (ii) directed flows, and (iii) heterogeneous node capacities. These experiments aim to evaluate whether the main conclusions—particularly the identification of critical nodes and the superiority of the proposed importance-capacity-state (ICS) redistribution strategy—remain consistent under more realistic conditions.

(1) Impact of Weighted Edges

In the baseline model, all edges are assumed to be unweighted. However, in real-world equipment support systems, transportation channels and supply routes often exhibit heterogeneous capacities or accessibility levels. To capture this effect, edge weights $w_{ij}$ are introduced to represent channel capacity or connection strength.

Three weighting schemes are considered:

(1) Uniform weights: $w_{ij}=1$, corresponding to the baseline case;

(2) Degree-based weights: $w_{ij}={\displaystyle \frac{k_i+k_j}{2}}$, reflecting higher capacity for connections between high-degree nodes;

(3) Random heterogeneous weights: $w_{ij}\sim U\left(0.5,1.5\right)$, simulating stochastic variability.

As shown in Figure 13, introducing edge weights changes the degradation trajectory only slightly. The degree-based weighting scheme yields the highest robustness among the three tested settings, whereas the random heterogeneous weighting scheme leads to a slightly faster decline. However, all three curves remain close overall, indicating that the proposed method is insensitive to moderate edge-weight perturbations and preserves stable robustness conclusions.

(2) Impact of Directed Flows

The baseline model assumes undirected edges. To better approximate real-world flow constraints, partial directionality is introduced into the network.

Specifically, the storage–supply layer is assigned directed edges representing upstream-to-downstream flows, while 20% and 30% of edges in the transportation layer are randomly converted to directed links. The results are shown in Figure 14.

However, the key conclusions remain unchanged. Coupled nodes continue to exhibit the highest vulnerability, and targeted attacks still result in significantly faster performance degradation than random failures. Figure 14 confirms that the relative ordering of robustness across attack scenarios remains stable. These findings indicate that the proposed model is not dependent on the undirected assumption and retains its validity under directional flow constraints.

(3) Impact of Heterogeneous Node Capacities

In the baseline model, node capacity parameters $\alpha$ and $\beta$ are assumed to be homogeneous. To reflect real-world variability, heterogeneous capacity distributions are introduced.

Two schemes are considered:

(1) Random assignment:

$${\alpha }_i\sim U\left(0.2,0.5\right),{\beta }_i\sim U\left(1.05,1.25\right)$$

(2) Type-based assignment:

Central depots > regional centers > forward warehouses

Figure 15 shows that the proposed ICS redistribution strategy maintains the best robustness performance across all scenarios, with the slowest decline in global efficiency. Compared with uniform and single-factor strategies, ICS more effectively suppresses cascading failures by jointly considering node importance, residual capacity, and operational state. Interestingly, the type-based capacity assignment slightly improves overall robustness, suggesting that differentiated resource allocation for critical nodes can further enhance system resilience in practical applications.

4.8. Multi-Metric Robustness Evaluation

To provide a comprehensive assessment of network robustness and to avoid potential bias arising from single-indicator evaluation, this study conducts a multi-metric robustness analysis under targeted attack scenarios. Four complementary metrics are considered: largest connected component ratio $G\left(t\right)$, global efficiency $E_{glob}\left(t\right)$, service-level proxy $SL\left(t\right)$, and the area under the robustness curve (AURC).

The dynamic evolution of these metrics is illustrated in Figure 16, where four redistribution strategies—Uniform Allocation (UA), Degree-based Allocation (DB), Residual Capacity-based Allocation (RC), and the proposed Importance-Capacity-State (ICS) strategy—are compared.

As shown in Figure 16a, all strategies exhibit a decreasing trend in the largest connected component ratio as the number of attacked nodes increases. However, the proposed ICS strategy consistently maintains a larger connected component, indicating stronger structural resilience compared to baseline methods. Figure 16b presents the evolution of global efficiency. It can be observed that efficiency degrades more rapidly than structural connectivity, reflecting the sensitivity of transport performance to network disruption. Among the compared strategies, ICS shows the slowest decline, demonstrating its effectiveness in preserving network functionality under cascading failures. Figure 16c illustrates the service-level proxy, which reflects the proportion of demand that can still be satisfied. The results show that ICS maintains the highest service level throughout the attack process, indicating that it not only preserves connectivity but also sustains system operational capability more effectively. Figure 16d provides a comparison of the AURC values. The ICS strategy achieves the largest AURC, indicating superior overall robustness across the entire attack process. In contrast, the UA strategy exhibits the smallest AURC, confirming its vulnerability under cascading failures.

