A network key node identification method based on global redundant paths

Abstract

The application relates to a network key node identification method based on global redundant paths, and belongs to the technical field of network science and complex network analysis. The method provided by the application simultaneously fuses global redundant path information (captured through an explosive percolation model) of a macro scale and community structure information (obtained through a community detection algorithm) of a meso scale, and comprehensively describes the functional importance of nodes in network topology from multiple scales. Specifically, the explosive percolation model evaluates the influence of node removal on network connectivity from a global perspective, and community decomposition reveals the functional role of nodes in a local community from a meso perspective, and the two are complementary to each other and overcome the limitations of single-scale analysis. Experimental verification on nine real network datasets shows that, compared with the existing optimal method, the method improves the identification accuracy by 10.99%, and the overall evaluation effect is improved by 46.68%, thereby verifying the superiority of the multi-scale fusion strategy.

Description

Technical Field

[0001] This invention belongs to the fields of network science and complex network analysis technology, and specifically relates to a method for identifying key network nodes based on globally redundant paths. Background Technology

[0002] 1. Overview of Network Robustness and Critical Node Identification Network robustness refers to a network's ability to maintain its basic functions and structural integrity when faced with disturbances such as node or edge failures, malfunctions, or malicious attacks. This concept holds a central position in the field of complex network research and is widely used in the security assessment and optimization design of various practical network systems, including transportation networks, power networks, communication networks, and social networks. Quantitative analysis of network robustness helps to understand the mechanisms by which network structure supports functional stability and provides a theoretical basis for the formulation of network protection strategies.

[0003] Key Node Identification is a core issue in network robustness assessment and optimization. This technique aims to identify a sequence of nodes within a network that decisively impacts network connectivity and functional integrity; the absence of these nodes will lead to the greatest possible degradation of network connectivity. Key Node Identification results can provide decision support for prioritized protection, resource deployment, and risk warning of network systems. In practical applications, accurate identification of key nodes is crucial for ensuring the security of critical infrastructure networks, suppressing the spread of harmful information in social networks, and enhancing the stability of biological networks.

[0004] In the research and practice of critical node identification, the relative size of the Giant Connected Component (GCC) is often used as a core indicator to measure network connectivity. GCC refers to the number of nodes contained in the largest connected subgraph remaining in the network after node removal. By monitoring the decay of GCC during the sequential removal of nodes, the effectiveness of different critical node identification methods can be quantitatively evaluated. The pioneering research of Albert et al. and Holme et al. found that scale-free networks are highly robust to random node removal, meaning that randomly removing a small number of nodes does not significantly affect the overall connectivity of the network; however, such networks exhibit significant vulnerability to targeted attacks based on degree centrality, meaning that preferentially removing nodes with high degrees can lead to rapid network fragmentation. This discovery reveals a deep correlation between network topology characteristics and robustness, laying an important foundation for subsequent research on critical node identification methods.

[0005] 2. A Key Node Identification Method Based on Centrality Indicators Centrality-based critical node identification methods are the most widely used approach in network robustness assessment. The core idea of ​​these methods is that different nodes in a complex network have varying degrees of impact on network connectivity due to their different topological locations. By designing appropriate centrality metrics, the importance of nodes is quantified and ranked, thereby identifying the sequence of critical nodes. Several representative centrality metric methods are introduced below.

[0006] Degree centrality is the most basic measure of centrality, defined as the number of edges directly connected to a target node, i.e., the number of a node's direct neighbors. Degree centrality reflects the importance of a node from the perspective of local topology, assuming that nodes with more direct connections contribute more to network connectivity. This method is computationally simple and highly interpretable, but it struggles to effectively distinguish the true importance of nodes when dealing with networks with rich, redundant path structures.

[0007] Betweenness centrality measures a node's mediating role in the shortest paths between all pairs of nodes in a network. Specifically, a node's betweenness centrality is defined as the proportion of shortest paths passing through that node to the total number of shortest paths in the network. Betweenness centrality assesses a node's importance from the perspective of global information flow propagation, arguing that nodes located on a large number of shortest paths play a crucial role in information transmission within the network. This metric is suitable for identifying "bridge" nodes in a network, but its computational complexity is high, limiting its application in large-scale networks.

[0008] K-shell decomposition is a method for evaluating node importance based on the core-edge hierarchy of a network. This method recursively and iteratively removes nodes with a degree less than k and their edges, dividing the network into different levels of shells. Nodes with higher shell values ​​are considered to be located in the core of the network and play a more important role in maintaining the integrity of the network structure. While the K-shell method can effectively identify core node groups in a network, its ability to distinguish the importance of nodes within the same shell layer is limited.

[0009] Leverage centrality is a centrality measure that considers the connectivity of a node relative to its neighbors. This metric quantifies a node's importance by comparing its degree to the average degree of its neighbors: a node is considered to have a high leverage centrality value if its degree is significantly higher than the average degree of its neighbors. This method overcomes, to some extent, the shortcomings of traditional degree centrality, which ignores neighbor structure information.

[0010] Cluster Rank quantifies node importance by comprehensively considering both the node's clustering coefficient and the influence of its neighbors. The clustering coefficient measures the tightness of connections between a node's neighbors, reflecting the clustering characteristics of the local region where the node resides. Cluster Rank incorporates the clustering coefficient as a penalty factor in its importance calculation, arguing that the actual importance of nodes in highly clustered local structures may be overestimated. This method, to some extent, reflects the impact of redundant connections on node importance measurement.

[0011] Information entropy centrality, based on entropy theory in information theory, assesses a node's importance by aggregating the degree information of its neighbors. This method calculates the information entropy value of the degree distribution of neighboring nodes to eliminate redundant information provided by obscure neighbors, thus obtaining a more accurate ranking of node importance. Its core idea is that the more uniform the degree distribution of neighboring nodes, the richer the structural information carried by that node.

[0012] The cycle ratio measures the impact of redundant connections on node importance by examining the cycle structure in a network. This method counts the number of shortest cycles containing the target node and constructs a cycle matrix to characterize the degree of node participation in the network's redundant path structure. Nodes with higher cycle ratios are typically located in multiple redundant loops, and their removal has a more significant impact on the network's local connectivity.

[0013] Multi-Spanning Tree-based Degree Centrality (MSTDC) assesses node importance by eliminating metric biases caused by redundant connections. This method randomly selects a small number of root nodes from the network to construct multiple spanning trees, and then aggregates the degree values ​​of nodes across these trees. Since spanning trees eliminate loop structures in the original network, MSTDC effectively avoids interference from redundant relationships in the measurement of node importance, thus providing a more accurate ranking of node importance.

[0014] 3. The role of community structure in network analysis Community structure, also known as community topology, is an important topological feature prevalent in complex networks. It refers to the fact that nodes in a network can be divided into several groups (or communities), with dense connections between nodes within a group and sparse connections between nodes between groups. Community structure provides crucial structural information in distinguishing the heterogeneity of nodes and edges, and is an important perspective for understanding the principles and functional mechanisms of network organization.

[0015] From a network topology perspective, nodes within the same community typically exhibit high internal connectivity, forming tightly connected local substructures; while different communities are interconnected only through a few connecting edges. This "dense inside, sparse outside" connection pattern makes community structure an important tool for understanding network hierarchical organization. In identifying key nodes, community structure information helps distinguish the different roles nodes play in the network: nodes within a community primarily participate in local information exchange, while nodes connecting different communities act as bridges in global information transmission.

[0016] The Weak Ties Hypothesis provides crucial theoretical support for understanding the importance of inter-community connections. This hypothesis posits that while the number of edges connecting different communities (i.e., weak ties) is relatively small, they play an irreplaceable and critical role in facilitating the cross-community dissemination of information and achieving global network connectivity. In social networks, information or influence can spread globally through bridging edges connecting different communities; in infrastructure networks, cross-community connections are often the critical paths for maintaining overall network connectivity. Therefore, in the process of identifying key nodes, fully considering the node's community location and its cross-community connectivity characteristics is of great value for accurately assessing the node's global importance.

