Identification of Sparse Interdependent Edges in Heterogeneous Network Models via Greedy Module Matching

Abstract

The identification of interdependent edges plays a critical role in improving information propagation efficiency and enhancing network robustness in interdependent networks. However, existing methods exhibit significant limitations when identifying interdependent edges between networks with substantial differences in edge density. This paper proposes a greedy module matching-based method for sparse interdependent edge identification in similar-order heterogeneous networks. The method utilizes degree entropy and betweenness centrality as node characteristic values for sparse and dense networks, respectively. It first leverages structural differences between sparse and dense networks to determine the upper bound of interdependent edges. Then, a clustering algorithm is employed to identify modules in both networks that align with the estimated number of interdependent edges. Finally, a greedy algorithm is applied to infer interdependent edges between sparse and dense networks. The proposed method is validated using synthetic networks and power-communication networks, with network robustness and connection efficiency as evaluation metrics. Additionally, further validation is conducted through applications in problem–answer networks. Experimental results demonstrate that the proposed approach significantly improves the prediction of sparse interdependent relationships in heterogeneous complex networks and has broad applicability across multiple domains.

1. Introduction

In the study of complex network robustness, previous research has primarily focused on isolated single-layer networks [1,2]. In 2010, Buldyrev et al. [3] introduced an interdependent network model in Nature, revealing that mutually dependent networks are more susceptible to large-scale cascading failures when critical nodes malfunction. This study shifted academic attention from single-network analyses to the robustness of interdependent networks, aiming to mitigate cascading failure risks. To enhance the robustness of interdependent networks, the strategic design of interdependent edges has emerged as a central research focus. In recent years, researchers have made significant progress in network modeling, cascading failure analysis, and coupling strategies.

Traditional models of interdependent networks often simplify multiple networks into unweighted, undirected graphs and describe interdependencies using a fully one-to-one coupling mechanism [3,4]. While intuitive, this approach overlooks the intrinsic physical properties and heterogeneity of nodes. For instance, in power-communication networks, certain critical nodes exhibit higher resilience due to their importance and greater weight, making them more resistant to failures, whereas other nodes are more prone to failure due to overload or connectivity issues. Simplistic coupling mechanisms neglect these characteristics, resulting in imprecise robustness assessments. To address this limitation, some studies have incorporated physical attributes into network models. Yang et al. [5] extended betweenness-based methods to construct a more realistic nonlinear model, demonstrating that node weights and coupling relationships play a key role in enhancing robustness and attack resistance. Fang et al. [6] further proposed that interdependencies between networks need not be strictly one-to-one; instead, a partially interdependent coupling model provides a more flexible theoretical framework that better aligns with real-world interdependent systems.

Moreover, many studies have sought to enhance the robustness of interdependent networks by identifying key nodes [7,8], adding or removing network links [9,10], or optimizing the internal structures of individual network components [11,12]. However, these approaches often neglect coupling effects between networks. Early coupling strategies typically assumed that interdependent edges were randomly distributed across two networks to analyze their impact on system robustness [13]. While simple, random connections fail to capture intrinsic network structure characteristics, leading to inefficiencies and limiting real-world applicability. Researchers subsequently explored node centrality metrics, such as degree [14], betweenness centrality [15], and clustering coefficient [16], to optimize coupling mechanisms. High-centrality nodes are generally regarded as critical system nodes, and prioritizing connections to these nodes can significantly improve network connectivity and attack resilience. Results indicate that self-similar coupling strategies effectively enhance interdependent network robustness. Chen et al. [17] validated the positive correlation between average cluster size and system robustness, as well as the negative correlation between cluster count and defined cluster size thresholds. Dong et al. [18] examined the robustness of modular networks, identifying an optimal interconnection node ratio capable of sustaining greater structural damage without network collapse. Morteza et al. [19] investigated the implicit optimal trade-off relationships in balancing subnetworks. Zou et al. [20] highlighted that fragile inter-layer dependency links are primary contributors to cascading failures, underscoring the importance of accurate characterization of interdependencies to improve system robustness.

Several studies have explored more sophisticated coupling strategies, as shown in

Table 1. Comparison of existing methods.

No.	Authors	Network Model	Method Features	Problem Addressed
1	Wang et al. (2018) [21]	Random and scale-free interdependent networks	Neighbor-prioritized connection strategy	Enhancing robustness under cascading failure
2	Dong et al. (2019) [22]	Spatially embedded scale-free networks	Five coupling modes and hybrid coupling	Optimizing dependency link radius to improve attack resilience
3	Liu et al. (2022) [23]	Power-communication coupled networks	Multi-objective optimization and correlation coefficients	Balancing trade-offs between robustness and functionality
4	Chattopadhyay et al. (2017) [24]	Scale-free networks	Optimal interconnection design based on degree sequences	Maximizing robustness against random and targeted attacks
5	Marashi et al. (2021) [25]	IEEE power grid	Correlation analysis and neural network prediction	Fault propagation prediction and interdependence identification
6	Akbarzadeh & Katsikas (2021) [26]	MDSM graph model	Multi-order dependency indicators	Analyzing interdependent structures in complex CPS
7	Turalska & Swami (2021) [27]	BTW sandpile model	Greedy control strategy	Controlling cascading failures and exploring optimal coupling relations
8	Zhang et al. (2024) [28]	Edge-coupled random and scale-free networks	Reinforced inter-layer edges and percolation theory	Preventing abrupt collapses and enhancing system robustness

