Inverse link disturbance protection method and system based on maximum residual link entropy

Abstract

The invention discloses an inverse link disturbance protection method and system based on maximum residual link entropy. The disturbance protection method comprises the following steps: providing an original graph network and a sensitive link set in the original graph network; defining a disturbance range around the sensitive link set, and establishing m-hop sub-graphs in the disturbance range; calculating and searching a target link edge set which maximizes the normalized residual path entropy in the range of the m-hop sub-graph; and performing directional disturbance on the target link edge set to obtain an adversarial graph network. The maximum residual link entropy inverse link prediction disturbance defense method is provided on the basis of a microscopic topological structure of a graph network, quantitative indexes are unified, and objective evaluation standards are provided; meanwhile, the algorithm provided by the invention has model independence, and the protection effect on target sensitive connection is good under different link prediction algorithms; under the same disturbance budget, the performance of the inverse link prediction algorithm based on the maximum residual link entropy provided by the invention is better.

Description

[0001] This invention claims priority to Chinese invention patent application number 202511114517.8, filed on August 8, 2025, entitled "Inverse Link Prediction Method, System and Application Based on Maximum Residual Link Entropy". Technical Field

[0002] This invention belongs to the field of graph data mining technology, specifically relating to an inverse link disturbance protection method and system based on maximum residual link entropy. Background Technology

[0003] Link prediction, a core task in graph analysis, aims to infer potential or missing links based on existing network structures. With the development of deep learning, this task has achieved significant breakthroughs. However, while providing intelligent services, link prediction technology also introduces potential privacy risks.

[0004] Existing sensitive link protection mechanisms mainly employ network structure perturbation methods. By introducing perturbations with a certain budget into the graph network, the similarity between sensitive nodes is reduced, thereby decreasing the likelihood of sensitive links being detected. However, most existing defense methods are optimized for specific link prediction algorithms, making them unsuitable for different metrics or various link prediction methods.

[0005] Meanwhile, existing methods for protecting sensitive links typically rely on random perturbation strategies with a pre-set budget. These strategies disrupt the topological similarity between sensitive nodes by adding, deleting, or swapping edge structures, thereby reducing the success rate of predictive attacks. However, these methods have significant limitations: random perturbations lack specificity and may lead to a large number of redundant operations; large-scale link modifications can significantly alter the statistical characteristics of the network, potentially allowing attackers to identify traces of human intervention through anomaly detection techniques, further exposing the location of sensitive relationships. Summary of the Invention

[0006] The main objective of this invention is to provide an inverse link disturbance protection method and system based on maximum remaining link entropy, so as to overcome the shortcomings of the prior art.

[0007] To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0008] In a first aspect, the present invention provides an inverse link disturbance protection method based on maximum remaining link entropy, comprising:

[0009] Provide the original graph network and the set of sensitive links in the original graph network;

[0010] Define a perturbation range around the sensitive link set, and establish an m-hop subgraph within the perturbation range;

[0011] Within the range of the m-hop subgraph, calculate and find the set of target link edges that maximize the normalized residual path entropy;

[0012] By performing directional perturbation on the target link edge set, an adversarial graph network is obtained.

[0013] Secondly, the present invention also provides an inverse link disturbance protection system based on maximum residual link entropy, comprising:

[0014] The original network module is used to provide the original graph network and the set of sensitive links in the original graph network;

[0015] A perturbation range module is constructed to define a perturbation range around the sensitive link set and to establish an m-hop subgraph within the perturbation range;

[0016] The path entropy calculation module is used to calculate and find the set of target link edges that maximize the normalized residual path entropy within the range of the m-hop subgraph;

[0017] The perturbation optimization module is used to perform targeted perturbation on the target link edge set to obtain an adversarial graph network.

[0018] Compared with the prior art, the beneficial effects of the present invention include at least the following:

[0019] This invention proposes a maximum residual link entropy inverse link prediction perturbation defense method based on the micro-topology of graph networks. It not only unifies the quantitative indicators and provides an objective evaluation standard, but also the algorithm proposed in this invention is model independent and has good protection effect on target sensitive links under different link prediction algorithms. Compared with existing inverse link prediction algorithms, the maximum residual link entropy-based inverse link prediction algorithm provided in this invention performs better under the same perturbation budget.