[0017] Redundant connections are closely related to the formation of community structures. Studies have shown that the more redundant connections a network has, the clearer and more prominent its community structure tends to be. Redundant connections create rich local coupling structures within a community, making the community boundaries more defined and amplifying the heterogeneity differences between nodes and edges. This community heterogeneity enhanced by redundant connections provides additional structural criteria for distinguishing the importance of nodes at different locations in key node identification.

[0018] Several mature community detection algorithms have been developed for practical detection and analysis of community structures. The Louvain algorithm is a greedy community detection method based on modularity optimization. It achieves community partitioning by iteratively optimizing the network's modularity function, offering advantages such as high computational efficiency and applicability to large-scale networks. The Infomap algorithm, based on the random walk principle in information theory, discovers community structures by minimizing the encoding length describing the network's random walk trajectories. The Label Propagation algorithm is a community detection method based on semi-supervised learning. It achieves community partitioning by simulating the propagation process of labels between network nodes, exhibiting near-linear time complexity and applicability to ultra-large-scale networks.

[0019] 4. Seepage Theory and Network Robustness Analysis Percolation theory is a classic theoretical framework in statistical physics for studying connectivity phase transitions in random media, and it has been widely applied in recent years to the robustness analysis of complex networks. This theory describes the evolution of network connectivity as nodes or edges gradually increase or decrease, from the perspective of phase transitions and critical phenomena.

[0020] In traditional percolation models, networks start from an empty graph state and gradually add nodes or edges, studying the critical process of network evolution from a locally disconnected state to the emergence of a macroscopically connected whole (i.e., a giant connected component). As the connection density increases, the network undergoes a percolation phase transition at a certain critical threshold, suddenly resulting in a giant connected component covering a finite proportion of the network's nodes. Traditional percolation theory provides an important theoretical tool for understanding the formation mechanism of network connectivity, and its analytical framework can be extended to the identification of key nodes in network robustness assessment.

[0021] Explosive percolation is a significant extension of traditional percolation theory. Its core idea is to introduce a competitive edge selection mechanism during network evolution to suppress the rapid merging of giant connected components. Unlike the gradual increase in connectivity in traditional percolation, explosive percolation selects the optimal edge from multiple candidate edges according to specific rules at each step, resulting in abrupt phase transitions. This research reveals the significant impact of edge addition order and selection mechanisms on network connectivity evolution, providing a new perspective for understanding the orderly formation of network structures.

[0022] From a methodological perspective, the critical node identification problem in network robustness assessment can be viewed as a reverse evolution of the percolation process. Forward percolation studies the growth of network connectivity as nodes or edges are gradually added, while critical node identification studies the decay of network connectivity as nodes are gradually removed. This reverse percolation perspective provides a unified theoretical framework for analyzing the structural response of networks under attack or failure. Through reverse percolation analysis, the differences in the impact of different node removal strategies on network connectivity can be quantitatively characterized, thus providing theoretical guidance for the design and evaluation of critical node identification methods.

[0023] 5. Redundant paths and network robustness A redundant path is a network in which there are multiple distinct paths connecting any two nodes. The quantity and quality of redundant paths are key factors determining network robustness and play a central role in network robustness assessment and critical node identification research.

[0024] From a network robustness perspective, abundant redundant paths mean that the connections between nodes in the network have multiple backup characteristics. When some nodes or edges fail, information or resources can still be transmitted through alternative paths, thus maintaining the basic functions of the network. Highly redundant networks typically exhibit high clustering coefficients and low average shortest path lengths, known as small-world properties. For example, social networks often have significant small-world properties, allowing most nodes to be reached from any other node with relatively few hops. These properties not only facilitate efficient communication between nodes but also significantly enhance the overall robustness of the network. Adding edges to the original network to increase the number of redundant paths is widely recognized as an effective means of improving network connectivity and enhancing its robustness.

[0025] However, the existence of redundant paths poses a significant challenge to centrality-based key node identification methods. When a large number of redundant connections exist in a network, traditional methods such as degree centrality and betweenness centrality exhibit obvious limitations in identifying truly influential nodes. Redundant connections provide multiple alternative paths for related nodes, forming a tightly coupled local structure (called local coupling). In this local coupling structure, multiple nodes are interconnected through abundant redundant connections, sharing a large number of neighbors and overlapping influence ranges. This makes these nodes exhibit similar importance scores under traditional centrality measures, making it difficult to effectively distinguish their true differences in importance.

[0026] Of particular note is that nodes in locally coupled structures often share the same K-kernel value, but their actual importance can vary significantly. Traditional centrality metrics treat nodes and edges in a network as homogeneous elements, ignoring the amplification effect of node heterogeneity caused by redundant connections. Networks with more redundant connections often have clearer community structures. This heterogeneity information generated by the combined effects of redundant connections and community structure, if rationally utilized in key node identification, is expected to significantly improve identification accuracy. Therefore, accurately identifying the key nodes that truly affect network robustness in complex networks rich in redundant paths has become a crucial technical problem urgently needing to be solved in the field of network science.

[0027] Existing technologies for identifying critical nodes in complex networks have the following four main shortcomings in applications such as network robustness assessment and critical infrastructure protection: 1. The lack of a unified multi-scale network structure analysis method makes it difficult to comprehensively assess the importance of nodes. Existing critical node identification methods typically assess node importance at a single scale, focusing either on micro-scale node attributes (such as degree centrality, betweenness centrality, and other local or semi-local indicators) or relying on macro-scale network statistical features (such as network diameter and average path length). These methods fail to effectively integrate micro-neighborhood information, meso-level community structure information, and macro-level global path information within the same analytical framework. This single-scale analysis leads to one-sided node importance assessment results: methods relying solely on micro-attributes struggle to capture a node's strategic position within the global network topology, while methods relying solely on macro-level statistics fail to reflect local structural differences at the node level. In particular, existing technologies lack a collaborative analysis mechanism that uses meso-level community structure as a bridge connecting micro-level node attributes and macro-level global paths, making it difficult to effectively identify truly globally influential critical nodes and thus limiting the comprehensiveness and accuracy of network robustness assessments.

[0028] 2. Traditional centrality metrics ignore the heterogeneity of nodes and edges, leading to interference from redundant paths in the evaluation results. Traditional centrality methods such as degree centrality, betweenness centrality, and K-kernel treat all nodes and edges in a network as equally important during computation, ignoring the heterogeneity that is prevalent in real-world networks. However, in real networks, redundant connections provide multiple reachable paths for involved node pairs, forming a special local structure (i.e., local coupling) where nodes within this local coupling have similar or identical K-kernel values, but their actual importance can vary significantly. Although methods such as Cycle Ratio and MSTDC have begun to consider the impact of redundant relationships on node importance in recent years, these methods still fail to distinguish the differences in importance between different edges and nodes during computation, failing to effectively utilize the heterogeneity information provided by community structure. Furthermore, excessive redundant connections often lead to more complex community structures, further amplifying the differences between nodes and edges. This causes traditional centrality indices to produce significant biases in networks with dense redundant paths, making it impossible to accurately identify key nodes that have a decisive impact on network connectivity.

[0029] 3. Greedy strategies struggle to account for inter-node synergy. Traditional key node identification methods based on centrality metrics essentially employ a greedy strategy, evaluating node importance sequentially from highest to lowest single centrality metric value. This greedy ranking approach fails to fully consider the overlapping and synergistic effects of node influence. In real-world complex networks, the combined influence of multiple nodes is often not simply the sum of their individual influences: when two nodes have highly overlapping neighbor sets or share the same redundant path, removing one node alone may have a significant impact, but removing both may have a much smaller impact than expected. The prevalent presence of multiple redundant paths in networks further weakens the local effects of individual node evaluations, making greedy ranking methods prone to getting trapped in local optima when identifying node combinations. Since key node identification is essentially a combinatorial optimization problem, greedy strategies cannot effectively solve for the globally optimal node combinations, thus limiting identification accuracy.