. For example, the Nearest-Neighbor Prioritized Coupling (NPC) method [21], as well as a hybrid coupling model that incorporates spatial constraints into network topology, introduces Nearest Neighbor Coupling (NN) and Local Degree-Degree Similarity Coupling (LD) [22]. Liu et al. [23] investigated the trade-off between robustness and functionality in power network-coupled systems, proposing an optimization method based on degree centrality and betweenness to balance these competing demands. Chattopadhyay et al. [24] designed an interconnection structure based on node degree, optimizing interdependencies between networks to maximize resilience against random attacks. Using percolation theory, they analyzed optimal designs for fully and partially interdependent structures. Marashi et al. [25] employed correlation metrics and heuristic causal analysis to identify interdependencies among components in cyber-physical systems, aiming to reduce unexpected failures and mitigate malicious attacks. Akbarzadeh et al. [26] introduced a new method based on the Modified Dependency Structure Matrix (MDSM) to analyze interdependencies within critical infrastructure cyber-physical systems. Turalska et al. [27] proposed a greedy control strategy based on the Bak–Tang–Wiesenfeld sandpile model to mitigate cascading failure risks in interdependent networks. By coupling a network layer in the supercritical state with a layer in the subcritical state, the overall system becomes more stable, significantly reducing the occurrence of large-scale collapses. Zhang et al. [28], employing random and scale-free network models, investigated the percolation transition behavior in edge-coupled interdependent networks. Their study focused on the impact of reinforced inter-layer connections on system robustness. By introducing a certain proportion of reinforced inter-layer edges, they derived analytical expressions for the size of the giant connected component, the finite component distribution, and the percolation critical threshold.

Although existing studies have attempted to incorporate physical attributes into network modeling and explore more complex interdependent edge connections, these approaches often fail to adequately account for density variations and structural differences between heterogeneous networks. In particular, traditional coupling strategies struggle to effectively balance dependencies when coupling sparse and dense networks, leading to limited improvements in robustness. This paper builds upon traditional interdependent network models and proposes a sparse interdependent edge identification method to enhance the robustness of interdependent networks constructed from heterogeneous subnetworks.

The key innovations of this method include the following:

• Leveraging mid-level modular structural features as the foundation for interdependent edge identification, particularly in similar-order heterogeneous networks with significant edge density differences.
• Determining the range of interdependent edges based on structural differences between heterogeneous subnetworks and employing a greedy matching algorithm to iteratively identify sparse interdependent edges.

2. Materials and Methods

The proposed method targets heterogeneous complex networks with similar node counts but significant differences in edge density, aiming to identify the optimal interdependent connection pattern between sparse and dense networks. As illustrated in Figure 1 (with supplementary illustrations provided in Appendix A), the method first computes node characteristic values for both sparse and dense networks. Based on these values, the modular structures of the two networks are identified separately. Next, the upper bound of interdependent edges between the two networks is determined. Finally, the optimal interdependent pattern is identified using modular structural characteristics.

2.1. Calculation of Node Characteristic Values Based on Network Topology

To evaluate node importance, we compute the node characteristic values separately for sparse and dense networks. For nodes in sparse networks, degree entropy is employed as an importance indicator. Degree entropy [29] is derived from the concept of information entropy, describing the diversity or uncertainty in node degrees within a network. The degree entropy of a node i, denoted as Hi, is defined as

$$H_i=-{\displaystyle \sum _{j\in N_i}\frac{w_{ij}}{{\displaystyle \sum _{k\in N_i}{w}_{ik}}}}\log ({\displaystyle \frac{w_{ij}}{{\displaystyle \sum _{k\in N_i}{w}_{ik}}}}),$$

(1)

In this equation, N_i represents the set of nodes connected to node ii, w_ij denotes the edge weight between nodes i and j, and ${\displaystyle \sum _{k\in N_i}{w}_{ik}}$ is the total weight of all edges associated with node i. ${\displaystyle \frac{w_{ij}}{{\displaystyle \sum _{k\in N_i}{w}_{ik}}}}$ represents the normalized edge weight of edge (i, j), indicating the relative strength of node i’s connection to node j within its local neighborhood.

For nodes in dense networks, betweenness centrality [30], closeness centrality [30], and degree centrality [31] are adopted as the feature indicators. The node feature value TD is defined as shown in Equation (2):

$$TD=\phi BN(v)+\varsigma DG(v)+\lambda CO(v),$$

(2)

where BN(v)denotes the betweenness centrality of node v, DG(v) represents its degree centrality, and CO(v) corresponds to its closeness centrality. The parameters φ, ς, and λ indicate the relative weights assigned to each feature. Their values can be flexibly determined according to the characteristics of different networks, subject to the following constraints: (1) each parameter must lie within the range (0, 1); and (2) the sum of the weights must satisfy φ + ς + λ = 1.

2.2. Network Module Identification Based on Clustering

The modular structure of the sparse network is identified based on node characteristic values, and interdependent edges are determined at the module level, where the number of modules corresponds to the number of interdependent edges. To compute the maximum number of interdependent edges, we analyze the density characteristics of both sparse and dense networks. The maximum number of interdependent edges, K, is defined as:

$$K={\displaystyle \frac{(\alpha N_g+\beta N_c)\times E_g}{E_c}},$$

(3)

where N_g represents the number of nodes in Network A, N_c represents the number of nodes in Network B, E_g denotes the number of edges in Network A, and E_c denotes the number of edges in Network B. Parameters α and β represent the node characteristics of the sparse and dense networks, respectively. Its value is determined based on the difference in node counts between the two networks, serving to adjust the disparity in the number of nodes across subnetworks. The parameter must satisfy the following two conditions: (1) α + β = 1; and (2) 1 < αN_g/βN_c < 2, where N_g and N_c denote the number of nodes in the respective.

Equation (3) is derived from a proportional assumption linking the number of modules in sparse networks to potential node-level dependencies and considers the role of edge abundance in dense networks as a mechanism for mitigating redundant connectivity. By weighting and summing the node quantities of both networks and multiplying by the edge count of the sparse network, while using the edge count of the dense network as a “dilution factor,” the expression estimates the maximum number of interdependent edges.