[0020] The above description is merely an overview of the technical solution of the present invention. In order to enable those skilled in the art to better understand the technical means of this application and to implement it in accordance with the contents of the specification, the preferred embodiments of the present invention are described below in conjunction with detailed drawings. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0022] Figure 1This is a schematic diagram of the reverse link prediction method based on maximum remaining link entropy provided in a typical implementation case of the present invention;

[0023] Figure 2 This is a schematic diagram of the system structure and process of the inverse link prediction method based on maximum residual link entropy provided in a typical implementation case of the present invention. Detailed Implementation

[0024] In view of the shortcomings of the prior art, the inventors of this invention, through long-term research and extensive practice, have proposed the technical solution of this invention. The following will further explain and illustrate this technical solution, its implementation process, and its principles.

[0025] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.

[0026] The first aspect of this invention provides an inverse link disturbance protection method based on maximum remaining link entropy, which includes the following steps:

[0027] Provide the original graph network and the set of sensitive links in the original graph network;

[0028] Define a perturbation range around the sensitive link set, and establish an m-hop subgraph within the perturbation range;

[0029] Within the range of the m-hop subgraph, calculate and find the set of target link edges that maximize the normalized residual path entropy;

[0030] By performing directional perturbation on the target link edge set, an adversarial graph network is obtained.

[0031] In some implementations, the process of establishing the m-hop subgraph specifically includes:

[0032] Using the sensitive node as the current node, extract the direct neighbors of the current node;

[0033] Using the direct neighbor as the new current node, iteratively search for direct neighbors until the number of iterations reaches m.

[0034] As a typical embodiment of the above technical solution, the specific implementation process of the reverse link disturbance protection method provided by the present invention may include the following main steps:

[0035] S1. Create and obtain the graph network. Topology and set of sensitive links .

[0036] S2. Construct a set of sensitive nodes The maximum path length centered on the center does not exceed of Subgraph .

[0037] S3, For each pair of sensitive nodes Calculate the path entropy value separately and then aggregate them.

[0038] S4. Sort the candidate edges according to maximizing the remaining link entropy.

[0039] S5. Iteratively select candidate edges until the perturbation budget is reached and maintain the topology of the perturbation graph.

[0040] The process can be summarized as follows: obtain the graph network structure and the set of sensitive links to be protected, determine the scope of the disturbance, and obtain the partitioned network containing the sensitive links. -hop subgraph; calculates the path entropy of sensitive nodes in the network, based on The perturbation edge range within the hop subgraph is used to calculate and find the order of edges that maximize the normalized residual path entropy; by iteratively adding perturbation operations, the error prediction of the link prediction model is gradually realized, thus obtaining the final perturbation graph.

[0041] In practical applications, the specific process of the above method can be as follows:

[0042] 1) Obtaining the graph network Topological structure information and sensitive link set ,in For a set of nodes, It is an edge set;

[0043] 2) Construct a system based on sensitive nodes Centered - Jumping diagram Its node set satisfies:

[0044] 3) Calculate sensitive links Path entropy :

[0045] 4) Extract all lengths path ;

[0046] 5) According to the signal attenuation model Calculate single-path strength;

[0047] 6) Aggregate path entropy: ;

[0048] 7) Based on scoring function Select the perturbation edge;

[0049] 8) Iteratively add perturbation edges to maximize the normalized remaining path entropy. Generative Adversarial Networks .

[0050] In some implementations, the m-jump subgraph is represented as ,in:

[0051]

[0052]

[0053] in, This represents the m-jump subgraph; Indicates the target node to be linked And the maximum path length does not exceed The set of nodes; This indicates all linked target nodes with a maximum path length not exceeding [a certain value]. The set of edges; This represents the maximum path length between two nodes; Indicates the target node. Indicates the target node to be linked And the maximum path length does not exceed the length of any other node on path m. This represents all paths that meet the requirements.

[0054] The subgraph construction method borrows from a hierarchical breadth-first search (BFS) algorithm. First, sensitive node pairs are placed in the frontier of the current node. Then, all neighbors of that frontier are added to the frontier of the next node, resulting in first-order neighbors. Using the frontier of the next node as the new frontier of the current node yields second-order neighbors. Through this iterative process, the sensitive links are finally obtained. The entire subgraph contains all nodes and links, ensuring that perturbation edges that can effectively protect sensitive links are found within a relatively small perturbation range, and corresponding operations are performed on them.

[0055] In some implementations, the calculation process of the normalized residual path entropy specifically includes:

[0056] Calculate the residual entropy contribution of each path;

[0057] Summing up the remaining entropy contributions on all paths yields the total entropy contribution before the perturbation.