[0030] 4. Lack of a systematic identification framework based on globally redundant paths Most existing spatial reasoning methods rely on end-to-end local inference mechanisms, lacking an explicit analytical framework oriented towards the global network structure. Specifically, existing methods cannot explicitly evaluate and quantify the distribution of redundant paths between nodes, community boundary locations, and global connectivity within the entire network context. Most centrality indicators only focus on the local topology around a node or path information within a limited range, lacking a systematic mechanism to examine the contribution of a node to redundant paths in the entire network from a global perspective. This neglect of global structural information makes it difficult for existing methods to accurately measure the changing trend of network connectivity after node failure, and also fails to effectively utilize the crucial role of community boundary nodes in maintaining cross-community connectivity. Therefore, existing technologies have significant shortcomings in recognition accuracy, evaluation stability, and result interpretability, making it difficult to meet the needs of application scenarios with high reliability requirements for recognition results, such as critical infrastructure protection and cybersecurity assessment. Summary of the Invention

[0031] (a) Technical problems to be solved The technical problem to be solved by this invention is to provide a multi-scale network key node identification method, which overcomes the limitations of traditional methods that only evaluate the importance of nodes from a single scale, ignore the heterogeneity of network structure, and lack global optimization capabilities, and achieves accurate identification of key nodes with global influence in complex networks and effective evaluation of network robustness.

[0032] (II) Technical Solution To address the aforementioned technical problems, this invention provides a method for identifying critical network nodes based on globally redundant paths, comprising the following steps: First, by community detection and boundary region analysis, the network is decomposed into a community structure with internal connectivity features, and the set of boundary nodes across communities is identified, thereby achieving spatial dimensionality reduction for the key node identification task. Secondly, by using explosive seepage modeling based on global redundant paths and community detection and boundary region analysis results, a competitive node addition mechanism is constructed at the community sub-layer level. By simulating the abrupt phase transition behavior in the gradual evolution of the network, a candidate set of key nodes inside each sub-graph is generated. Finally, by reordering the identification results, and based on the optimization criterion of maximizing the unit disintegration benefit, the generated key node candidate set is reordered globally to generate a reordered key node identification sequence.

[0033] (III) Beneficial Effects Compared with the prior art, the present invention has at least the following beneficial effects: (1) The accuracy advantage of multi-scale information fusion.

[0034] Existing techniques (such as BPD, MinSum, and CoreHD) assess node importance only at a single scale (microscopic node attributes or macroscopic global statistical features), neglecting the mesoscopic community structure information prevalent in complex networks. This results in limited identification accuracy in networks with significant community structures. The proposed method in this invention simultaneously integrates macroscopic-scale global redundant path information (captured through an explosive percolation model) and mesoscopic-scale community structure information (obtained through a community detection algorithm), comprehensively characterizing the functional importance of nodes in the network topology from multiple scales. Specifically, the explosive percolation model evaluates the impact of node removal on network connectivity from a global perspective, while community decomposition reveals the functional role of nodes in local communities from a mesoscopic perspective. The two methods complement each other, overcoming the limitations of single-scale analysis. Experimental validation on nine real-world network datasets shows that this method improves identification accuracy by 10.99% and overall evaluation performance by 46.68% compared to the best existing methods, validating the superiority of the multi-scale fusion strategy.

[0035] (2) Significant improvement in global optimization capability.

[0036] Traditional greedy methods (such as BPD and MinSum) select locally optimal nodes at each step. However, due to the nonlinear interactions between nodes in complex networks, locally optimal decisions are difficult to accumulate into a globally optimal solution. This invention introduces an explosive seepage model, modeling the critical node identification problem as a competitive connected component merging process. This model considers the global impact of future multi-step merging during the node addition stage, achieving a global perspective on node importance assessment through four node selection rules: the rule of minimizing the maximum connected component increment ensures priority control of large-scale connected component expansion; the rule of minimizing the increase in the number of connected components avoids premature network structure splitting; and the rules of merging priority and isolation priority balance selection strategies under different topological scenarios. This global optimization mechanism fundamentally overcomes the local optima deficiency of traditional greedy strategies, and can more effectively control information propagation and structural stability in the network while retaining the same number of critical nodes.

[0037] (3) Significant improvement in computational efficiency.

[0038] Existing global optimization methods (such as BPD) have high time complexity due to the need for iterative computation across the entire network, making them difficult to scale to large-scale networks. This invention employs a "divide and conquer" strategy. First, it uses the Louvain community detection algorithm to decompose the network into several communities, transforming the global key node identification problem into local subproblems on multiple community subgraphs. Each subgraph is much smaller than the original network, allowing for independent solutions at the community level. Simultaneously, the identification and preprocessing of community boundary nodes ensure the complete preservation of cross-community connection information. The overall time complexity of this method is approximately O(log n). (in (where the total number of network nodes is 1), significantly outperforming the computational complexity of traditional global methods. Experimental results demonstrate that this method can efficiently handle large-scale networks containing tens of thousands of nodes, exhibits good scalability, and is suitable for online real-time key node identification scenarios.

[0039] (4) Wide applicability and robustness.

[0040] This method demonstrates stable recognition performance across different types and sizes of networks. Experiments on model networks (including BA scale-free networks, ER random networks, and WS small-world networks) show a 25.44% improvement in recognition accuracy and a 44.22% improvement in overall performance compared to the current state-of-the-art methods. Experiments on nine real-world networks (covering multiple fields such as social, biological, technical, and infrastructure) show a 10.99% improvement in recognition accuracy and a 46.68% improvement in overall performance. Furthermore, in a systematic comparison with state-of-the-art (SOTA) methods such as BPD (belief propagation), MinSum (minimum sum algorithm), CoreHD (kernel decomposition), GNDR (network disintegration method), and FINDER (graph neural network-based method), this method achieves optimal or near-optimal performance on the vast majority of test networks and evaluation metrics, validating the broad applicability and robustness of this method. Detailed Implementation

[0041] To make the objectives, contents, and advantages of the present invention clearer, the specific embodiments of the present invention will be described in further detail below with reference to examples.

[0042] This invention proposes a Community Node Percolation (CNP) method for robustness assessment of complex networks. This method takes the topological information of the complex network as input and transforms the global topological information of the original network into an ordered sequence of critical node identifications through three core technical steps: community detection and boundary region analysis, explosive seepage modeling based on global redundant paths, and reordering of identification results. This provides a unified and computable technical solution for network security protection, infrastructure resilience assessment, and network structure optimization.

[0043] In this method, the community detection and boundary region analysis module decomposes large-scale complex networks into community structures with high-density internal connections and identifies the set of boundary nodes across communities, thereby achieving spatial dimensionality reduction for the key node identification task. The explosive seepage modeling module based on global redundant paths constructs a competitive node addition mechanism at the community sub-layer level based on the processing results of the community detection and boundary region analysis module. By simulating the abrupt phase transition behavior during the gradual evolution of the network, it generates a candidate set of key nodes within each sub-graph. The identification result reordering module performs a global consistency reordering of the generated key node candidate set based on the optimization criterion of maximizing unit disintegration benefits, thereby improving the accuracy of the key node identification order.

[0044] Compared with existing key node identification methods based on a single centrality metric or local network information, this method achieves the following technical advantages by introducing global topological constraints of community structure, competitive node selection mechanism of explosive percolation model, and reordering optimization strategy based on connected component contribution: First, it decomposes the global key node identification task to the community sublayer level, significantly reducing the computational complexity in large-scale network scenarios; Second, it captures the structural role of nodes in network connectivity by measuring global redundant paths, avoiding the limitation of traditional centrality metrics that only focus on local topological features; Third, the reordering mechanism optimizes the identification order of key nodes by quantifying the marginal contribution of each node to global connectivity, making the ranking results more in line with the actual needs of network robustness assessment.