This formula was heuristically abstracted from extensive observations of simulated network structures and demonstrates strong structural prediction consistency across varying density regimes. Under fixed node quantities, the influence of E_g and E_c on the resulting value of K is illustrated in Figure 2. As shown in the figure, when E_c is large and E_g is relatively small, the sensitivity of K to node ratio variation becomes lower, indicating favorable predictive controllability.

After computing the number of interdependent edges, we extract graph-based features from the networks. An unsupervised clustering algorithm (k-means) [32] is then applied in the feature space to group networks with similar topological characteristics. This clustering approach is distinct from modularity-based community detection methods, which operate directly on network structure. The clustering process is as follows:

Step 1: Initialization

Randomly select Q initial cluster centers from the node set, where Q is a user-defined parameter representing the upper bound of potential modules. Typically, Q ≥ K, where K is the estimated optimal number of modules determined using modularity maximization or silhouette analysis.

Step 2: Distance Calculation and Assignment

For each node, compute the similarity (or distance) to each cluster center based on a feature vector derived from node characteristics. To highlight the structural differences between networks of varying density, the feature construction is layer-specific: in sparse networks, degree entropy is employed as the dominant feature to capture local structural diversity, while in dense networks, betweenness centrality serves as the primary feature to reflect global structural properties. Nodes are then assigned to the module of the nearest cluster center, forming an initial partition into Q modules.

Step 3: Cluster Center Update

For each module, recompute the cluster center as the mean of the feature vectors of all member nodes. In our implementation, this reduces to computing the average entropy for nodes within the module, which reflects the average information uncertainty of the module.

Step 4: Iteration and Convergence

Repeat Steps 2 and 3 iteratively, updating node assignments and cluster centers, until one of the following conditions is met:

(a) the variation in cluster centers falls below a predefined convergence threshold ϵ or (b) a maximum number of iterations T_max is reached.

This iterative process ensures that the final clustering is stable and captures inherent structural patterns in the network.

Step 5: Output and Module Determination

The algorithm outputs a final partition where each node is classified into one of Q modules based on feature similarity. If over-clustering is detected (e.g., small or redundant modules), a post-processing step merges modules based on modularity gain or inter-module entropy similarity.

Through clustering, network nodes are divided into multiple categories, forming the class set $H=\left\{H_1,H_2,\dots H_{K_H}\right\}$. Here, H_K represents the class k in the network, and K_H denotes the number of clusters in Network A. After module identification is completed, the largest node characteristic value within each module is designated as the module’s representative characteristic value, as defined in Equation (4):

$${\lambda }_{\max }^{(H_k)}=\max \left(\left\{{\lambda }_i^{\left(H_k\right)}\right\}\right),$$

(4)

where ${\lambda }_i^{\left(H_k\right)}$ represents the characteristic value of the node i in the k-th class $H_k$. This maximum characteristic value is used as a measure of node importance within the cluster.

2.3. Identification of Interdependent Patterns Based on the Greedy Algorithm

After determining the number of interdependent edges, the greedy algorithm is employed to identify interdependencies between sparse and dense networks based on their module characteristic values. The greedy algorithm is an approach that makes locally optimal choices at each step in order to achieve a globally optimal solution [33]. We use the eigenvalue sorting results as the selection criterion. In the pairing process, we first select the modules with the largest eigenvalues from the sparse network layer $H_i$ and the dense network layer $C_k$, forming an initial module-to-module pair. Within module $H_i$, we randomly select a node iii based on the eigenvalue ranking, with the selection probability governed by the distribution P. Then, from module $C_k$, we select the node $j$ with the highest eigenvalue. The resulting dependency edge is $e=(i,j)$. This edge $e=(i,j)$ is added to the dependency edge set $E_C=\left\{e_1,e_2,\dots e_m\right\}$, where $m=\min (H_k,C_k)$, and the process continues until all pairings are completed. The probability of selecting a node is proportional to its characteristic value, defined as

$$P(i)={\displaystyle \frac{H_i}{{\displaystyle \sum _{i^{\prime }\in H_i}{H}_{i^{\prime }}}}},$$

(5)

where $H_i$ represents the eigenvalue of node $i$ in module $H_i$, and ${\displaystyle \sum _{i^{\prime }\in H_i}{H}_{i^{\prime }}}$ is the total sum of the eigenvalues of all nodes in module $H_i$.

We incorporate a constraint-aware adjustment mechanism into the greedy matching process to enhance the diversity of node selection, with a core principle that each node is selected only once. This strategy aims to improve the structural resilience of the system under adversarial conditions. By restricting the repeated matching of high-centrality nodes, the approach effectively mitigates functional redundancy, negative synergy, and the concentration of vulnerabilities within modules, thereby strengthening the system’s robustness against both targeted attacks and random perturbations.

2.4. Network Robustness Evaluation

To assess the effectiveness of the proposed method, we select natural connectivity [31], geodesic vulnerability [32], network efficiency [33], and the largest connected component fraction as robustness evaluation metrics. Natural connectivity measures the overall connectivity between nodes in a graph, describing the direct connectivity degree between a node and all other nodes, as defined in Equation (6).

$$\eta =\ln \left({\displaystyle \frac{1}{N}}{\displaystyle \sum }_{i=1}^N{e}^{{\lambda }_i}\right),$$

(6)

where η represents natural connectivity, λ_i is the ith eigenvalue of the network adjacency matrix, and N denotes the number of nodes in the network.