4.9. Recovery Strategy Evaluation Under Budget Constraints

To evaluate recovery strategy effectiveness, this study implements a budget-constrained framework that compares the proposed importance-based strategy (ICS) against betweenness, k-shell, and random recovery through a 20-step sequential restoration process, measuring performance via the largest connected component ratio ($G$), global efficiency ($E_{glob}$), and service-level proxy ($SL$).

As shown in Figure 17, the proposed ICS strategy consistently outperforms the alternatives, exhibiting the highest connectivity ($G$), the fastest recovery of global efficiency ($E_{glob}$), and superior functional restoration ($SL$) by integrating structural and functional node importance. While betweenness-based recovery shows moderate effectiveness by targeting flow-central nodes, it and other baselines like k-core and random recovery underperform compared to ICS, underscoring the critical advantage of guided, multi-dimensional restoration in resource-constrained scenarios.

5. Conclusions

This paper investigates the equipment support system by constructing a two-layer equipment transportation network model. Utilizing multi-layer complex network theory, it analyzes the topological characteristics of this model and proposes a node-importance metric calculation method that comprehensively considers both intra-node layer importance and relative network layer importance. Based on this, a load-capacity coupled cascade failure vulnerability model is established. Through simulation experiments, the dynamic evolution patterns of network robustness under different disturbance factors are explored. The findings provide theoretical foundations and decision references for structural optimization, critical node identification, and emergency recovery strategy formulation in multilayer equipment transportation networks. Key conclusions are as follows:

(1)

The two-layer equipment transportation network exhibits typical scale-free properties characteristic of complex networks.

Node connections exhibit pronounced non-uniform distribution, with a small number of core nodes bearing primary storage, supply, and transportation responsibilities. Tight inter-node connections and dense information/material flow paths reveal strong clustering and inter-layer coupling. This finding provides structural guidance for optimizing equipment storage–supply system layouts and transportation route planning.

(2)

The proposed multi-layer node-importance metric shows stronger capability in identifying critical nodes than representative baseline ranking methods in the studied network.

By integrating network-level relative importance with node-level intra-layer influence, the proposed metric provides a cross-layer comparable ranking that is particularly suitable for functionally asymmetric and strongly coupled equipment support systems. Benchmarking experiments based on targeted attacks further show that the proposed metric leads to faster robustness degradation and smaller AUC values than degree, betweenness, multiplex PageRank, and TOPSIS-style multilayer ranking, indicating a stronger ability to identify cascade-sensitive nodes.

(3)

The cascading failure model based on node capacity and status reveals the evolution patterns of network vulnerability.

Simulation results indicate that network vulnerability under targeted attacks is significantly higher than under random attacks, where failure of only a few critical nodes can cause a sharp decline in overall system performance. Increasing both the safety tolerance coefficient and the limit coefficient enhances a node’s overload tolerance and reduces cascade failure risk. Coupled nodes serve as bridges for transport and information exchange between different layers; their damage leads to significant declines in network connectivity and efficiency. Therefore, targeted protection and redundancy configuration strategies should be adopted in practical systems.

(4)

Load redistribution strategies critically impact system robustness.

Comparative analysis reveals that traditional uniform or single-factor distribution methods fail to adequately account for node-importance metric and capacity constraints, potentially exacerbating load-capacity–state coupled cascading failure model. In contrast, the proposed comprehensive allocation strategy—based on node importance, capacity, and status—significantly enhances the system’s resilience to disturbances, demonstrating superior stability and controllability.

Furthermore, this study employs a hyper-adjacency matrix to construct a two-layer equipment transportation network model. Future research may further incorporate tensor modeling approaches to compare the applicability and computational efficiency of different multilayer modeling frameworks in dynamic evolution and cross-layer propagation. Concurrently, leveraging real-world equipment transportation data, in-depth studies can be conducted on the dynamic load evolution and recovery strategy optimization for various node types (e.g., warehousing centers, highway hubs, railway nodes) under multiple attack patterns. This will enhance the resilience and security assurance capabilities of multilayer equipment transportation systems.