[0045] 1. Community Detection and Boundary Area Analysis Module The community detection and boundary region analysis module is the first core technical component of this invention. Its function is to utilize the modular structural characteristics of complex networks to decompose the large-scale key node identification task in the global scope into multiple relatively independent community sub-layer level tasks, thereby reducing the computational complexity of the problem, and quickly locating the bridging structure that has an important impact on the global connectivity of the network through priority analysis of boundary nodes.

[0046] (1) Club test This invention selects the Louvain algorithm, optimized based on modularity, as the fundamental method for community detection. Modularity is a quantitative indicator of the quality of network community partitioning, defined as the ratio of the difference between the actual number of edges and the expected number of edges within a community to the total number of edges in the network. Given an undirected graph... ,in For a set of nodes, Let be the set of edges. , For dividing the network into The scheme for dividing the associations, i.e. the results of the association division. C k Represents the k-th club, where each club is a set of nodes, with a modularity of [missing information]. The calculation formula is:

[0047] in, For the image The adjacency matrix elements, when the node With nodes When there are edges between them ,otherwise ; and They are nodes and nodes The degree; Represents a node The tag associated with the organization; For indicator functions, when The value is 1 if it is true, and 0 otherwise.

[0048] The Louvain algorithm uses a greedy iterative strategy to optimize the modularity index. The algorithm flow is as follows: Step 1: Initialization Phase. Each node in the network is considered an independent community; that is, the initial number of communities equals the number of nodes. ,at this time , .

[0049] Step 2: Local Optimization Phase. This involves sequentially traversing each node in the network. Try moving it to the community to which all its neighbor nodes belong, and calculate the increment of modularity after each move. If there exists such that The movement of the node will... Move to make The largest club; otherwise, keep the node. The current community affiliation remains unchanged. Repeat this process until no node movement can further improve modularity.

[0050] Step 3: Network Aggregation Phase. Each existing community is aggregated into a super node. The edges connecting communities are aggregated into weighted edges between super nodes, with the weight of each edge equal to the total number of original edges between the two communities. This constructs a new aggregated network.

[0051] Step 4: Iterative convergence check. Repeat steps 2 and 3 until the network's modularity no longer increases significantly, at which point the algorithm terminates.

[0052] The computational complexity of Louvain's algorithm is... It exhibits good computational efficiency in large-scale sparse network scenarios and is suitable for complex network analysis tasks with a node scale of up to millions.

[0053] (2) Community Boundary Node Identification After the community testing is completed and the network's community division results are obtained. Subsequently, this invention further identifies the set of boundary nodes for a community. A community boundary node is defined as: if a node... There is at least one neighbor node (Right now Belongs to and Different communities are called nodes. For the club Boundary nodes.

[0054] In a formal sense, the club set of boundary nodes Defined as:

[0055] Among them, V\C i This means removing C. i The set of nodes other than the internal nodes; The union of the boundary node sets of all communities constitutes the global boundary node set. :

[0056] This is the output result of the community boundary node identification step; Boundary nodes play a bridging role in maintaining global network connectivity. Specifically, edges connecting different communities are called bridge edges, and their set is denoted as . :

[0057] The two endpoints of a bridging edge are called boundary nodes. Boundary nodes play a crucial role in the global connectivity of a network because there are usually multiple redundant paths between nodes within a community to maintain local connectivity, while the connectivity between communities is highly dependent on a limited number of bridging edges and their endpoint nodes. Therefore, prioritizing the analysis of boundary nodes during the critical node identification process can quickly assess the vulnerability characteristics of the network at the modular structure level.

[0058] By analyzing community boundary regions, this invention decomposes the task of identifying key nodes in the original network into the following two levels: First, the set of boundary nodes The nodes in the graph are directly selected as high-priority key node candidates because the failure or removal of these nodes may cause disconnection between the corresponding community subgraphs. Second, for each club The key identification task for internal non-boundary nodes is restricted to the community subgraph. (Depend on This is performed within the scope of the subgraph induced by the nodes and edges within the community, thereby encompassing the global scope. The problem of scale is reduced to multiple dimensions Sub-problems of scale.

[0059] (3) Determination of community subgraphs exceeding the threshold This invention introduces an identification threshold. This serves as a parameter for determining whether to perform deep critical node analysis on the community subgraph. For each community... If its node size If the threshold is exceeded, the community is called an over-threshold community and needs to enter the subsequent critical node identification process based on the explosive seepage model, that is, the execution process of the explosive seepage modeling module based on the global redundant path; otherwise, the non-boundary nodes within the community are assigned a lower critical priority.

[0060] The threshold value is set based on the actual application scenario and computational resource constraints. In general complex network analysis tasks, Recommended value range is Larger A smaller value can reduce the number of subgraphs entering the explosive seepage modeling stage, thereby reducing the overall computational cost; This ensures that more communities undergo in-depth analysis, improving the comprehensiveness of key node identification.

[0061] 2. Explosive seepage modeling module based on global redundant paths The explosive seepage modeling module based on global redundant paths is the second core technical component of this invention. Its function is to construct a generative model that simulates the abrupt phase transition behavior during the gradual evolution of the network, namely the explosive seepage model, for the subgraph of the superthreshold community. It generates a set of key node candidates inside the subgraph of the superthreshold community through a competitive node selection mechanism.

[0062] (1) Basic concepts of explosive seepage and basic design ideas for identifying key nodes Explosive percolation (EP) is a special type of percolation phase transition phenomenon. In traditional stochastic percolation models, nodes or edges are added to the network in a uniform and random manner, and the size of the largest connected component (GCC) of the network continuously increases with the probability of addition, exhibiting second-order phase transition characteristics. In contrast, the explosive percolation model introduces a competitive selection mechanism: in each evolution step, nodes are selected for addition from the candidate set according to specific optimization rules, thereby suppressing the premature formation of the largest connected component and causing a sudden (first-order phase transition) growth in the size of the connected component near the phase transition point.

[0063] This invention applies the competitive selection concept of explosive seepage to the critical node identification task. The basic idea is as follows: by simulating the process of gradually adding nodes to an empty graph, nodes that contribute the least to the growth of the maximum connected component size are added first; when the maximum connected component size is about to exceed a set threshold, the addition process is terminated. The nodes that were not added at this point constitute the set of nodes that play a crucial role in maintaining network connectivity, i.e., the candidate set of critical nodes.

[0064] (2) Global Redundancy Path Measurement The core innovation of the explosive seepage model proposed in this invention, which distinguishes it from existing technologies, lies in the introduction of a global redundant path measurement mechanism. The global redundant path measurement not only considers the direct impact of nodes on the current maximum connected component of the network, but also comprehensively considers the indirect impact of adding nodes on the reachable paths and shortest path structure in the network.

[0065] Specifically, for candidate nodes and the currently built network (in For the set of nodes that have been added, (for the recovered edge set), nodes Connectivity Adds Gain In other words, the global redundant path gain is defined as the overall gain measure of the network's global connectivity after the node is added. This invention evaluates the added gain from the following two dimensions: Dimension 1: Reachability Path Gain. Nodes Add to Subsequently, it and its neighboring nodes may join the largest connected component, thus increasing the node size of the largest connected component. Reachable path gain. Quantified as:

[0066] in, Indicates adding a node And restore the network after it is connected. , Represents network , The number of nodes in the largest connected component.

[0067] Dimension Two: Structural Redundancy Gain. Nodes The addition of [a] may be achieved by merging the current [edges]. Multiple connected components in the array. If the node... Connected If there are 10 connected components, then these connected components will be merged into a larger connected component. Structural redundancy gain. Quantified as the total number of nodes in the merged connected components:

[0068] in, For nodes exist The connection in Different connected components.

[0069] The global redundancy path gain of the present invention Combining the above two dimensions, it is defined as a weighted sum of reachability path gain and structural redundancy gain:

[0070] in, and For the weight parameters, satisfying , , Under the default configuration, .