Geodesic vulnerability is employed to evaluate the fragility of nodes or edges within a network. It quantifies the importance of nodes or edges in influencing overall network connectivity, as defined in Equation (7).

$$\kappa =1-{\displaystyle \frac{{\displaystyle \sum _{i\ne j}1/d_{ij}^{LC}}}{{\displaystyle \sum _{i\ne j}{\left(1/d_{ij}^{BC}\right)}^{\prime }}}},$$

(7)

where κ represents geodesic vulnerability, $d_{ij}^{LC}$ denotes the geodesic distance between node pairs after the removal of a node, and $d_{ij}^{BC}$ represents the geodesic distance between node pairs in the original network.

Network efficiency is a metric used to evaluate the speed and cost of information propagation within a network. It provides insight into the overall performance and effectiveness of the network and is defined in Equation (8).

$$E={\displaystyle \frac{1}{N\left(N-1\right)}}{\displaystyle \sum _{v_i\ne v_j\in V}\frac{1}{d_{ij}}},$$

(8)

where E represents network efficiency, and d_ij denotes the shortest path length between nodes v_i and v_j.

The Largest Connected Component Fraction is an important metric in network analysis, used to quantify the size of the largest connected component relative to the total number of nodes in the network.

$$R={\displaystyle \frac{N^{\prime }}{N}},$$

(9)

where R represents the LCC fraction, representing the ratio of the size of the largest connected component to the total number of nodes in the network, and N′ denotes the number of nodes contained in the largest connected subgraph.

2.5. Interdependent Edge Configuration Efficiency Index

To quantitatively evaluate the effectiveness of sparse interdependent edge recognition methods, this paper proposes the Interdependent Edge Configuration Efficiency Index (IECEI), denoted as Γ. This index integrates both the connectivity efficiency of interdependent networks and the structural costs incurred in the construction and maintenance of interdependent edges. The metric is defined in Equation (10).

$$\Gamma ={\displaystyle \frac{\frac{1}{N_{PC}\left(N_{PC}-1\right)}{\displaystyle \sum _{v_i\ne v_j\in V}\frac{1}{d_{ij}}}}{a{\displaystyle \sum _{(u,v)\in E_{PC}}{\displaystyle \sum _{u\ne v}\frac{{\sigma }_{uv}\left(e\right)}{{\sigma }_{uv}}+(1-a){\displaystyle \sum _{(u,v)\in E_{PC}}\left(\frac{{\displaystyle \sum _{k=1}^{N_P}{P}_{uk}}}{10\left(N_P-1\right)}+\frac{{\displaystyle \sum _{l=1}^{N_C}{P}_{vl}}}{10\left(N_C-1\right)}\right)}}}}}$$

(10)

where

• N_P represents the number of nodes in the sparse network.
• N_C represents the number of nodes in the dense network.
• N_PC denotes the total number of nodes in the interdependent network.
• d_ij is the shortest path length between nodes i and j in the interdependent network.
• V represents the set of nodes in the interdependent network.
• A is a weight parameter.
• E_PC denotes the set of interdependent edges in the interdependent network.
• u and v represent nodes in the sparse and dense networks, respectively.
• (u, v) represents an interdependent edge.
• σ_uv is the total number of shortest paths between nodes u and v in the interdependent network.
• σ_uv(e) represents the number of shortest paths passing through edge (u, v).

The structural cost consists of two components: the operating load cost and the construction cost. The operating load cost is measured by the betweenness centrality of the interdependent edge, which indicates the extent to which the edge participates in global path routing. A higher betweenness centrality reflects stronger transmission dependency within the system, implying a greater risk of failure and increased maintenance overhead. The construction cost is assessed based on the structural importance of the nodes connected by the interdependent edge within their respective subnetworks. If an interdependent edge connects core nodes, it may exacerbate fault propagation, increase deployment complexity, and elevate maintenance costs. In summary, the IECEI reflects the system efficiency achieved per unit of structural cost. A higher IECEI value indicates that the corresponding configuration of interdependent edges achieves greater efficiency with lower deployment and risk-related costs, thereby offering better economic performance and robustness.

3. Results and Discussion

To verify the effectiveness of the proposed method, we apply it to synthetic interdependent network models, power-communication interdependent network models, and question–answer interdependent network models for interdependent edge identification. These experiments are based on the assumption of a strongly coupled interdependent network. Our selection of synthetic networks (40 nodes per layer) and real systems (IEEE 30-bus, Cisco 31-node topology) reflects trade-offs between simulation tractability and interpretability. While these networks are indeed smaller than some large-scale simulations, they capture key structural features observed in interdependent infrastructures.

3.1. Validation on Synthetic Interdependent Networks

3.1.1. Identification of Interdependent Edges in Synthetic Networks

In the synthetic interdependent network model, the sparse network adopts a scale-free topology [34], consisting of 40 nodes and 55 edges, while the dense network follows a small-world topology [35], containing 40 nodes and 460 edges.

Applying the proposed method to this network, the sparse network is partitioned into ten modules, and the dense network is divided into eight modules, resulting in the identification of eight interdependent edges. The results are illustrated in Figure 3.

In Figure 3, the left panel (“Small-World”) and the right panel (“Scale-Free”) display the module identification results for nodes in the dense and sparse networks, respectively. The labels “Cluster” and “NO”. denote the module index and the node index within each module, respectively. Blue nodes represent the identified interdependent nodes, while red edges connecting these nodes indicate the identified interdependent edges.

The scale-free small-world synthetic interdependent network is illustrated in Figure 4, where the upper layer represents the small-world dense network, and the lower layer corresponds to the scale-free sparse network. The red dashed lines indicate the eight identified interdependent edges connecting nodes between the two networks.

We computed the degree centrality, betweenness centrality, and closeness centrality values for all nodes in the dense network and extracted the top 10 nodes with the highest values for each centrality metric, as shown in

Table 2. Top 10 nodes ranked by centrality metrics in the dense network.

