A method and system for network disintegration based on redundant links

Abstract

This invention provides a network disintegration method and system based on redundant links. The method includes the following steps: obtaining the redundant connection relationships of all nodes in the network structure to be disintegrated and the number of nodes in each node; analyzing the neighbor dissimilarity of all nodes based on the redundant connection relationships and calculating the centrality index of each node; deleting the target node with the largest centrality index and calculating the maximum network connectivity index of the network structure to be disintegrated after the target node is deleted; calculating the disintegration evaluation index of the network structure to be disintegrated by combining the number of nodes and the maximum network connectivity index; repeating the centrality index calculation step and the target node deletion step until the disintegration evaluation index reaches a preset index threshold, thereby completely disintegrating the network structure to be disintegrated. This invention improves the efficiency of network disintegration.

Description

Technical Field

[0001] This invention belongs to the field of network science and technology, specifically relating to a network disintegration method and system based on redundant links. Background Technology

[0002] Network robustness has become an increasingly important research area in network science due to its wide range of applications. Network robustness refers to a network's ability to maintain functionality while resisting malicious attacks, such as controlling the spread of viruses and ensuring the resilience of transportation and power systems against malicious attacks. In network science, there is a consensus that the number of redundant paths is one of the key factors determining network robustness, and increasing the number of edges in the network structure can significantly improve network connectivity.

[0003] Furthermore, network edges and paths are widely considered typical structures beneficial for enhancing network robustness, and the number of redundant paths is a key factor determining network robustness. Adding edges to the original network is considered a very effective way to improve network connectivity and enhance its robustness. In other words, high redundancy contributes to improved network robustness. For example, small-world networks, such as social networks, tend to have high clustering coefficients and low diameters, allowing most nodes to be reached from any other node with relatively few links. These properties allow for efficient communication between nodes and contribute to the overall robustness of the network.

[0004] However, excessive redundant connections can interfere with the efficiency of centrality-based network disintegration strategies. Network disintegration aims to identify a sequence of nodes (or edges) whose absence would cause the greatest disruption to network connectivity, thus profoundly damaging the network structure and negatively impacting network function. Related research confirms that network structure is a key factor influencing the effectiveness of disintegration strategies. The community structure of a network also significantly impacts the design of network disintegration strategies. On the one hand, a growing body of empirical research suggests that community structure provides important information in distinguishing the heterogeneity of nodes and edges. On the other hand, nodes within the same community are densely connected, meaning that if any node in the network becomes infected, the infection may only spread locally within its own community. However, the infection can also spread globally by creating new infections through edges connecting different communities. Redundant connections also play a crucial role in the formation of network community structure. Networks with more redundant connections tend to have clearer community structures, which helps in identifying and designing network disintegration strategies for specific communities.

[0005] Network disintegration is essentially a combinatorial optimization problem, meaning there is no efficient algorithm to identify the optimal sequence of nodes (or edges) in indeterminate polynomial time. Among various decomposition strategies, methods based on centrality indices are widely used due to their strong interpretability and ease of implementation. These methods assume that in complex networks, different nodes have varying impacts on the network's robustness and fault vulnerability. Centrality-based disintegration strategies progressively remove nodes and their edges in descending order of their centrality values.

[0006] However, traditional methods such as degree centrality cannot effectively identify influential nodes in some scenarios, especially when redundant connections exist. While redundant connections enhance network robustness, they can negatively impact both centrality identification and node effectiveness. For example, redundant connections provide multiple paths to the involved node pairs, forming a special structure called local coupling where internal nodes are tightly connected. Nodes in local coupling may share the same k-core, but their importance can vary significantly, raising the question of how redundant connections affect centrality.

[0007] In existing technologies, the role of redundant relationships in identifying influential nodes in a network has become an increasingly important area of ​​research. Progress has been made in this field, verifying the side effects of clustering coefficients on node importance and confirming the strong influence of redundant connections on node importance. Furthermore, a degree centrality index based on multiple spanning trees has been proposed to eliminate redundant relationships. This index is calculated by aggregating the degrees of a small number of spanning trees constructed from randomly selected root nodes. While these methods represent significant improvements over traditional approaches, they initially treat network nodes and edges as equal, ignoring their heterogeneity. Excessive redundant connections can lead to more complex community structures, amplifying the differences between nodes and edges. Ignoring the heterogeneity of nodes and edges in the network structure makes it difficult to accurately identify central nodes during network disintegration based on centrality indices, resulting in low disintegration efficiency. Summary of the Invention

[0008] This invention provides a network disintegration method and system based on redundant links to solve the problem that it is difficult to accurately find the central node in the network structure during the network disintegration process based on centrality indicators, which leads to low disintegration efficiency.

[0009] In a first aspect, the present invention provides a network disintegration method based on redundant links, the method comprising the following steps:

[0010] Obtain the redundant connections of all nodes in the network structure to be dismantled and the number of nodes in each node;

[0011] Based on the redundant connection relationship, the dissimilarity of the neighbors of all the nodes is analyzed, and the centrality index of each node is calculated.