[0058] Each perturbation edge within the perturbation range is taken as the target edge, and the contribution of the total entropy after perturbation is calculated after perturbing the target edge.

[0059] The normalized residual path entropy is obtained by calculating the ratio of the difference between the total entropy contribution before the perturbation and the total entropy contribution after the perturbation to the total entropy contribution before the perturbation.

[0060] To calculate the graph Medium-sensitive links Signal strength, defined For the node To the node And the path length does not exceed The set of all paths. Each path is denoted as . ,in This represents the path number. Assume it starts from node... Send a uniform amplitude "semaphore" along the path As the signal passes through each intermediate node in sequence, the attenuation of the signal is inversely proportional to the degree of each intermediate node.

[0061] In some implementations, the contribution of the residual entropy is calculated as follows:

[0062]

[0063]

[0064] in, d represents the remaining semaphore strength; d(·) represents the degree of the node; Indicates a path; Indicates the index marker of the path; This represents the contribution of the remaining entropy. to This refers to all intermediate nodes on the path, excluding the first and last nodes. The higher the degree, the stronger the attenuation of the transmitted signal by the node.

[0065] In some implementations, the calculation method for the pre-disturbance total entropy contribution is expressed as follows:

[0066] Based on the above signal strength, to characterize the "uncertainty" of a path in information transmission, the entropy contribution of a single path is further defined as: ;

[0067] Because when When the maximum value does not exceed 0.5, The range of values ​​is within Therefore, the larger the entropy value, the richer the "information" retained during transmission, which also means a higher signal arrival rate between node pairs. Furthermore, in complex networks, to comprehensively consider the impact of multiple possible paths on node pairs, the entropy contributions on all paths are summed:

[0068] This summation represents the target node pair The entropy contribution is used to measure the probability that there is no direct link between any pair of nodes in a network. All sensitive links in the network are grouped into a set, and... The total entropy contribution of the sensitive link set is expressed as follows, according to Formula 3.5:

[0069] ;

[0070] in, Represents node pairs Residual path entropy in a network without disturbance; Represents the set of all paths from node i to node j; This represents the pre-disturbance total entropy contribution from node i to node j. Considering that information flow has no specific direction in an undirected network, it is calculated separately from the perspective of balance. and .

[0071] In some implementations, the calculation process of the normalized residual path entropy includes:

[0072] By perturbing a network, deleting or adding a certain number of perturbed edges, the resulting new network is called an adversarial network. It means that, in The sensitive node pairs are calculated in the middle. The path entropy is denoted as: Therefore, the perturbation is defined as follows. The total entropy contribution is:

[0073]

[0074] To measure the impact of added perturbations on the entropy contribution of sensitive nodes in the network, normalized residual path entropy is introduced. , is represented as:

[0075]

[0076] in, This represents the contribution of the total entropy after perturbation of the target edge; This represents the total path entropy contribution from node i to node j after the disturbance. This represents the normalized residual path entropy.

[0077] A larger value indicates that the addition of perturbation operations is more effective and provides more protection for sensitive links; conversely, if the value is close to 0, it means that adding or deleting edges has failed to effectively disrupt the predictable path of sensitive node pairs.

[0078] In some implementations, the disturbance protection method may further include:

[0079] Based on the maximum selection of candidate perturbation edges using the scoring function;

[0080] The target link edge set is obtained by iteratively adding from the candidate perturbation edges based on the objective of maximizing the normalized residual path entropy.

[0081] The scoring function is expressed as follows:

[0082]

[0083] in, Represents the scoring function; This represents the shortest distance between two nodes; The number of endpoints of the node is represented by ; a and b represent the index labels of the node.

[0084] In a typical embodiment, analysis of the path entropy calculation formula reveals that when the path length is shorter and the degree of nodes in the path is smaller, the node... The signal strength received through this path will be greater, and thus the entropy value contributed by this path to the system will also be higher.

[0085] Therefore, when selecting the undetermined perturbation operation edge When performing this task, both the relative position of an edge within the network and the degree of the nodes it connects to must be considered. Given a limited perturbation budget, to minimize the risk of the link prediction algorithm accurately identifying sensitive links, an importance scoring metric is further proposed: And it is calculated from the following two aspects:

[0086] First, examine the candidate edges. The two endpoints With sensitive nodes Does a shortest path exist between them? If it does and the path length is not zero, then calculate the shortest path between the nodes. With nodes Find the shortest distance and sum the reciprocals, the formula is defined as:

[0087]

[0088] in, Represents a node With nodes The shortest distance.