Sorting	DC	BC	CC
1	234	234	234
2	201	211	211
3	211	230	201
4	209	209	230
5	210	233	209
6	214	236	233
7	215	214	236
8	227	215	214
9	230	232	215
10	231	231	231

. In the table, DC, BC, and CC denote the node rankings based on degree centrality, betweenness centrality, and closeness centrality, respectively. As observed, the top 10 nodes identified by the three metrics exhibit substantial overlap; however, the ranking based on degree centrality differs more noticeably from those based on the other two metrics. Accordingly, we set the weights φ, ς, and λ to 0.9, 0.1, and 0, respectively.

3.1.2. Robustness Analysis of Synthetic Interdependent Networks

To validate the effectiveness of the proposed method, we applied DED, NPC, and RL methods to identify interdependent edges in the scale-free small-world interdependent network. Subsequently, we subjected the constructed interdependent network to random attacks and targeted attacks. The robustness of the network was evaluated using natural connectivity and network efficiency, as introduced in Section 2.4. The results are presented in Figure 5.

In Figure 5, the labels “SF-WS-DED,” “SF-WS-NPC,” “SF-WS-RL,” and “SF-WS-CBG” respectively represent the robustness metrics of the synthetic interdependent network under attack when constructed using the following:

• Degree-Electric Degree (DED) coupling method;
• Nearest-Neighbor Prioritized Coupling (NPC) method;
• Random Linking (RL) method;
• The proposed Greedy Strategy method (CBG).
• Figure 5 illustrates the changes in robustness under different attack scenarios;
• Figure 5a: Variation in natural connectivity under random node attacks;
• Figure 5b: Variation in natural connectivity under random edge attacks;
• Figure 5c: Variation in network efficiency under high-betweenness node attacks;
• Figure 5d: Variation in network efficiency under high-betweenness edge attacks.

The robustness of the four methods under random node attacks and random edge attacks is shown in Figure 6a,b. The natural connectivity trends indicate that CBG significantly slows down the decline in natural connectivity. As observed in Figure 6a, after the fourth random node attack, CBG maintains 58.2% natural connectivity, whereas DED, NPC, and RL retain only 23.2%, 25.4%, and 30.3%, respectively. In later attack stages, subnetwork disconnections suppress failure propagation, leading to smaller reductions in natural connectivity. Figure 6b further illustrates that after the fifth random edge attack, CBG preserves 12% more natural connectivity than the other methods. Under targeted attacks on high-betweenness nodes and edges, the robustness of the four methods is depicted in Figure 6c,d. CBG effectively mitigates the impact of high-betweenness node or edge failures on the overall network. Figure 6c shows that after the 16th high-betweenness node attack, CBG maintains network efficiency at 28.6%, while NPC and DED coupling strategies drop to 24.1% and 19.8%, respectively. Due to randomized link distribution, RL exhibits better attack resistance than NPC and DED, keeping network efficiency at 27.8%, though still slightly below CBG. Figure 6d confirms that under high-betweenness edge attacks, CBG maintains stable network efficiency, showing minimal decline, whereas the other three methods experience significant efficiency reductions.

Based on these findings, a comprehensive evaluation of robustness metrics under different attack types is conducted, utilizing median, mean, and minimum values for normalized data distribution analysis, as illustrated in Figure 6. Figure 6a compares robustness metrics across methods under random node attacks. Figure 6b compares robustness metrics under random edge attacks. Figure 6c compares robustness metrics under high-betweenness node attacks. Figure 6d compares robustness metrics under high-betweenness edge attacks.

From the mean value analysis, the proposed method exhibits higher average robustness across different attack scenarios compared with the other three methods, maintaining stronger resilience throughout the attack process. From the minimum value analysis, the proposed method demonstrates greater ability to sustain the lowest performance level under various attack scenarios. This advantage is particularly evident under high-betweenness node attacks, highlighting the method’s effectiveness in preventing rapid network collapse under intentional attacks. From the median value analysis, the proposed method consistently achieves higher median network efficiency across all attack scenarios, maintaining strong robustness in most attack phases.

3.1.3. Evaluation of Connection Efficiency in Synthetic Interdependent Networks

To further validate the effectiveness of the proposed method, the IECEI introduced in Section 2.5 is employed to evaluate the interdependent networks constructed using four different methods. In the experiment, the parameter α = 0.5 is set, and the calculation results are presented in Figure 7. As the number of interdependent edges increases, the IECEI values exhibit different trends. An increase in interdependent edges leads to higher system complexity, and the sensitivity of different methods to edge additions varies accordingly. By comparing the IECEI curves of the four methods, we can clearly observe the effectiveness variation of each method as network complexity increases.

As shown in Figure 7, the CBG method consistently maintains eight interdependent edges, while the other three methods exhibit a gradual increase in interdependent edge count. When the number of interdependent edges reaches eight, the IECEI performance of CBG is comparable to that of RL, but CBG outperforms NPC and DED by approximately 3%. This advantage arises because the CBG strategy deliberately biases early connections toward nodes with high degree or betweenness centrality. Although such links may involve slightly higher structural costs than random placements, they provide disproportionate efficiency gains since central nodes dominate communication flows. Consequently, in the low-link regime, CBG avoids inefficiency and instead achieves rapid performance improvement due to this centrality bias. As the number of interdependent edges increases, the IECEI values show different variation patterns across methods. When the interdependent edge count reaches 20, the IECEI decreases by approximately 30.7%, 29.3%, and 31.6%, respectively. However, when the edge count rises to 40, the IECEI exhibits a diminishing rate of change. These experimental results reveal a strong correlation between the number of interdependent edges and the IECEI. Reducing unnecessary interdependent edges not only lowers network complexity but also improves system efficiency. By optimizing the selection of interdependent edges, the CBG method effectively enhances efficiency while preserving network robustness, demonstrating superior performance compared with the other methods.