[0012] Delete the target node with the largest centrality index, and calculate the maximum network connectivity index of the network structure to be dismantled after the target node is deleted;

[0013] The collapse assessment index of the network structure to be collapsed is calculated by combining the number of nodes and the maximum network connectivity index.

[0014] Repeat the centrality index calculation step and the target node deletion step until the disintegration evaluation index reaches the preset index threshold, and the network structure to be disintegrated is completely disintegrated.

[0015] Optionally, the step of analyzing the neighbor dissimilarity of all nodes based on the redundant connection relationship and calculating the centrality index of each node includes the following steps:

[0016] Analyze the neighbor dissimilarity of all nodes based on the redundant connection relationship, and generate a neighborhood dissimilarity matrix;

[0017] The neighborhood dissimilarity matrix is ​​used to calculate the neighborhood dissimilarity of each node using the neighborhood dissimilarity centrality algorithm;

[0018] Identify the community tags to which all the nodes belong;

[0019] By combining the community tags and preset penalty coefficients to optimize the neighborhood dissimilarity, a centrality index for each node is obtained.

[0020] Optionally, the formula for calculating the neighborhood dissimilarity of each node based on the neighborhood dissimilarity matrix and using the neighborhood dissimilarity centrality algorithm is as follows:

[0021]

[0022] In the formula: NDC(i) represents the neighborhood dissimilarity of node i, k i τ represents the number of neighbors of node i. i Let τ represent the set of neighbors of node i. j Let τ represent the set of neighbors of node j. i \τ j " represents a set of dissimilar nodes, which includes nodes that are not in τ j But in τ i The node, "|τ i \τ j " |" indicates the number of dissimilar nodes in the set of dissimilar nodes.

[0023] Optionally, the step of combining the community label and a preset penalty coefficient to optimize the neighborhood dissimilarity and obtain the centrality index of each node includes the following steps:

[0024] An adjustment coefficient is generated by combining the community tags and the preset penalty coefficient;

[0025] The number of nodes in the neighborhood dissimilarity centrality algorithm is optimized based on the adjustment coefficient to obtain the neighborhood dissimilarity community centrality algorithm;

[0026] The centrality index of each node is calculated using the neighborhood difference community centrality algorithm, and the calculation formula for the centrality index is as follows:

[0027]

[0028] In the formula: NDCC(i) represents the centrality index of node i, α represents the adjustment coefficient, and c i c represents the community tag to which node i belongs. j The community label to which node j belongs is represented, and β is the penalty coefficient.

[0029] Optionally, the method further includes the following steps:

[0030] Calculate the Jaccard coefficient between any two of the nodes;

[0031] The neighborhood difference community centrality algorithm is optimized using the Jaccard coefficient to obtain complete neighborhood information for all nodes.

[0032] The optimized centrality index of each node is calculated using the optimized neighborhood differential community centrality algorithm. The calculation formula for the optimized centrality index is as follows:

[0033]

[0034] In the formula: NDJC(i) represents the centrality index of node i after optimization.

[0035] Secondly, the present invention also provides a network disintegration system based on redundant links, the system comprising:

[0036] A network node information acquisition subsystem is used to acquire the redundant connection relationships of all nodes in the network structure to be dismantled and the number of nodes of the nodes.

[0037] A centrality index calculation subsystem is used to analyze the neighbor dissimilarity of all the nodes based on the redundant connection relationship, and calculate the centrality index of each node.

[0038] The node deletion subsystem is used to delete the target node with the largest centrality index and calculate the maximum network connectivity index of the network structure to be dismantled after the target node is deleted.

[0039] A subsystem for calculating disintegration index is used to calculate the disintegration assessment index of the network structure to be disintegrated by combining the number of nodes and the maximum network connectivity index.

[0040] The network disintegration subsystem controls the node deletion subsystem and the disintegration index calculation subsystem to repeat the centrality index calculation steps and the target node deletion steps until the disintegration evaluation index reaches a preset index threshold, thereby completely disintegrating the network structure to be disintegrated.

[0041] Optionally, the centrality index calculation subsystem includes:

[0042] A matrix generation module is used to analyze the dissimilarity of the neighbors of all the nodes based on the redundant connection relationship and generate a neighborhood dissimilarity matrix.

[0043] The difference calculation module is used to calculate the neighborhood difference of each node based on the neighborhood dissimilarity matrix and using the neighborhood dissimilarity centrality algorithm;

[0044] The tag recognition module is used to identify the community tags to which all the nodes belong;

[0045] The centrality index calculation module is used to combine the community label and the preset penalty coefficient to optimize the neighborhood dissimilarity and obtain the centrality index of each node.

[0046] Optionally, the dissimilarity calculation module calculates the neighborhood dissimilarity of each node based on the neighborhood dissimilarity matrix and using the neighborhood dissimilarity centrality algorithm, using the following formula:

[0047]

[0048] In the formula: NDC(i) represents the neighborhood dissimilarity of node i, k i τ represents the number of neighbors of node i. i Let τ represent the set of neighbors of node i. j Let τ represent the set of neighbors of node j. i \τ j " represents a set of dissimilar nodes, which includes nodes that are not in τ j But in τ i The node, "|τ i \τ j " |" indicates the number of dissimilar nodes in the set of dissimilar nodes.