[0089] Secondly, examine the candidate edges The sum of the degree measures of the two endpoints:

[0090]

[0091] Based on the above two aspects, candidate edges are defined. Importance rating:

[0092]

[0093] when When the value is high, selecting this edge for perturbation is more effective in reducing the overall residual entropy and the similarity between target nodes. Therefore, if the budget allows, the edge with the highest score should be deleted first, ultimately generating a perturbed network. .

[0094] In some implementations, the disturbance protection method specifically includes:

[0095] Using a link prediction model For all node pairs The existence of links between them is predicted, and the prediction probability matrix is ​​finally obtained. ,in This refers to the set of sensitive links;

[0096] For each pair of nodes The predicted information entropy is calculated separately to quantify the uncertainty of the model's prediction of the existence of the link.

[0097] As a typical example, the steps of an inverse link disturbance protection method based on maximum residual link entropy provided by an embodiment of the present invention are as follows:

[0098] 1) Obtaining the graph network Structural information and construct adjacency matrix Extract its micro-topological features, including node degree, clustering coefficient, path distribution, etc.

[0099] 2) Use a link prediction model For all node pairs in the graph The existence of links between them is predicted, and the prediction probability matrix is ​​finally obtained. ,in This represents the set of target links that have been selected for protection.

[0100] 3) For each pair of nodes Calculate the predicted information entropy of each link to quantify the uncertainty of the model's prediction of the link's existence: in, Indicates model prediction nodes With nodes The probability that an edge exists between them;

[0101] 4) For each edge in the network graph E, assuming that the graph after removing the perturbation is denoted as E. Therefore, its predictive information entropy needs to be reassessed as follows: in, The graph represents the process of adding perturbation operations. Medium model prediction nodes With nodes The probability that an edge exists between them;

[0102] 5) Define the link disturbance contribution (i.e., the maximum remaining link entropy) as: If after deleting a specific edge If the value is the largest, then this link is considered a candidate perturbation edge;

[0103] 6) Given a disturbance budget ratio Under the condition of (defined as the product of perturbation rate and original network link count), prioritize the selection that makes Largest set of links And perform a perturbation operation, here using edge deletion to perturb the graph. This forms the perturbed graph structure. ;

[0104] 7) Repeat steps 2) through 6) until the disturbance budget is reached.

[0105] In the above method, the micro-topological structure information of the graph network also includes the local path centrality of nodes and the m-hop neighborhood statistics of the relevant paths linking target nodes.

[0106] Link disturbances are achieved only by deleting specific links, thereby minimizing the overall entropy increase. The link prediction model represented includes not only traditional node similarity-based methods, but also graph neural network-based methods, where node embeddings are represented as: in, For graph attention weights, For a trainable parameter matrix, For the first Layer input representation.

[0107] The above method can significantly reduce the predictability of the selected target edge set while maintaining full graph connectivity and average path length to the greatest extent.

[0108] A second aspect of this invention also provides an inverse link disturbance protection system based on maximum residual link entropy, comprising:

[0109] The original network module is used to provide the original graph network and the set of sensitive links in the original graph network;

[0110] A perturbation range module is constructed to define a perturbation range around the sensitive link set and to establish an m-hop subgraph within the perturbation range;

[0111] The path entropy calculation module is used to calculate and find the set of target link edges that maximize the normalized residual path entropy within the range of the m-hop subgraph;

[0112] The perturbation optimization module is used to perform targeted perturbation on the target link edge set to obtain an adversarial graph network.

[0113] A third aspect of the present invention also provides a defense network graph trained by the above method, which can be used as an input graph or a published graph for defense link prediction attack tasks, and is applicable to graph security and privacy protection tasks.

[0114] A fourth aspect of the present invention also provides an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the computer program, when executed by the processor, performs the steps in the above-described disturbance protection method or runs the above-described reverse link prediction algorithm model.

[0115] A fifth aspect of the present invention also provides a readable storage medium storing a computer program, which, when run, performs the steps in the above-described disturbance protection method or runs the above-described image classification model.

[0116] The technical solution of the present invention will be further described in detail below through several embodiments and in conjunction with the accompanying drawings. However, the selected embodiments are only for illustrating the present invention and do not limit the scope of the present invention.