3.2. Real-World Power-Communication Network

3.2.1. Identification of Interdependent Edges in Power-Communication Networks

We constructed two real-world networks, a power network and a communication network, using IEEE 30-bus system data [36] and Cisco network data [37]. The power network consists of 30 nodes and 41 edges, with edge weights representing the active power flow, which reflects the actual power transmission across the transmission lines in the power network. If the power flow through a particular line is high, it may indicate that the line occupies a critical position, carrying a substantial amount of energy flow, and thus significantly influencing the robustness and overall efficiency of the power grid. The communication network consists of 31 nodes and 349 edges, classified as an unweighted dense network. The edges between the nodes simply represent basic connectivity, without considering the strength or capacity of communication between them. To integrate the two layers for efficiency or shortest path calculations, we employed a cross-layer normalization approach. This normalization process is based on the maximum power transmission capacity of the power network, ensuring that both the power and communication layers have consistent dimensions and scales during cross-layer analysis. Applying the proposed method, the networks were partitioned into eight modules, identifying eight interdependent edges. The results are illustrated in Figure 8. The left panel of Figure 8 represents node assignments within modules in the IEEE 30-bus power network. The right panel represents node assignments within modules in the Cisco communication network. Blue squares denote selected interdependent nodes, while red lines indicate identified interdependent edges.

The obtained power-communication interdependent network model is shown in Figure 9, where the upper layer represents the Cisco communication network, and the lower layer corresponds to the IEEE 30-node power network. The red dashed lines indicate the interdependent edges, with a total of eight interdependent edges.

3.2.2. Robustness Analysis of the Power-Communication Interdependent Network

We applied CBG, DED, NPC, and RL methods to identify interdependent edges in the real-world power-communication network. Subsequently, the constructed interdependent network was subjected to random attacks and targeted attacks. The robustness of the network was evaluated using four metrics introduced in Section 2.4: natural connectivity, geodesic vulnerability, the LCC fraction, and network efficiency. These are commonly used in general network analysis, but they do not capture power system-specific factors. As a result, the conclusions drawn from these metrics should be considered preliminary for power system analysis. The results are shown in Figure 10.

In Figure 10, the labels “IEEE30-SK-DED,” “IEEE30-SK-NPC,” “IEEE30-SK-RL,” and “IEEE30-SK-CBG” represent the robustness metrics of the real interdependent network under attack when constructed using the degree-electric degree (DED) coupling method, the nearest-neighbor prioritized coupling (NPC) method, the random linking (RL) method, and the proposed greedy strategy method (CBG). Figure 10 illustrates the changes in robustness under different attack scenarios: Figure 10a shows the variation in natural connectivity under random node attacks, Figure 10b shows the variation in network efficiency under high-betweenness node attacks, Figure 10c shows the variation in geodesic vulnerability under high-betweenness node attacks, Figure 10d shows the variation in natural connectivity under random edge attacks, Figure 10e shows the variation in network efficiency under high-betweenness edge attacks, and Figure 10f shows the variation in the Largest Connected Component Fraction under high-betweenness edge attacks.

According to Figure 10a, after the third attack, the natural connectivity of the DED and RL methods decreased by 82.5% and 89.7%, respectively. Under the same conditions, the decline rates of NPC and CBG were nearly identical at approximately 32.2%. However, after the fourth attack, the declining trend of NPC became significantly higher than that of CBG. As shown in Figure 10b, the initial differences among methods were minimal. However, after 16 high-betweenness node attacks, the network efficiency of CBG remained at 25.4%, while NPC and DED dropped to 10.8% and 6%, respectively. Figure 10c indicates that after the eighth attack, CBG maintained a geodesic vulnerability of approximately 20%, whereas DED, NPC, and RL recorded values of 16%, 14.4%, and 17%, respectively. According to Figure 10d, after the third attack, CBG outperformed the other three methods significantly. In Figure 10e, after 20 attacks, CBG still retained 42% network efficiency, whereas NPC and DED experienced sharp declines after the 12th and 14th attacks, respectively. This suggests that using degree alone as a criterion for network interdependency offers weaker protection against high-betweenness edge failures, leading to accelerated network collapse and the fastest efficiency reduction. Although NPC and DED performed better than CBG during early attack stages, they exhibited greater efficiency deterioration in later stages under high-betweenness edge attacks. Simulation results indicate that modular partitioning disperses interdependencies, preventing localized failures from propagating extensively. Even if high-betweenness edges are removed, remaining edges can still distribute loads and preserve network efficiency. As seen in Figure 10f, the LCC fraction in NPC begins to decline after the fifth attack, while DED and RL experience declines from the sixth attack onward. However, CBG maintains its stability until the seventh attack, demonstrating stronger resilience compared with the other three strategies under equal attack conditions.

Based on these results, we conducted a comprehensive robustness evaluation of the four methods under different attack types in the power-communication interdependent network. Using median, mean, minimum, and inverse range values, we performed normalized data distribution analysis, as shown in Figure 11.

From the mean value analysis, the CBG method exhibits higher robustness metrics across different attack scenarios compared with the other three methods, particularly in node attack scenarios, where it demonstrates superior resilience. In contrast, NPC and DED methods fail to adequately distribute connections among key nodes, resulting in lower mean robustness values. From the minimum value analysis, the CBG method effectively maintains the lowest performance level under extreme attack conditions, with its advantage being particularly evident in high-betweenness node attack scenarios. From the median value analysis, the CBG method consistently achieves higher median network efficiency across all attack scenarios, maintaining strong robustness throughout most attack phases. As shown in Figure 11e, under high-betweenness edge attacks, CBG and RL methods exhibit similar mean, minimum, and median values, but CBG shows less variation between the initial and final attacks, leading to more stable overall network efficiency.