[0049] Optionally, the centrality index calculation module includes:

[0050] A coefficient generation unit, pre-set with a penalty coefficient, is used to generate an adjustment coefficient by combining the community tag and the penalty coefficient;

[0051] The algorithm optimization unit is used to optimize the number of nodes in the neighborhood dissimilarity centrality algorithm based on the adjustment coefficient, so as to obtain the neighborhood dissimilarity community centrality algorithm.

[0052] The first centrality index calculation unit is used to calculate the centrality index of each node using the neighborhood difference community centrality algorithm. The calculation formula for the centrality index is as follows:

[0053]

[0054] In the formula: NDCC(i) represents the centrality index of node i, α represents the adjustment coefficient, and c i c represents the community tag to which node i belongs. j The community label to which node j belongs is represented, and β is the penalty coefficient.

[0055] Optionally, the centrality index calculation module further includes:

[0056] A coefficient calculation unit is used to calculate the Jaccard coefficient between any two nodes.

[0057] The neighborhood information acquisition unit is used to optimize the neighborhood difference community centrality algorithm using the Jaccard coefficient to obtain complete neighborhood information for all the nodes.

[0058] The second centrality index calculation unit is used to calculate the optimized centrality index of each node using the optimized neighborhood difference community centrality algorithm. The calculation formula for the optimized centrality index is as follows:

[0059]

[0060] In the formula: NDJC(i) represents the centrality index of node i after optimization.

[0061] The beneficial effects of this invention are:

[0062] The network disintegration method based on redundant links in this invention mainly includes the following steps: obtaining the redundant connection relationships of all nodes in the network structure to be disintegrated and the number of nodes in each node; analyzing the neighbor dissimilarity of all nodes based on the redundant connection relationships and calculating the centrality index of each node; deleting the target node with the largest centrality index and calculating the maximum network connectivity index of the network structure to be disintegrated after the target node is deleted; calculating the disintegration evaluation index of the network structure to be disintegrated by combining the number of nodes and the maximum network connectivity index; repeating the centrality index calculation step and the target node deletion step until the disintegration evaluation index reaches a preset index threshold, thus completely disintegrating the network structure to be disintegrated. Because the neighbor dissimilarity of nodes in a network structure with redundant connections is considered during the centrality index calculation process, the calculated centrality index is more accurate, thus allowing for more accurate identification of central nodes in the network structure, ultimately improving network disintegration efficiency. Attached Figure Description

[0063] Figure 1 This is a flowchart illustrating the network disintegration method based on redundant links in this invention.

[0064] Figure 2 This is a schematic diagram of the network structure to be dismantled in one embodiment of the present invention.

[0065] Figure 2 (a) is a schematic diagram of the network structure to be dismantled in one embodiment of the present invention.

[0066] Figure 2 (b) is a schematic diagram of the neighborhood dissimilarity matrix of the network structure to be dismantled in one embodiment of the present invention. Detailed Implementation

[0067] This invention discloses a network disintegration method based on redundant links.

[0068] Reference Figure 1 The network disintegration method based on redundant links specifically includes the following steps:

[0069] S101. Obtain the redundant connections of all nodes in the network structure to be dismantled and the number of nodes in each node.

[0070] The process begins by constructing an adjacency matrix of the network structure to be dismantled. Nodes in the network are represented as rows and columns of this matrix, and the elements represent the connections between nodes. The adjacency matrix is ​​then preliminarily processed as follows: diagonal elements are set to 0, and all off-diagonal elements are set to 1, indicating that the node is connected to other nodes. Matrix multiplication is then performed on the processed adjacency matrix to obtain a new matrix. This new matrix is ​​further processed by setting all elements greater than 1 to 1, indicating that the two nodes at that position have multiple connections, i.e., redundant connections. The degree of each node in the new matrix is ​​then counted. Through these steps, the redundant connections and the number of nodes in the network structure to be dismantled can be obtained.

[0071] S102. Analyze the neighbor dissimilarity of all nodes based on the redundant connection relationship, and calculate the centrality index of each node.

[0072] For each node, the number of distinct neighboring nodes is counted, i.e., the dissimilarity of the neighboring nodes. Measures such as degree, nearest neighbor degree, and second-order nearest neighbor degree can be used to count the number of distinct neighboring nodes. Based on the dissimilarity of the neighboring nodes, a centrality index is calculated for each node. Commonly used centrality indices include betweenness centrality, closeness centrality, and eigenvector centrality.

[0073] Betweenness centrality refers to the frequency with which a node acts as a mediator in a network, i.e., the number of times a node appears on the shortest path between different pairs of nodes. Betweenness centrality can be calculated for each node using fast algorithms such as Brandes' algorithm. Tight centrality is the reciprocal of the average distance from a node to other nodes. Tight centrality can be calculated for each node using fast shortest path algorithms such as Dijkstra's algorithm. Eigenvector centrality refers to the importance of a node in the network, which is related to the importance of its neighbors. Eigenvector centrality can be calculated for each node using algorithms such as power iteration. Through these steps, the dissimilarity of all nodes' neighbors can be analyzed based on redundant connections, and the centrality index of each node can be calculated.