[0071] (3) Node addition rule design This invention designs four selectable node addition rules to select nodes to be added from the candidate node set at each evolution step. In practical applications, users can choose one of these rules based on network characteristics and analysis needs.

[0072] Rule 1: Minimum Connectivity Component Increment Rule (Min-GCC-Growth Rule, denoted as...) ) In each evolutionary step, select the node that minimizes the increase in the size of the largest connected component after the addition:

[0073] This rule directly suppresses the growth rate of the largest connected component, making the unadded nodes the key nodes for maintaining the separation state of smaller connected components.

[0074] Rule 2: Minimum Connectivity Component Merge Count Rule (hereinafter referred to as...) ) In each evolutionary step, select the node that minimizes the number of connected components merged:

[0075] in, Represents a node In the current network The number of connected components in a network. This rule tends to select nodes with fewer connected components, thus slowing down the global connectivity process of the network.

[0076] Rule 3: Minimum Connected Component Size Rule (Min-Merge-Size Rule, denoted as...) ) In each evolutionary step, select the node that minimizes the total number of nodes in the connected component:

[0077] This rule tends to select nodes that connect small connected segments, avoiding premature merging between large connected segments.

[0078] Rule 4: Min-Product Rule (hereinafter referred to as...) ) In each evolution step, the node that minimizes the product of the number of merged connected components and the total size of the connected components is selected:

[0079] This rule is a compromise between rule two and rule three, and in practice it usually shows a more balanced effect in identifying key nodes.

[0080] Of the four rules mentioned above, rule one uses a competitive selection based on reachability path gain, while rules two, three, and four use a competitive selection based on structural redundancy gain. Users can flexibly choose the rule based on network topology characteristics and the focus of the identification task.

[0081] (4) Explosive seepage modeling algorithm flow For a given superthreshold community subgraph (in The complete process of the explosive seepage modeling algorithm based on global redundant paths is as follows: Input: Community subgraph Identification threshold (Subgraph-level termination threshold), node addition rules

[0082] Output: Subgraph Key node candidate set

[0083] Initialization phase: Step S201: Construct an empty graph ,in , .

[0084] Step S202: Initialize the candidate node set Node set has been added .

[0085] Step S203: Calculate the subgraph modularity This serves as a reference indicator for subsequent termination conditions.

[0086] Iterative evolution stage: Step S204: Traverse the candidate node set Each node in Add rules based on the selected nodes. compute nodes Add priority score The scoring methods for each rule are as follows: like :

[0087] like :

[0088] like :

[0089] like :

[0090] Step S205: Select the node with the lowest priority score as the node to be added in the current step:

[0091] If multiple nodes have the same minimum score, one of them is randomly selected.

[0092] Step S206: Transfer the node Add to an existing network: , , .

[0093] Step S207: Based on the original subgraph Restore the adjacency relationship of nodes. With the current network All existing edges between nodes:

[0094] Step S208: Update the current network Given a connected component structure, calculate the maximum connected component size. .

[0095] Step S209: Determine the termination condition. If any of the following conditions are met, terminate the iteration and proceed to the output stage: Condition 1: The size of the largest connected component exceeds the subgraph termination threshold, i.e. ; Condition 2: The candidate node set is empty, i.e. ; Condition 3: Adding any remaining candidate nodes will cause the maximum connected component size to exceed the threshold, i.e.:

[0096] Output phase: Step S210: The set of candidate nodes that have not been added is the subgraph. Key node candidate set:

[0097] Step S211: Output the candidate set of key nodes The algorithm ends.

[0098] (5) The theoretical significance of global redundant paths The global redundancy path metric introduced in this invention has clear theoretical significance in network science. In complex networks, the importance of a node depends not only on its local topological properties (such as degree centrality and betweenness centrality), but also on its structural role in global connectivity. The global redundancy path metric comprehensively evaluates the structural importance of a node from two dimensions: path diversity and connected component merging effect. From the perspective of path diversity, if the addition of a node can introduce a large number of new reachable paths to the network and significantly expand the size of the maximum connected component, then the node has a high network integration effect, and its reverse inference identification priority is low (because removing the node contributes relatively little to the network collapse).

[0099] From the perspective of the connectivity merging effect, if the addition of a node can connect multiple separate connected components and merge them into a larger connected component, then the node plays a pivotal role in the global connectivity process of the network, and its unadded state means that the node is a key node in maintaining the network's separate structure.

[0100] Through a competitive selection mechanism for globally redundant paths, the explosive seepage model of this invention can effectively identify those nodes that are "most needed" in the process of gradual network connectivity. These nodes are precisely the key structural elements that maintain the current isolated state of the network.

[0101] 3. Reordering Module for Recognition Results The identification result reordering module is the third core technical component of this invention. Its function is to perform a global consistency reordering of the key node candidate set generated by the explosion seepage modeling module based on the optimization criterion of maximizing maximum unit disruption gain, so as to optimize the identification order of key nodes and make them more accurately reflect the marginal contribution of each node to the global connectivity of the network.

[0102] (1) Definition of unit disintegration revenue In the task of identifying critical nodes in a network, an intuitive optimization goal is that the number of nodes removed in each identification step should result in the maximum possible loss of network connectivity. However, due to the differences in topology across different regions (connected segments) of the network, the same number of node removal operations may have drastically different effects on different regions.

[0103] To address the aforementioned problems, this invention proposes the concept of unit disintegration benefit. It is assumed that after community detection and explosive seepage modeling, the set of connected components of the remaining network (the subgraph formed by unidentified nodes) is... For each connected component Suppose that completely isolating it from the network (i.e., removing all critical structures associated with it) requires removing Each node. Importance score of the connected component. Defined as the ratio of its incremental contribution to global connectivity to the number of nodes that need to be removed:

[0104] in, Indicates the removal of connected components. The change in the global maximum connected component size after the corresponding key node.

[0105] Unit Disintegration Revenue The physical meaning is: the maximum connected component size decay rate that can be caused by removing a node. The larger the value of a connected component, the higher the recognition priority should be given to its corresponding key node.

[0106] (2) Reordering algorithm flow The complete process of the reordering algorithm based on maximizing unit collapse gains is as follows: Input: Original network The set of key nodes to be reordered in, For the set of boundary nodes, For each threshold community subgraph, a candidate set of key nodes is provided. Output: Reordered key node identification sequence

[0107] Step S301: Construct the remaining network That is, a subgraph induced by all nodes that are not identified as key nodes and the edges between them.

[0108] Step S302: Calculate the remaining network connected component set And calculate each connected component Node size .

[0109] Step S303: For each connected component Statistics from key node set It needs to be removed from the middle. Only a certain number of nodes are needed to completely isolate the connected component from the global connectivity structure, where... To and The number of connected critical nodes.

[0110] Step S304: Calculate the unit disintegration gain score for each connected component:

[0111] Here, Connecting pieces Node size As an approximate estimate.

[0112] Step S305: Select the connected component with the highest current score. ,in .

[0113] Step S306: From the set of key nodes Selecting connected components All connected key nodes are added to the output sequence according to their priority in the original sequence. and from Remove these nodes.

[0114] Step S307: Update the remaining network and connected component structure. Repeat steps S302 to S306 until the set of critical nodes is reached. Empty.

[0115] Step S308: Output the reordered key node identification sequence .

[0116] (3) Analysis of the optimization effect of reordering The core optimization idea of ​​the reordering module is to dynamically evaluate the unit disintegration benefit of each connected component in the current network state, and prioritize the processing of those regions with the "highest cost-effectiveness" (i.e. connected components that can bring the greatest connectivity loss for each node removed), so that the identification order of key nodes is more in line with the global optimal strategy.

[0117] Compared with static sorting methods (i.e., identification according to a fixed order output by the explosive seepage model), the dynamic reordering mechanism of this invention has the following advantages: First, adaptability. The reordering module optimizes its decision-making at each step based on the latest topology of the remaining network, enabling it to adaptively adjust its identification strategy to cope with dynamic changes in the network structure.