[0117] Specifically, the effectiveness of the inverse link prediction method based on maximum residual link entropy was verified using known network data. It was also compared with existing inverse link prediction methods, and the generalization ability of the proposed method was further verified as follows:

[0118] Example 1

[0119] This embodiment demonstrates a comparison between the training method provided by this invention and existing inverse link prediction methods:

[0120] This embodiment constructs four basic models: the RLD model, which randomly deletes some links in a network where sensitive links have already been removed to generate a perturbation graph. This random strategy disrupts the network structure, increasing the difficulty of predicting sensitive links; the HP model, which heuristically adjusts the network structure to protect sensitive links, making them difficult to predict accurately; and the EDA model, an optimization method based on statistical learning. Unlike traditional genetic algorithms that require crossover and mutation operations, EDA directly learns from samples which links should be deleted or added to continuously approach the optimal network structure perturbation, making it even more difficult for link prediction algorithms to accurately identify sensitive links.

[0121] For different methods, set the fitness coefficients for the GA and EDA models. The RPEM method sets the maximum path length between target node pairs. , and the proportion of the disturbance budget The value is 6%.

[0122] Tables 1 and 2 below show the comparison results of the inverse link prediction method described in this example and existing inverse link prediction methods on three different datasets using different inverse link prediction models. The experimental results are mainly evaluated using AUC and Precision values.

[0123] Table 1. AUC prediction results of perturbation graphs for Lesmis, Throne, and Jazz datasets and different inverse link prediction models.

[0124]

[0125] Table 2. Precision prediction results of perturbation maps for Lesmis, Throne, and Jazz datasets and different inverse link prediction models.

[0126]

[0127] As shown in Tables 1 and 2, the identification method described in this embodiment effectively improves the hiding effect of target sensitive links in the perturbation map.

[0128] Example 2

[0129] This embodiment demonstrates a comparison of the transferability of the training method provided by this invention with existing inverse link prediction methods:

[0130] This embodiment constructs four basic models: the RLD model, which randomly deletes some links in a network where sensitive links have already been removed to generate a perturbation graph. This random strategy disrupts the network structure, increasing the difficulty of predicting sensitive links; the HP model, which heuristically adjusts the network structure to protect sensitive links, making them difficult to predict accurately; and the EDA model, based on statistical learning optimization methods. Unlike genetic algorithms which require crossover and mutation operations, EDA directly learns from samples which links should be deleted or added to continuously approach the optimal network structure perturbation, making it more difficult for link prediction algorithms to accurately identify sensitive links. The target link hiding effect of the perturbation graphs generated by the four models is tested under eight different link prediction algorithms.

[0131] Set the parameters of LP for different link prediction methods. For the two network embedding methods, DeepWalk and HOPE, the dimension of the node embedding vector is set to 32.

[0132] Tables 3, 4, and 5 below show a comparison of the perturbation maps generated by the inverse link prediction method described in this example and existing inverse link prediction methods on three different datasets and under different link prediction models. The experimental results are mainly evaluated using AUC and Precision values.

[0133] Table 3 shows the performance of perturbation maps generated by different models on different link prediction methods on the Lesmis dataset.

[0134]

[0135]

[0136] Table 4 shows the performance of perturbation maps generated by different models on different link prediction methods on the Throne dataset.

[0137]

[0138] Table 5 shows the performance of perturbation maps generated by different models on different link prediction methods on the Jazz dataset.

[0139]

[0140] Based on the above embodiments, it is clear that the training method provided by the embodiments of the present invention effectively improves the hiding effect of sensitive links and has good generalization ability, achieving good performance in different link prediction methods.

[0141] Based on the above embodiments, it is clear that the embodiments of the present invention provide a method, system, and application for inverse link prediction based on the maximum residual link entropy from an information theory perspective in graph tasks. The method includes: acquiring micro-topological information of the target network; dividing the perturbation range; defining the difference in residual link entropy before and after the perturbation from an information theory perspective, and using this difference as a reference index for selecting the perturbation link; selecting the link that maximizes the residual link entropy for targeted perturbation operations, effectively reducing the probability of sensitive links being accurately identified by the link prediction algorithm within a limited perturbation budget; iterating the perturbation operation until a predetermined perturbation budget is reached. The method provided by the present invention accurately quantifies the impact of link perturbation on the prediction of the target sensitive link by the link prediction algorithm through the maximum residual link entropy index, thereby significantly improving the privacy protection effect for the target sensitive link; it has higher perturbation efficiency and lower perturbation budget, while possessing good versatility and portability, and can effectively achieve the privacy protection goal while maintaining the overall stability of the network structure.