3.2.3. Analysis of Network Connection Efficiency in the Power-Communication Interdependent Network

We applied the IECEI introduced in Section 2.5 to evaluate real-world interdependent networks constructed using the four methods. In the experiment, the parameter α = 0.5 was set, and the calculation results are presented in Figure 12. The x-axis represents the number of interdependent edges, while the y-axis denotes the IECEI values. The different curves correspond to the four methods—CBG, DED, NPC, and RL—each distinguished by a unique color.

In the experiment, the CBG method consistently maintained eight interdependent edges, while the other three methods (DED, NPC, and RL) gradually increased the number of interdependent edges. Despite the lower number of interdependent edges, the IECEI value of the CBG method remained significantly higher than those of the other three methods at eight interdependent edges. Specifically, when eight interdependent edges were established, the IECEI value of CBG exceeded those of DED, NPC, and RL by 21%, 19%, and 12%, respectively. As the number of interdependent edges increased to 20, the IECEI values of the other methods declined significantly, falling below that of CBG by 28%, 31%, and 31%, respectively. When the interdependent edge count reached 30, the downward trend of the IECEI values in the other methods began to stabilize. These experimental results demonstrate that the CBG method achieves higher efficiency while preserving network robustness, proving more effective than the other methods even when using only eight interdependent edges.

3.3. Identification of Interdependent Edges in the Question–Answer Network

Intent keyword recognition in question texts is a crucial research direction in intelligent question–answering systems, as certain intent keywords do not appear explicitly in the question but are indirectly reflected through interdependent relationships. By analyzing the interdependencies between the question–answer network models, the semantic associations between them can be examined to identify intent keywords. Therefore, we construct an answer network model using historical answer data and a question network model using subsequent question data. The proposed method is then employed to identify interdependent edges between the two networks, treating the interdependent words in the answer network as the intent keywords of the question.

This study utilizes the QuAC (Question Answering in Context) dataset [38], jointly released by New York University and Facebook AI Research in 2018, to advance research on conversational question–answering systems. The dataset is based on Wikipedia articles, containing over 14,000 dialogues and more than 100,000 question–answer pairs. The dataset construction relies on large-scale conversational QA scenarios, compiling dialogue records from multiple domains, ranging from everyday conversations to specialized knowledge topics. During dataset preparation, natural language processing techniques were applied to structure dialogue data, extracting questions and corresponding answers, and annotating contextual information to ensure data integrity and consistency. Additionally, the dataset includes speaker role information and sentiment labels, enriching its multi-dimensional features.

The QuAC dataset is characterized by its complexity and diversity in conversational QA. It not only contains direct question–answer pairs but also provides contextual dialogue information, enabling models to better understand discourse structure and logical relationships. Furthermore, the dataset includes various question types, ranging from factual inquiries to reasoning-based questions, posing significant challenges for model training. The sentiment labels and role information also serve as valuable resources for sentiment analysis and speaker identification.

Following the approach outlined in [39], we construct a complex network model for text data. Each word in the text is treated as a node, and semantic relations between words are represented as edges, specifically co-occurrence edges. The co-occurrence edge addition process involves defining an adjacency radius R and connecting all words within the adjacency radius as nodes with edges. To ensure that the constructed text network model is suitable for the proposed CBG, i.e., meeting the conditions of similar node counts and significantly different edge densities, in this study, R = 4, and an adjacent window slides across the text to incorporate all edges. During sliding, if an edge already exists in the network, its weight is incremented by 1. Upon adding co-occurrence edges, an undirected weighted complex network model is obtained. Further processing is conducted: In the question network model, edges with weights below two are removed, and nodes with degree 0 are deleted. In the answer network model, nodes with degree 0 are removed.

The final structural attributes of the question–answer network models are presented in

Table 3. Structural attributes of the question network model and answer network model.

	Number of Nodes	Number of Edges	Density	Average Clustering Coefficient	Average Closeness
Question Network Model	378	663	0.0093	0.2558	0.2389
Answer Network Model	379	1773	0.0248	0.6282	0.2399

, showing that the number of nodes is nearly identical, but the density of the answer network is significantly higher than the question network, classifying it as a heterogeneous network of the same order.

Prediction of Interdependent Edges

Following the methodology outlined in Section 2.1, we computed the feature values for each node in both the question network and the answer network. Based on Section 2.2, we determined the number of modules and identified the modular structures within both network models.

Table 4. Top 10 nodes with the highest feature values and their corresponding modules in the question-and-answer network models.

Question Network Model			Answer Network Model
Node	Entropy	Module	Node	Entropy	Module
happened	3.8937	63	KARVELAS	0.154788357	7
album	3.449835	114	GRAHAM	0.149007681	13
article	3.221585	32	CLAIMED	0.148506505	2
interesting	3.142876	33	ALBUM	0.11580243	10
work	2.872722	99	FAMILY	0.104835427	1
aspects	2.770941	31	TV	0.103442594	23
world	2.667937	177	LATER	0.096974915	5
win	2.610799	90	ROCK	0.091662983	8
second	2.58643	126	COURT	0.086465684	18
successful	2.533686	125	VISSI	0.083930992	18

presents the top 10 nodes with the highest feature values along with their corresponding module assignments.

Following the methodology outlined in Section 2.3, the interdependent relationships between the question network and the answer network were identified, as presented in

Table 5. Identification results of interdependent edges between the question-and-answer networks.