[0118] Second, global consistency. Through global benefit evaluation at the connected component level, the reordering module ensures the consistency of the identification order across different community subgraphs, avoiding the local optima problem that may result from independent analysis of each subgraph.

[0119] Third, refined measurement. Unit disintegration benefit score. It provides a quantitative priority evaluation standard, making the identification and ranking of key nodes have clear optimization goals and interpretability.

[0120] 4. Complete algorithm flow of the CNP method Based on the above three core technical aspects, the complete algorithm flow of the network critical node identification method (CNP method) based on globally redundant paths proposed in this invention is as follows: Input: The complex network graph to be analyzed Community sub-identification threshold Adding rules to explosion seepage nodes

[0121] Output: Ordered key node identification sequence

[0122] Step 1: Club Assessment The Louvain algorithm was used to test the input network. Conduct club assessments to obtain club classification results. Calculate and record the node size of each community. And the number of sides within the club .

[0123] Step 2: Community Boundary Node Identification Traverse the set of bridge edges between all clubs Identify the global boundary node set Boundary nodes Marked as a set of high-priority key nodes The study also analyzed the number of communities connected to each boundary node and their bridging role in the global connectivity of the network.

[0124] Step 3: Explosive seepage modeling of subgraphs exceeding the threshold For each club Determine whether it satisfies If the condition is met, then the subgraph of communities exceeding the threshold induced by this community will be defined. As input, the explosive seepage modeling algorithm based on globally redundant paths is executed (see Section 2 for details), and the candidate set of key nodes for this subgraph is output. Merge the candidate sets of key nodes in all subgraphs of communities that exceed the threshold, denoted as . .

[0125] Step 4: Merging the set of key nodes The key set of boundary nodes and the candidate set of key nodes of the subgraph are merged to form a preliminary key node identification set:

[0126] Step 5: Reordering and Optimizing Recognition Results The merged set of key nodes The input identification result reordering module (see Section 3 for details) executes a reordering algorithm based on maximizing unit disintegration gains, and outputs the final ordered key node identification sequence. .

[0127] Step Six: Output Results Return the final key node identification sequence The nodes in the sequence are sorted from highest to lowest according to their criticality to the global connectivity of the network.

[0128] 5. Computational Complexity Analysis The computational complexity analysis of the CNP method proposed in this invention is as follows: First, the community group evaluation process. The time complexity of Louvain's algorithm is... ,in Let be the total number of network nodes. The space complexity is O(n). ,in Let be the number of network edges.

[0129] Second, the boundary node identification stage. Traversing all edges to identify bridging edges has a time complexity of O(log n). The construction complexity of the boundary node set is The total time complexity of this step is... .

[0130] Third, the explosion seepage modeling stage. For the first... Subgraph of communities exceeding the threshold ,in , In the worst-case scenario, explosive seepage modeling requires... Each iteration requires traversing all candidate nodes and calculating their addition priority score. The update of the connected component structure requires... Therefore, the time complexity of a single subgraph is O(n log n). .

[0131] Assume the number of community subgraphs exceeding the threshold is The total time complexity of this step is:

[0132] In real-world networks, community sizes typically follow a power-law distribution (a few large communities and a majority of small communities), and only those exceeding a threshold... The community needs to perform explosive seepage modeling, therefore (Total number of clubs) The actual computational cost of this step is far below the theoretical upper limit of the worst-case scenario.

[0133] Fourth, the reordering process. The time complexity of counting and sorting connected component sets is O(n log n). ,in This represents the number of connected components in the remaining network. In the worst case, Therefore, the time complexity of this step is O(n). .

[0134] Overall Complexity. Combining the time complexities of each of the above steps, the total time complexity of the CNP method of this invention is:

[0135] For sparse networks ( ), and the number of community subgraphs exceeding the threshold In the case of smaller values, the total time complexity is approximately:

[0136] When a network has a significant community structure and the community sizes are relatively evenly distributed, ( (total number of clubs), and The computational cost of the second term is effectively controlled at a low level. Under these conditions, the overall time complexity of the CNP method of this invention is approximately O(n). It has the same complexity as the Louvain community detection algorithm and has good scalability in large-scale complex network scenarios.

[0137] In addition, the following alternative implementations exist for this method: Alternative Solution 1: Alternative Choices for Community Detection Algorithms. In the community structure detection step 1, in addition to the Louvain algorithm, other community detection algorithms can be selected, including but not limited to: (1) Infomap algorithm – based on the random walk coding principle in information theory, it discovers community structures by minimizing the coding length describing the random walk trajectory of the network, and is particularly suitable for directed networks and weighted networks; (2) Label Propagation algorithm – based on the local information propagation mechanism, each node selects the label with the highest frequency among its neighbors as its own label, with a near-linear time complexity OE, suitable for ultra-large-scale networks; (3) Walktrap algorithm – based on the node similarity measurement of random walks, it discovers community structures using hierarchical clustering methods; (4) Fast Newman algorithm – a modularity optimization method based on a greedy strategy. Different community detection algorithms are suitable for networks with different topological characteristics, and can be flexibly selected according to the actual application scenario.

[0138] Alternative Solution 2: Variant Design of Explosive Seepage Node Selection Rules. In the global redundancy path analysis in step two, the four node selection rules can be designed with variants based on different network topology characteristics. For example: (1) Rule variant based on node degree—introducing node degree as a weighting factor when comparing candidate nodes, prioritizing connected components associated with edges connecting nodes with high height; (2) Rule variant based on clustering coefficient—introducing local clustering coefficient as a constraint, prioritizing connected components with low clustering coefficient characteristics for merging control; (3) Rule variant based on betweenness centrality—introducing betweenness centrality as an evaluation index, prioritizing the merging process of connected components connected by high betweenness edges; (4) Rule variant based on eigenvector centrality—using the eigenvector centrality of nodes to evaluate the overall importance of connected components. The above rule variants can be combined or used individually according to the specific characteristics of the target network.

[0139] Alternative Solution 3: Alternatives to the node reordering strategy. In the node reordering in step four, in addition to the sorting strategy based on maximizing unit collapse benefits, the following alternative strategies can also be selected: (1) Sorting based on node centrality indicators - using indicators such as degree centrality, betweenness centrality, proximity centrality or eigenvector centrality to sort the importance of nodes within a community; (2) Weighted sorting based on community importance - first assess the importance of each community in the overall network topology (such as the number of nodes, edges, and connections to external communities in the community), and then sort the internal nodes by combining the community importance weights; (3) Sorting based on multi-indicator fusion - comprehensively evaluate multiple node importance indicators through weighted fusion, and the weights can be determined by grid search or adaptive learning; (4) Sorting based on random walk - using random walk algorithms such as PageRank and LeaderRank to sort the global importance of nodes.

[0140] Alternative Solution 4: Variations in the value strategy of parameter η. The identification threshold parameter η used in step 3 to determine the community exceeding the threshold can adopt different value strategies according to different application scenarios: (1) Fixed threshold strategy - take η=50 as a fixed threshold, which is suitable for general network analysis scenarios; (2) Adaptive threshold strategy - adaptively determine the value of η according to the overall size of the network, for example, take the percentage of the total number of network nodes as the threshold benchmark; (3) Multi-threshold hierarchical strategy - use multiple η values ​​to perform hierarchical key node identification and construct key node sets under different strictness levels; (4) Threshold search strategy based on optimization algorithm - take the final effect of key node identification as the objective function and automatically optimize the value of η through grid search or evolutionary algorithm.

[0141] Alternative Solution 5: Joint Node-Edge Criticality Assessment. Building upon the identification of only critical nodes, this solution can be extended to a joint assessment scheme that simultaneously identifies critical nodes and critical edges. Specifically, during the global redundant path analysis in step two, in addition to recording the importance ranking of nodes, the criticality scores of edges participating in connected component merging are also recorded. During the reordering in step four, the joint impact of node removal and edge removal on network robustness is considered. This extended solution provides a more comprehensive network vulnerability assessment and is suitable for network security scenarios requiring the simultaneous identification of critical nodes and critical links.