[0074] S103. Delete the target node with the largest centrality index, and calculate the maximum network connectivity index of the network structure to be disintegrated after the target node is deleted.

[0075] The process involves removing the target node from the network structure to be dismantled, resulting in a new network structure. Connectivity analysis is then performed on this new network structure to identify the largest connected subgraph. The number of nodes and edges in the largest connected subgraph is calculated to determine its size. Dividing these numbers by the original number of nodes and edges yields the node coverage and edge coverage of the subgraph. Multiplying these values ​​together gives the Global Connectivity Criterion (GCC) of the subgraph.

[0076] S104. Calculate the disintegration assessment index of the network structure to be disintegrated by combining the number of nodes and the maximum network connectivity index.

[0077] Where C(Q) represents the number of nodes in the largest connected subgraph after removing node Q, and N represents the number of nodes, then the formula for calculating the breakdown evaluation index is:

[0078] S105. Repeat the centrality index calculation step and the target node deletion step until the disintegration evaluation index reaches the preset index threshold, and the network structure to be disintegrated is completely disintegrated.

[0079] The preset threshold value is usually 1. Steps S102 to S104 are repeated until the evaluation index S = 1 is dismantled. At this point, the network structure to be dismantled is completely dismantled.

[0080] The implementation principle of this method is as follows:

[0081] The process involves obtaining the redundant connections of all nodes in the network structure to be dismantled and the number of nodes in each node; analyzing the neighbor dissimilarity of all nodes based on the redundant connections and calculating the centrality index of each node; deleting the target node with the highest centrality index and calculating the maximum network connectivity index of the network structure after the target node is deleted; calculating the dismantling evaluation index of the network structure to be dismantled by combining the number of nodes and the maximum network connectivity index; repeating the centrality index calculation step and the target node deletion step until the dismantling evaluation index reaches a preset index threshold, thus completely dismantling the network structure to be dismantled. Because the neighbor dissimilarity of nodes in a network structure with redundant connections is considered during the centrality index calculation, the calculated centrality index is more accurate, thus allowing for more accurate identification of central nodes in the network structure, ultimately improving network dismantling efficiency.

[0082] In one implementation, step S102, which involves analyzing the neighbor dissimilarity of all nodes based on redundant connectivity and calculating the centrality index of each node, specifically includes the following steps:

[0083] Analyze the dissimilarity of neighbors for all nodes based on redundant connectivity and generate a neighborhood dissimilarity matrix;

[0084] The neighborhood dissimilarity matrix is ​​used to calculate the neighborhood dissimilarity of each node using the neighborhood dissimilarity centrality algorithm;

[0085] Identify the community tags to which all nodes belong;

[0086] By combining community tags and preset penalty coefficients to optimize neighborhood differences, a centrality index for each node is obtained.

[0087] In this embodiment, the calculation formula for the neighborhood dissimilarity of each node based on the neighborhood dissimilarity matrix and using the neighborhood dissimilarity centrality algorithm is as follows:

[0088]

[0089] In the formula: NDC(i) represents the neighborhood dissimilarity of node i, k i τ represents the number of neighbors of node i. i Let τ represent the set of neighbors of node i. j Let τ represent the set of neighbors of node j. i \τ j " represents a set of dissimilar nodes, which contains nodes that are not in τ j But in τ i The node, "|τ i \τ j The "|" symbol indicates the number of dissimilar nodes in a set of dissimilar nodes.

[0090] In one implementation, the steps of combining community tags and a preset penalty coefficient to optimize neighborhood dissimilarity and obtain the centrality index of each node specifically include the following steps:

[0091] An adjustment coefficient is generated by combining community tags and preset penalty coefficients;

[0092] Based on the adjustment coefficient to optimize the number of nodes in the neighborhood dissimilarity centrality algorithm, a neighborhood dissimilarity community centrality algorithm is obtained.

[0093] The centrality index of each node is calculated using the neighborhood difference community centrality algorithm. The formula for calculating the centrality index is as follows:

[0094]

[0095] In the formula: NDCC(i) represents the centrality index of node i, α represents the adjustment coefficient, and c i c represents the community label to which node i belongs. j β represents the community label to which node j belongs, and β is the penalty coefficient.

[0096] For example, refer to Figure 2 Assuming Figure 2 (a) is a schematic diagram of a network structure to be dismantled, where circles represent network nodes, black lines between nodes indicate connections, and dashed boxes represent different network communities. Therefore, there are 8 network nodes and 2 different network communities in the diagram. For each network node, its set of neighboring nodes is determined. For each neighboring node, its neighborhood dissimilarity is calculated, which can be obtained by calculating the difference between their distances or similarity scores. A neighborhood dissimilarity matrix is ​​obtained by constructing a vector from the neighborhood dissimilarity of each network node's neighbors. This neighborhood dissimilarity matrix is ​​then normalized to obtain the neighborhood dissimilarity matrix.