No.	Question Network Node	Answer Network Node	No.	Question Network Node	Answer Network Node
1	happened	KARVELAS	13	important	ORGANIZATION
2	album	GRAHAM	14	influence	HIRED
3	article	CLAIMED	15	war	PLAYED
4	interesting	ALBUM	16	say	MCMAHON
5	work	MANHUNT	17	play	AGE
6	aspects	NEW	18	make	POLICE
7	world	LATER	19	happen	VOW
8	win	ROCK	20	john	NIKOS
9	second	COURT	21	receive	PEPYS
10	successful	PLAGUE	22	awards	CERTAINLY
11	role	CONVICTED	23	life	UNUSUAL
12	songs	THOMPSON

. In Table 5, lowercase letters denote nodes in the question network, while uppercase letters represent nodes in the answer network. Each row lists a pair of interdependent nodes, indicating their correspondence between the two networks.

To evaluate the effectiveness of interdependent edge identification, we treated interdependent nodes as intent keywords associated with user queries. A set of queries was randomly extracted from the QuAC dataset, and for each query, interdependent terms—identified via sparse edge analysis—were designated as its corresponding intent keywords. We then submitted both the original queries and their keyword-augmented counterparts to Bing AI and retrieved the generated responses. Each response was evaluated against the ground-truth answers provided in the QuAC dataset by calculating recall scores.

Illustrative example: Original query: “What happened in 1983?” Identified intent keyword: “KARVELAS” Response without intent keyword: “The video game crash of 1983 led to a major recession in the North American gaming industry”. Response with intent keyword: “In 1983, Greek musician Nikos Karvelas released his second studio album, Taxi… He married Anna Vissi, a renowned Greek singer, in May”. Ground-truth answer: “In May 1983, she married Nikos Karvelas”.

The computed recall without the intent keyword was 28.57%, whereas the recall with the added keyword reached 85.71%.

Recall serves as a critical metric for assessing a model’s retrieval completeness, especially in tasks such as information retrieval, text classification, and named entity recognition. In total, we evaluated 22 queries, and the recall performance—visualized in Figure 13—demonstrates that keyword-augmented inputs consistently yielded higher recall scores compared with unaugmented queries. These findings suggest that the sparse interdependency recognition method effectively leverages historical data to infer intent-relevant keywords for enhanced semantic comprehension. To reduce subjective bias, we calculated Cohen’s Kappa coefficient [40] to assess inter-rater agreement. In the evaluation of 22 queries, the Kappa value reached 0.82, indicating a high level of consistency between reviewers.

4. Conclusions

This paper proposes a greedy module-matching approach for sparse interdependent edge identification in heterogeneous networks of the same order, effectively addressing the challenge of optimal sparse interdependent edge recognition in networks with similar node counts but significant density differences. By incorporating network physical properties, we compute node features and identify modular structures for weighted sparse networks and unweighted dense networks, determining the optimal number of interdependent edges. A greedy algorithm is then used to integrate local and global network features, accurately identifying and optimizing interdependent edge selection, thereby enhancing the performance of interdependent networks.

Applying this method to scale-free small-world artificial interdependent networks and real-world power-communication interdependent networks, experimental results demonstrate that the proposed approach significantly improves network robustness under both random and targeted attacks, with particularly pronounced advantages in targeted node attacks.

Additionally, to further assess the effectiveness of the method, we introduce IECEI and conduct comparative analyses, confirming that the proposed method effectively balances system robustness and connection efficiency. For artificial interdependent networks, under a sparse interdependent connection model, the connection efficiency value of CBG exceeds DED, NPC, and RL by approximately 3%, while surpassing the one-to-one full interdependent connection model by about 50%. For real-world power-communication interdependent networks, under sparse interdependent edge connections, CBG improves connection efficiency by 21%, 19%, and 12% compared with DED, NPC, and RL, respectively, while surpassing the one-to-one full interdependent edge connection model by approximately 56%.

To further verify the applicability of this method, we extend its use to intent keyword recognition in intelligent question–answering systems. The results indicate that when intent keywords identified by the proposed method are added to the original query, the answer relevance improves by 25%, demonstrating the method’s effectiveness in extracting sparse interdependencies for intent recognition.

Despite its demonstrated effectiveness, the proposed method has certain limitations. It assumes the availability of reliable modular structures in both networks, which may be sensitive to the choice of clustering algorithm and the estimation of module numbers. Moreover, while the greedy matching strategy effectively reduces node redundancy by enforcing single-selection constraints, it may still favor structurally proximate nodes—such as adjacent nodes or those within the same cluster—potentially compromising topological diversity and system robustness. To address this, future work will incorporate topological analysis metrics such as modularity, node dispersion, and embedding-space coverage to quantitatively assess the spatial distribution of selected nodes and their impact on system performance. Additionally, the method currently operates on static network topologies; its extension to dynamic or time-evolving interdependent networks remains unexplored. Although node features were integrated into the clustering and matching processes, their definition and selection may influence the results, and further investigation is needed to ensure robustness across diverse datasets. Finally, the method’s performance may be affected by noise or missing data in real-world scenarios, which warrants further study. We plan to explore multi-metric fusion strategies to better balance diversity, stability, and computational efficiency in future research.

Future work will focus on addressing these limitations and extending the applicability of the method. Specifically, we plan to explore adaptive versions of the approach that can handle dynamic interdependent networks with time-varying structures. We also aim to enhance the robustness of the method under noisy or incomplete data by integrating denoising techniques or probabilistic models. Moreover, we intend to apply this method to other critical domains such as transportation-social interdependencies, bioinformatics networks, and financial systems, to further evaluate its generalizability and practical utility. Finally, integrating explainable clustering techniques and interactive visualization tools may improve the transparency and interpretability of the edge selection process, especially in real-world decision-making contexts.