[0142] In summary, the network critical node identification method (CNP method) based on globally redundant paths proposed in this invention has the following three core technological innovations: Innovation Point 1: Task Decomposition Mechanism for Community Structure Constraints. By introducing community detection and boundary region analysis, this invention effectively decomposes the task of identifying global key nodes to the community sub-layer level, significantly reducing the problem scale in large-scale network scenarios. Furthermore, by prioritizing the analysis of boundary nodes, it quickly locates the key bridging structures for global network connectivity.

[0143] Innovation Point Two: Explosive Seepage Competitive Selection Model Based on Global Redundant Paths. This invention combines the competitive node selection concept of explosive seepage with global redundant path metrics, designing four optional node addition rules. It comprehensively evaluates the network connectivity role of nodes from two dimensions: reachability path gain and structural redundancy gain, effectively overcoming the limitation of traditional centrality indices that only focus on local topological features.

[0144] Innovation Point 3: Dynamic Reordering Optimization Mechanism Based on Unit Disintegration Benefit. This invention proposes the concept of unit disintegration benefit and designs a dynamic reordering algorithm based on the evaluation of connected component contribution. By prioritizing the identification of key nodes that have the largest marginal contribution to the global connectivity of the network, it optimizes the global consistency of the identification order and the actual evaluation accuracy.

[0145] This invention can be applied to scenarios such as security protection and vulnerability assessment of critical infrastructure networks (power grids, transportation networks, and communication networks), identification of key users and governance of information dissemination in social networks, precise location of prevention and control nodes in disease transmission networks, and resilience assessment and enhancement of information networks and supply chain networks. It provides network managers with scientific and efficient technical support for key node identification and network robustness assessment.

[0146] This invention has been systematically validated through experiments on the following nine real-world network datasets. These datasets cover multiple application domains and exhibit varying sizes, densities, and topological characteristics, enabling a comprehensive evaluation of the method's applicability and robustness. Specifically, these include: social network datasets (such as Twitter political topic retweet networks), infrastructure network datasets (such as the US power transmission network Grid and the Chicago transportation network), crime-related network datasets (such as the collaborative crime network Crime), and fiber optic communication network datasets (such as the US fiber optic network Fiber). These datasets are all derived from publicly available complex network data repositories (such as SNAP and Network Repository), which are widely used standard benchmarks in the field of network science.

[0147] As can be seen, this invention proposes a multi-scale network key node identification method based on community structure and global redundant paths. This method simultaneously integrates global redundant path information at the macro-scale and community structure information at the meso-scale. Through a "divide and conquer" computational strategy, it decomposes the global key node identification problem into multiple independent subproblems on local community subgraphs, significantly reducing computational complexity while ensuring global optimization capabilities. This achieves collaborative analysis of multi-scale information and accurate identification of key nodes. This invention also proposes a global redundant path analysis method based on the Explosive Percolation (EP) model. This method introduces explosive percolation theory into the field of network key node identification. Based on a competitive node addition mechanism, it simulates the merging process of the largest connected component in the network. Four node selection rules (including the rule of minimum increment of the largest connected component, the rule of minimum increase in the number of connected components, the rule of priority merging, and the rule of priority isolation) are used to perform global redundant path analysis on candidate nodes, evaluating node importance from a global perspective and overcoming the shortcomings of traditional greedy strategies that get trapped in local optima. Finally, this invention proposes a node reordering mechanism based on maximizing unit disintegration gains. This method, based on the initial ranking of nodes within the community, introduces a connected component importance scoring function. (in Indicates the removal of connected components. The decrease in the GCC (giant connected component) after the node in the middle. Indicates connected components The number of nodes in the network is used to globally reorder the node identification order by calculating the connectivity benefit of each node, ensuring that each identified node is the key node with the greatest unit input benefit in the current network state. The present invention also proposes a collaborative analysis method for community boundary node identification and subgraph decomposition. This method identifies the boundary nodes (Bridge Nodes) and their neighborhoods of cross-community connections, and decomposes the global network into multiple local subgraphs by combining the community partitioning results. Key nodes are identified independently at the subgraph level, and cross-community connection information is not lost through boundary node analysis, thus achieving an organic unity of local solution and global consistency.

[0148] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for identifying key network nodes based on globally redundant paths, characterized in that, Includes the following steps: First, by community detection and boundary region analysis, the network is decomposed into a community structure with internal connectivity features, and the set of boundary nodes across communities is identified, thereby achieving spatial dimensionality reduction for the key node identification task. Secondly, by using explosive seepage modeling based on global redundant paths and community detection and boundary region analysis results, a competitive node addition mechanism is constructed at the community sub-layer level. By simulating the abrupt phase transition behavior in the gradual evolution of the network, a candidate set of key nodes inside each sub-graph is generated. Finally, by reordering the identification results, and based on the optimization criterion of maximizing the unit disintegration benefit, the generated key node candidate set is reordered globally to generate a reordered key node identification sequence.

2. The method as described in claim 1, characterized in that, Community detection and boundary region analysis utilizes the modular structure of networks to decompose the large-scale key node identification task on a global scale into multiple relatively independent community sub-layer level tasks, thereby reducing the computational complexity of the problem. Furthermore, by prioritizing the analysis of boundary nodes, it locates bridging structures that affect the global connectivity of the network.

3. The method as described in claim 1, characterized in that, Community detection and boundary area analysis specifically include the following sub-steps: (1) Club test The Louvain algorithm, based on modularity optimization, is chosen as the community detection method. Modularity is a quantitative indicator that measures the quality of network community partitioning; it is defined as the ratio of the difference between the actual number of edges and the expected number of edges within a community to the total number of edges in the network. Given an undirected graph... ,in For a set of nodes, Let be the set of edges. , For dividing the network into The division scheme of the associations, i.e. the results of the association division. Modularity The calculation formula is: Among them, C k This represents the k-th club, where each club is a set of nodes. For the image The adjacency matrix elements, when the node With nodes When there are connecting edges ,otherwise ; and They are nodes and nodes The degree; Represents a node The tag associated with the organization; For indicator functions, when The value is 1 if it is true, and 0 otherwise. The Louvain algorithm uses a greedy iterative strategy to optimize the modularity index, and its process is as follows: Step 1: Initialization Phase: Treat each node in the network as an independent community, i.e., the initial number of communities equals the number of nodes. ,at this time , ; Step 2: Local Optimization Phase: Traverse each node in the network sequentially. Try moving it to the community to which all its neighbor nodes belong, and calculate the increment of modularity after each move. If it exists that makes The movement of the node will... Move to make The largest club; otherwise, keep the node. The current community affiliation remains unchanged; the process of moving nodes and calculating and judging is repeated until no node movement can further improve the modularity; Step 3: Network Aggregation Phase: Aggregate each formed community into a super node, and aggregate the edges between communities into weighted edges between super nodes. The weight of each edge is equal to the total number of original edges between the two communities, thereby constructing a new aggregated network. Step 4: Iterative convergence judgment: Repeat steps 2 and 3 until the improvement in the modularity of the network is less than the preset threshold. (2) Community Boundary Node Identification After the community testing is completed and the network's community division results are obtained. Next, the set of boundary nodes of the community is further identified; the boundary node of the community is defined as: if a node There is at least one neighbor node Then it is called a node. For the club Boundary nodes; In a formal sense, the club set of boundary nodes Defined as: Among them, V\C i This means removing C. i The set of nodes other than the internal nodes; The union of the boundary node sets of all communities constitutes the global boundary node set. : This is the output result of the community boundary node identification step; Boundary nodes play a bridging role in maintaining global network connectivity. Specifically, the edges connecting different communities are called bridging edges, and their set is denoted as . : The two endpoints of the bridging edge are the boundary nodes; By analyzing community boundary regions, the task of identifying key nodes in the original network is decomposed into the following two levels: First, the set of boundary nodes The nodes in the list are directly selected as high-priority candidate key nodes; Second, for each club The key identification task for internal non-boundary nodes is restricted to the community subgraph. Perform within the scope, thereby affecting the global scope. The problem of scale is reduced to multiple dimensions Sub-problems of scale; (3) Determination of community subgraphs exceeding the threshold Introducing recognition thresholds This serves as a parameter for determining whether a deep critical node analysis needs to be performed on the community subgraph; for each community If its node size If the threshold is exceeded, the community is called an over-threshold community, and the subsequent critical node identification process based on the explosive seepage model, i.e., the explosive seepage modeling process based on global redundant paths, is then initiated; otherwise, the non-boundary nodes within the community are assigned a preset lower criticality priority.