[0097] Figure 2 (a) The neighborhood dissimilarity matrix of the network structure to be disintegrated is as follows: Figure 2 As shown in (b), in the neighborhood dissimilarity matrix, each row from top to bottom represents the dissimilarity score of network nodes 1 to 8. The formula for calculating the centrality index of network node 1 based on the neighborhood dissimilarity matrix is ​​as follows:

[0098]

[0099] The formula for calculating the centrality index of network node 6 based on the neighborhood dissimilarity matrix is ​​as follows:

[0100]

[0101] In one implementation, it is assumed Figure 2 For any node in (a) to experience a viral outbreak, it is first necessary to attempt to identify the most critical node to stop its spread. Analysis shows that if any node other than v1, v2, v6, and [other nodes] is immune, the virus can still spread globally regardless of which node is infected, thus generating [a situation where...]. The average infection size of a node. However, if v1 is vaccinated, the virus is confined to a localized area of ​​transmission, thus reducing the scale of infection. For {v2,v3,v4,v5} or When initiated, {v6,v7,v8}. Therefore, the expected infection scale resulting from immunization v1 is approximately The calculation formula is (4×4+3×3) / 7. Similarly, when v1 and v6 are immunized, the expected infection size is approximately 5.29 and 4.14, respectively. Therefore, v1, v6, and v2 become the most influential nodes in controlling the spread of the disease.

[0102] As shown in Table 1, only the method based on the neighborhood dissimilarity matrix can accurately identify key nodes in the network. This is because other measurement methods treat all edges in the network as equally important, while the method based on the neighborhood dissimilarity matrix considers the network's community information. Specifically, this method observes that the network contains two distinct communities: node 1 has edges connecting to the other community, while node 2 has edges within its own community. Furthermore, node 6 has a larger set of dissimilar nodes compared to node 2.

[0103] Generally, cross-community connections facilitate network connectivity and information dissemination. If a disease outbreak occurs at node i, and its neighboring nodes are closely connected and belong to the same community, the spread may be limited to a local area. Conversely, if the neighbors belong to different communities, the spread will quickly expand globally. A simple formula, β = constant, can be used to measure the penalty coefficient. More complex forms, such as exponential functions, can also be applied by introducing different mathematical functions. However, this adds little value to the ranking of nodes and complicates the analysis.

[0104] Table 1. Different centrality index values ​​of nodes

[0105]

[0106] In one implementation, after calculating the centrality index of each node using the neighborhood difference community centrality algorithm, the following step is further included:

[0107] Calculate the Jaccard coefficient between any two nodes;

[0108] The neighborhood difference community centrality algorithm is optimized using the Jaccard coefficient to obtain complete neighborhood information for all nodes.

[0109] The optimized centrality index of each node is calculated using the optimized neighborhood difference community centrality algorithm. The formula for calculating the optimized centrality index is as follows:

[0110]

[0111] In the formula: NDJC(i) represents the centrality index of node i after optimization.

[0112] In this embodiment, the Jaccard coefficient is defined as:

[0113]

[0114] In one implementation, the effectiveness of the method disclosed in this invention in disrupting network structures is evaluated through an experimental setup, assuming a set of nodes V and a set of edges. Let G = (V, E) be an undirected network, where N is the number of nodes and M is the number of edges. Network disintegration refers to identifying a sequence of nodes (or edges) from which deleting them would drastically reduce network connectivity. This disintegration process leads to a decline in network performance. A key metric is the disintegration evaluation metric. Assuming C(Q) is the number of nodes in the largest connected subgraph after removing node Q, the formula for calculating the disintegration evaluation metric is:

[0115] In this implementation, a critical node removal ratio can be selected to characterize the ability to respond to a complete crash. The number N of nodes removed is recorded. fc Until the network almost completely collapses, C(Q) < N fc f c The threshold is calculated as follows:

[0116]

[0117] f c The smaller the value, the higher the performance of the disintegration strategy.

[0118] In the real world, network collapse can occur due to a variety of attacks. In most cases, attack strategies cannot completely dismantle the network, meaning that... c This approach may have certain limitations. Considering that the network is severely compromised but not completely destroyed, R is used to represent the network's ability to withstand attacks of any strength, and the network's response to disintegration is captured throughout the process, defined as:

[0119]

[0120] Where s(Q) is the size of the GCC remaining after removing node Q, it has a definite physical meaning and a simple mathematical formula. It can be obtained by calculating the area of ​​the robustness curve with the horizontal axis representing the proportion of nodes removed and the vertical axis representing the size of the GCC.

[0121] During the experiment, nodes were removed one by one for networks smaller than 1000. For networks larger than 1000, 1% of the total number of nodes was removed each time.

[0122] Because many real-world systems are far more complex than model networks, experiments were conducted on nine real-world networks covering a wide range of domains, including infrastructure networks, collaborative networks, protein-protein interaction (PPI) networks, and social networks. Specifically, they are:

[0123] Powergrid: The power grid network of the western states of the United States. Nodes can be generators, transformers, or substations. Edges represent power lines.

[0124] Parsed_Jazz: A network of collaborations between jazz musicians and bands, performed between 1912 and 1940. A node represents a jazz musician, and an edge represents the collaboration of two musicians in a band.

[0125] Netscience: A network of co-authors among scientists working in the field of network science since 2006. This network is a single-modal projection of a bipartite graph of authors and their scientific publications.

[0126] Parsed_Tap: A yeast protein-protein binding network generated through tandem affinity purification experiments. Nodes represent proteins, and edges represent their interactions.

[0127] OpenFlights is a global network of scheduled flights from airports, extracted from the openflights.org dataset. A node represents an airport, and an edge represents a scheduled commercial flight operated by an airline from one airport to another.

[0128] Facebook: A self-sustaining social network, data is collected from surveys of Facebook users. Each node represents a user, and each edge represents their friendship.

[0129] Circuit: A biological network, a type of electronic circuit network. Nodes represent network motifs, forming a group and an edge between two groups, rather than within a group.

[0130] Collins_Yeast: Protein-protein interaction networks in Saccharomyces cerevisiae (budding yeast), measured by cocomplex associations identified through high-throughput affinity purification and mass spectrometry (AP / MS).

[0131] Treating all networks as undirected networks and removing self-loops, the maximum connected component of the network is extracted, and its basic statistics are shown in Table 2.

[0132] Table 2 Basic Structural Characteristics of Real Networks

[0133]

[0134] The method disclosed in this invention is compared with existing state-of-the-arts methods based on redundancy relationships. Since these methods have been proven to outperform traditional metrics such as intermediateness, proximity, and centrality, these traditional metrics are not repeatedly compared for ease of visualization. The methods compared can be categorized as degree centrality (leverage centrality, cluster-rank, and information entropy) and path centrality (cycle ratio, and mstdc).

[0135] Leverage Centrality: Considers the degree of connectivity of a node relative to its neighbors. Nodes with higher connectivity among their neighbors are likely to have higher leverage centrality. The leverage centrality of a node is defined as follows:

[0136]

[0137] Where k(i) represents the degree of node i, and N i It is the set of neighboring nodes of node i.

[0138] Cluster Rank: Quantifies the influence of a node by considering the clustering coefficient and the effect of its neighbors. This reflects, to some extent, the impact of redundant connections on the importance of a node. The Cluster Rank of a node is defined as follows:

[0139]

[0140] Among them, c i Yes, the i-clustering coefficient, f(c) i The '+1' represents the influence of the clustering coefficient of node i, and the term '+1' represents the influence of node j itself. This implementation uses the same format as the original Cluster Rank paper.

[0141] Information Entropy: Based on entropy theory, it sorts nodes by aggregating information from neighboring nodes. It reflects average information and excludes redundant information provided by neighboring nodes, and is defined as:

[0142]

[0143] Among them, P ij =k i / A j ; That is, the second degree of the node.

[0144] Multi-Spanning Tree-based Degree Centrality (MSTDC): Measures the importance of nodes by eliminating redundant connections and local coupling. It is calculated by aggregating the degrees of a small number of spanning trees constructed from a few randomly selected root nodes, and is defined as:

[0145]

[0146] Where k represents the root node of the spanning tree, σ represents the randomly selected set of nodes, and T represents the number of spanning trees. In this implementation, the form T = 30 is used, the same as in the original MSTDC paper.

[0147] Cycle Ratio: Measures the effectiveness of redundant connections by considering the cyclic structure of the network. It defines the cyclic matrix C = [c ij ] N×N The formula for representing the shortest cycle is:

[0148]

[0149] Among them, c ii It is the number of cycles in the set of all shortest cycles containing the node, c ij This represents the number of cycles in the set of all shortest cycles passing through nodes i and j.

[0150] The Neighborhood Dissimilarity Matrix Algorithm (NDCC) and its optimized counterpart (NDJC) have two adjustable parameters that control the information aggregation weights. The appropriate community detection algorithm can be selected based on specific circumstances, making it more suitable for practical applications. This implementation selects four community detection algorithms: Infomap, Greedy Modularity, Louvain, and Label Propagation. The penalty coefficient β is set to a range of 0–1 with an interval of 0.05. By exploring combinations of these two parameters, the parameter with the best disintegration effect is selected for comparison.

[0151] In this embodiment, the proposed method is compared with a series of state-of-the-art methods on empirical networks, and a metric is defined to measure the efficiency improvement of the proposed method. Table 3 shows the f-values ​​of attacks based on different centrality metrics on different datasets. cResults. As can be seen from the table, NDCC and NDJC achieve better attack performance on all datasets. Meanwhile, the improvement rate of the algorithm in this application ranges from 1.75% to 22.677%. In particular, the most fundamental metric, NDC, significantly outperforms other centrality metrics on the Parsed Tap, Netscience, Openflights, Facebook, Hamster, and Collins Yeast networks. Furthermore, for f... c NDCC outperforms NDJC on all datasets except the Circuit network, meaning that NDCC is more suitable for causing complete crashes.

[0152]

[0153]

[0154] Table 3. F values ​​for different attack strategies in real-world networks c value

[0155]

[0156] Note: Bold lines represent the best values ​​for the method in this application and for state-of-arts, respectively.