4. The method as described in claim 3, characterized in that, Explosive seepage modeling based on global redundant paths is a generative model for superthreshold community subgraphs that simulates abrupt phase transition behavior during the gradual evolution of the network. The explosive seepage model generates a candidate set of key nodes within the superthreshold community subgraph through a competitive node selection mechanism.

5. The method as described in claim 3, characterized in that, When modeling explosive flow based on global redundant paths, the competitive selection concept of explosive flow is applied to the key node identification task. The design idea is to simulate the process of gradually adding nodes to an empty graph and prioritize adding those nodes that contribute the least to the growth of the maximum connected component size. The addition process terminates when the maximum connected component size is about to exceed the set threshold. The nodes that have not been added at this time constitute the set of nodes that play a key role in maintaining network connectivity, namely the key node candidate set.

6. The method as described in claim 5, characterized in that, The explosive seepage model introduces a global redundancy path measurement mechanism. The global redundancy path measurement not only considers the direct impact of nodes on the current maximum connected component of the network, but also comprehensively considers the indirect impact of adding nodes on the reachable paths and shortest path structure in the network. Specifically, for candidate nodes and the currently built network ,in For the set of nodes that have been added, For the recovered edge set, nodes Connectivity Adds Gain In other words, the global redundant path gain is defined as the overall gain measure of the network's global connectivity after the node is added; the gain is evaluated from the following two dimensions: Dimension 1: Reachable Path Gain: Nodes Add to Subsequently, if it and its neighboring nodes are added to the maximum connected component, the node size of the maximum connected component increases; reachable path gain Quantified as: in, Indicates adding a node And restore the network after it is connected. , Represents network , The number of nodes in the largest connected component; Dimension Two: Structural Redundancy Gain: Nodes The addition of merges the current edges through its connection. Multiple connected components in the array; if nodes Connected If there are 10 connected components, then these connected components will be merged into a larger connected component; structural redundancy gain Quantified as the total number of nodes in the merged connected components: in, For nodes exist The connection in A number of different connected components; Global Redundancy Path Gain Combining the above two dimensions, it is defined as a weighted sum of reachability path gain and structural redundancy gain: in, and For the weight parameters, satisfying , , .

7. The method as described in claim 6, characterized in that, When modeling explosive seepage based on global redundant paths, four optional node addition rules were designed to select the node to be added from the candidate node set in each evolution step. Rule 1: Minimum Connectivity Component Increment Rule (Min-GCC-Growth Rule), denoted as... In each evolutionary step, select the node that minimizes the increase in the size of the largest connected component after the addition: The minimum connected component increment rule directly suppresses the growth rate of the maximum connected component. Rule 2: The Min-Merge-Count Rule, denoted as... In each evolution step, select the node that minimizes the number of connected components merged: in, Represents a node In the current network The number of connected components connected in the network; this minimum number of connected components is used to delay the global connectivity process of the network; Rule 3: The Min-Merge-Size Rule, denoted as... In each evolutionary step, select the node that minimizes the total number of nodes in the connected component: Rule 4: Min-Product Rule, denoted as... In each evolution step, the node that minimizes the product of the number of merged connected components and the total size of the connected components is selected: Of the four rules mentioned above, rule one is based on reachable path gain for competitive selection, while rules two, three, and four are based on structural redundancy gain for competitive selection.

8. The method as described in claim 7, characterized in that, For a given superthreshold community subgraph ,in The complete process of explosive seepage modeling based on global redundant paths is as follows: Initialization phase: Step S201: Construct an empty graph ,in , ; Step S202: Initialize the candidate node set Node set has been added ; Step S203: Calculate the subgraph modularity , as a reference indicator for subsequent termination conditions; Iterative evolution stage: Step S204: Traverse the candidate node set Each node in Add rules based on the selected nodes. compute nodes Add priority score The scoring method for each node's added rules is as follows: like : like : like : like : Step S205: Select the node with the lowest priority score as the node to be added in the current step: If multiple nodes have the same minimum score, then one of them is randomly selected; Step S206: Transfer the node Add to an existing network: , , ; Step S207: Based on the original subgraph Restore the adjacency relationship of nodes. With the current network All existing edges between nodes: Step S208: Update the current network Given a connected component structure, calculate the maximum connected component size. ; Step S209: Determine the termination condition; if any of the following conditions are met, terminate the iteration and proceed to the output stage: Condition 1: The size of the largest connected component exceeds the subgraph termination threshold, i.e. ; Condition 2: The candidate node set is empty, i.e. ; Condition 3: Adding any remaining candidate nodes will cause the maximum connected component size to exceed the threshold, i.e.: Output phase: Step S210: The set of candidate nodes that have not been added is the subgraph. Key node candidate set: Step S211: Output the candidate set of key nodes .

9. The method as described in claim 8, characterized in that, The concept of unit disintegration benefit is introduced in the identification result reordering step. It is assumed that after community detection and explosive flow modeling, the set of connected components of the remaining network and the subgraph formed by the unidentified nodes is... For each connected component If it needs to be completely isolated from the network, it needs to be removed. Each node, the importance score of the connected component. Defined as the ratio of its incremental contribution to global connectivity to the number of nodes that need to be removed: in, Indicates the removal of connected components. The change in the size of the global maximum connected component after the corresponding key node; Unit Disintegration Revenue The physical meaning is: the maximum connected component size decay rate that can be caused by removing a node. The larger the value of the connected component, the higher the recognition priority should be given to the corresponding key node; The complete process of the recognition result reordering step is as follows: Step S301: Construct the remaining network That is, the subgraph induced by all nodes that are not identified as key nodes and the edges between them; Step S302: Calculate the remaining network connected component set And calculate each connected component Node size ; Step S303: For each connected component Statistics from key node set It needs to be removed from the middle. Only a certain number of nodes are needed to completely isolate the connected component from the global connectivity structure, where... To and The number of connected critical nodes; Step S304: Calculate the unit disintegration gain score for each connected component: Here, Connecting pieces Node size As an approximate estimate; Step S305: Select the connected component with the highest current score. ,in ; Step S306: From the set of key nodes Selecting connected components All connected key nodes are added to the output sequence according to their priority in the original sequence. and from Remove these nodes from the process; Step S307: Update the remaining network and connected component structure: Re-execute steps S302 to S306 until the set of critical nodes is reached. Empty; Step S308: Output the reordered key node identification sequence , The nodes in the network are sorted from highest to lowest according to their criticality to the global connectivity of the network.

10. A network critical node identification system based on globally redundant paths, characterized in that, include: The community detection and boundary region analysis module is used to decompose the network into a community structure with internal connectivity features and identify the set of boundary nodes across communities, thereby achieving spatial dimensionality reduction for the key node identification task. The explosive seepage modeling module based on global redundant paths is used to build a competitive node addition mechanism at the community sub-layer level based on community detection and boundary region analysis results. By simulating the abrupt phase transition behavior in the gradual evolution of the network, it generates a candidate set of key nodes within each sub-graph. The identification result reordering module is used to perform global consistency reordering on the generated key node candidate set based on the optimization criterion of maximizing unit disintegration benefits, and generate a reordered key node identification sequence.

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