[0157] To explore the impact of attacks on networks that are severely compromised but not completely collapsed, the R-value of a general index can be compared. This index describes the collapse effect of a network under any attack strength. Table 4 shows the R-values ​​for attacks based on different centrality indices on different datasets. As can be seen from the table, the proposed method outperforms other indices in all scenarios, with improvements ranging from 5.042% to 38.182%. In particular, the most basic index, NDC, significantly outperforms the benchmark in Powergrid, Netscience, Openflights, and Hamster networks. Meanwhile, for R, NDJC shows better attack performance than NDCC in Parsed Jazz, Celegants, and Circuit networks, while there is no significant difference in the collapse effect in the remaining networks. This means that NDJC is more suitable for generalized network collapse attacks. Therefore, it can be concluded that in practical applications, NDJC is a more effective method for decomposing complex networks than existing methods.

[0158] Table 4. R-values ​​for different attack strategies in real-world networks

[0159]

[0160]

[0161] Note: Bold lines represent the best values ​​of the method in this application and the state-of-arts standards, respectively.

[0162] Experimental results from the above embodiments show that high redundancy helps improve network robustness, but it interferes with the effectiveness of attacks based on centrality metrics. The role of redundant connections in identifying key nodes is increasingly valued; however, treating each node and edge as equally important and ignoring the impact of heterogeneity will affect the identification of key nodes. The network disintegration method based on redundant links disclosed in this invention takes into account the impact of heterogeneity and can more accurately locate key nodes in the network structure.

[0163] This invention also discloses a network disintegration system based on redundant links, the system comprising:

[0164] The network node information acquisition subsystem is used to acquire the redundant connection relationships and the number of nodes of all nodes in the network structure to be dismantled.

[0165] The centrality index calculation subsystem is used to analyze the neighbor dissimilarity of all nodes based on redundant connection relationships and calculate the centrality index of each node.

[0166] The node deletion subsystem is used to delete the target node with the highest centrality index and calculate the maximum network connectivity index of the network structure to be disintegrated after the target node is deleted.

[0167] The Disintegration Index Calculation Subsystem is used to calculate the disintegration assessment index of the network structure to be disintegrated by combining the number of nodes and the maximum network connectivity index.

[0168] The network disintegration subsystem controls the node deletion subsystem and the disintegration index calculation subsystem to repeat the centrality index calculation steps and the target node deletion steps until the disintegration evaluation index reaches the preset index threshold, thus completely disintegrating the network structure to be disintegrated.

[0169] The implementation principle of this method is as follows:

[0170] The network node information acquisition subsystem obtains the redundant connections and the number of nodes in the network structure to be dismantled. The centrality index calculation subsystem analyzes the neighbor dissimilarity of all nodes based on the redundant connections and calculates the centrality index of each node. The node deletion subsystem deletes the target node with the highest centrality index and calculates the maximum network connectivity index of the network structure to be dismantled after the target node is deleted. The dismantling index calculation subsystem calculates the dismantling evaluation index of the network structure to be dismantled based on the number of nodes and the maximum network connectivity index. The network dismantling subsystem controls the node deletion subsystem and the dismantling index calculation subsystem to repeat the centrality index calculation and target node deletion steps until the dismantling evaluation index reaches a preset index threshold, thus completely dismantling the network structure to be dismantled. Because the calculation of the centrality index takes into account the neighbor dissimilarity of nodes in a network structure with redundant connections, the calculated centrality index is more accurate, thus allowing for more accurate identification of central nodes in the network structure, ultimately improving the efficiency of network dismantling.

[0171] In one implementation, the centrality index calculation subsystem includes:

[0172] The matrix generation module is used to analyze the dissimilarity of all nodes' neighbors based on redundant connection relationships and generate a neighborhood dissimilarity matrix.

[0173] The dissimilarity calculation module is used to calculate the neighborhood dissimilarity of each node based on the neighborhood dissimilarity matrix and using the neighborhood dissimilarity centrality algorithm.

[0174] The tag recognition module is used to identify the community tags to which all nodes belong;

[0175] The centrality index calculation module is used to combine community tags and preset penalty coefficients to optimize neighborhood differences and obtain the centrality index of each node.

[0176] In one implementation, the difference calculation module calculates the neighborhood difference of each node based on the neighborhood dissimilarity matrix and using the neighborhood dissimilarity centrality algorithm, as follows:

[0177]

[0178] In the formula: NDC(i) represents the neighborhood dissimilarity of node i, k i τ represents the number of neighbors of node i. i Let τ represent the set of neighbors of node i. j Let τ represent the set of neighbors of node j. i \τ j " represents a set of dissimilar nodes, which contains nodes that are not in τj But in τ i The node, "|τ i \τ j The "|" symbol indicates the number of dissimilar nodes in a set of dissimilar nodes.

[0179] In one implementation, the centrality index calculation module includes:

[0180] The coefficient generation unit has a preset penalty coefficient, which is used to generate adjustment coefficients by combining community tags and the penalty coefficient.

[0181] The algorithm optimization unit is used to optimize the number of nodes in the neighborhood dissimilarity centrality algorithm based on the adjustment coefficient, so as to obtain the neighborhood dissimilarity community centrality algorithm.

[0182] The first centrality index calculation unit is used to calculate the centrality index of each node using the neighborhood difference community centrality algorithm. The calculation formula for the centrality index is as follows:

[0183]

[0184] In the formula: NDCC(i) represents the centrality index of node i, α represents the adjustment coefficient, and c i c represents the community label to which node i belongs. j β represents the community label to which node j belongs, and β is the penalty coefficient.

[0185] In one implementation, the centrality index calculation module further includes:

[0186] The coefficient calculation unit is used to calculate the Jaccard coefficient between any two nodes;

[0187] The neighborhood information acquisition unit is used to obtain complete neighborhood information for all nodes by optimizing the neighborhood difference community centrality algorithm using the Jaccard coefficient.

[0188] The second centrality index calculation unit is used to calculate the optimized centrality index of each node using the optimized neighborhood difference community centrality algorithm. The calculation formula for the optimized centrality index is as follows:

[0189]

[0190] In the formula: NDJC(i) represents the centrality index of node i after optimization.

[0191] In this embodiment, the Jaccard coefficient is defined as:

[0192]

[0193] Those skilled in the art should understand that the discussion of any of the above embodiments is merely exemplary and is not intended to imply that the scope of protection of this application is limited to these examples; under the concept of this application, the technical features of the above embodiments or different embodiments can also be combined, the steps can be implemented in any order, and there are many other variations of different aspects of one or more embodiments of this application as described above, which are not provided in detail for the sake of brevity.

[0194] One or more embodiments in this application are intended to cover all such substitutions, modifications, and variations that fall within the broad scope of this application. Therefore, any omissions, modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of one or more embodiments in this application should be included within the protection scope of this application.

Claims

1. A network disintegration method based on redundant links, characterized in that, Includes the following steps: Obtain the redundant connections of all nodes in the network structure to be dismantled and the number of nodes in each node; Based on the redundant connectivity analysis, the neighborhood dissimilarity of all nodes is determined, and the centrality index of each node is calculated. The formula for calculating the centrality index is as follows: In the formula: Represents the node The centrality index, Represents the node Neighborhood differences, Indicates the adjustment factor. Represents the node The number of neighbors, Represents the node The neighborhood group, Represents the node The neighborhood group, Represents a set of dissimilar nodes, wherein the set of dissimilar nodes contains nodes that are not present in the specified node list. But The node, This represents the number of dissimilar nodes in the set of dissimilar nodes. Represents the node Community tags associated with Represents the node Community tags associated with This is the preset penalty coefficient; Delete the target node with the largest centrality index, and calculate the maximum network connectivity index of the network structure to be dismantled after the target node is deleted; The collapse assessment index of the network structure to be collapsed is calculated by combining the number of nodes and the maximum network connectivity index. Repeat the centrality index calculation step and the target node deletion step until the disintegration evaluation index reaches the preset index threshold, and the network structure to be disintegrated is completely disintegrated.

2. The network disintegration method based on redundant links according to claim 1, characterized in that, The method further includes the following steps: Calculate the Jaccard coefficient between any two of the nodes; The centrality index is optimized based on the Jaccard coefficient, and the optimized centrality index is calculated using the following formula: In the formula: Represents the node The optimized centrality metric, This is the Jaccard coefficient.

3. A network disintegration system based on redundant links, characterized in that, The system includes: A network node information acquisition subsystem is used to acquire the redundant connection relationships of all nodes in the network structure to be dismantled and the number of nodes of the nodes. A centrality index calculation subsystem is used to analyze the neighborhood dissimilarity of all nodes based on the redundant connectivity and calculate the centrality index of each node. The formula for calculating the centrality index is as follows: In the formula: Represents the node The centrality index, Represents the node Neighborhood differences, Indicates the adjustment factor. Represents the node The number of neighbors, Represents the node The neighborhood group, Represents the node The neighborhood group, Represents a set of dissimilar nodes, wherein the set of dissimilar nodes contains nodes that are not present in the specified node list. But The node, This represents the number of dissimilar nodes in the set of dissimilar nodes. Represents the node Community tags associated with Represents the node Community tags associated with This is the preset penalty coefficient; The node deletion subsystem is used to delete the target node with the largest centrality index and calculate the maximum network connectivity index of the network structure to be dismantled after the target node is deleted. A subsystem for calculating disintegration index is used to calculate the disintegration assessment index of the network structure to be disintegrated by combining the number of nodes and the maximum network connectivity index. The network disintegration subsystem controls the node deletion subsystem and the disintegration index calculation subsystem to repeat the centrality index calculation steps and the target node deletion steps until the disintegration evaluation index reaches a preset index threshold, thereby completely disintegrating the network structure to be disintegrated.

4. The network disintegration system based on redundant links according to claim 3, characterized in that, The centrality index calculation module also includes: A coefficient calculation unit is used to calculate the Jaccard coefficient between any two nodes. The second centrality index calculation unit is used to optimize the centrality index based on the Jaccard coefficient. The optimized centrality index is calculated using the following formula: In the formula: Represents the node The optimized centrality metric, This is the Jaccard coefficient.

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