[0142] This embodiment also provides an electronic device, including: one or more processors; and a memory, wherein the memory is used to store one or more executable instructions; the one or more processors are configured to execute the one or more executable instructions to perform the training steps described in this embodiment or to run the image classification model provided in this embodiment of the invention.

[0143] Specific embodiments of the present invention may be systems, methods, and / or computer program products. Computer program products may include computer-readable storage media having computer-readable program instructions loaded thereon for causing a processor to implement various aspects of the present invention. Computer-readable storage media may be tangible devices that hold and store instructions used by instruction execution devices. Computer-readable storage media may include, for example, but not limited to, electrical storage devices, magnetic storage devices, optical storage devices, electromagnetic storage devices, semiconductor storage devices, or any suitable combination thereof. More specific examples of computer-readable storage media (a non-exhaustive list) include: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disc read-only memory (CD-ROM), digital multifunction disc (DVD), memory sticks, floppy disks, mechanical encoding devices, such as punch cards or recessed protrusions storing instructions thereon, and any suitable combination thereof.

[0144] It should be noted that although the steps are described in a specific order above, it does not mean that the steps must be executed in the above specific order. In fact, some of these steps can be executed concurrently, or even in a different order, as long as the required function can be achieved.

[0145] It should be understood that the embodiments described above are only some, not all, of the embodiments of the present invention. The detailed description of the embodiments of the present invention is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

Claims

1. A method for inverse link disturbance protection based on maximum residual link entropy, characterized in that, include: Provide original graph network and the set of sensitive links in the original graph network, wherein Represents a set of nodes. Represents the set of edges; Define a perturbation range around the sensitive link set, and establish an m-hop subgraph within the perturbation range; Within the range of the m-hop subgraph, calculate and find the set of target link edges that maximize the normalized residual path entropy; By performing directional perturbation on the target link edge set, an adversarial graph network is obtained.

2. The inverse link disturbance protection method based on maximum residual link entropy according to claim 1, characterized in that, The process of establishing the m-jump subgraph specifically includes: Using the sensitive node as the current node, extract the direct neighbors of the current node; Using the direct neighbor as the new current node, iteratively search for direct neighbors until the number of iterations reaches m.

3. The inverse link disturbance protection method based on maximum residual link entropy according to claim 1, characterized in that, The m-jump subgraph is represented as follows: ,in: ； ； in, This represents the m-jump subgraph; Indicates the target node to be linked And the maximum path length does not exceed The set of nodes; This indicates all linked target nodes with a maximum path length not exceeding [a certain value]. The set of edges; This represents the maximum path length between two nodes; Indicates the target node. Indicates the target node to be linked And the maximum path length does not exceed the length of any other node on path m. This indicates all roads that meet the requirements.

4. The inverse link disturbance protection method based on maximum residual link entropy according to claim 1, characterized in that, The calculation process of the normalized residual path entropy specifically includes: Calculate the residual entropy contribution of each path; Summing up the remaining entropy contributions on all paths yields the total entropy contribution before the perturbation. Each perturbation edge within the perturbation range is taken as the target edge, and the contribution of the total entropy after perturbation is calculated after perturbing the target edge. The normalized residual path entropy is obtained by calculating the ratio of the difference between the total entropy contribution before the perturbation and the total entropy contribution after the perturbation to the total entropy contribution before the perturbation.

5. The inverse link disturbance protection method based on maximum residual link entropy according to claim 4, characterized in that, The calculation method for the contribution of the residual entropy is expressed as follows: ； ； in, d represents the remaining semaphore strength; d(·) represents the degree of the node; Indicates a path; Indicates the index marker of the path; This represents the contribution of the remaining entropy.

6. The inverse link disturbance protection method based on maximum residual link entropy according to claim 5, characterized in that, The calculation method for the pre-perturbation total entropy contribution is expressed as follows: ； ； in, Represents node pairs Residual path entropy in a network without disturbance; Represents the set of all paths from node i to node j; This represents the contribution of the total entropy before the disturbance from node i to node j.

7. The inverse link disturbance protection method based on maximum residual link entropy according to claim 6, characterized in that, The calculation process of the normalized residual path entropy is expressed as follows: ； ； in, This represents the contribution of the total entropy after perturbation of the target edge; This represents the total path entropy contribution from node i to node j after the disturbance. This represents the normalized residual path entropy.

8. The inverse link disturbance protection method based on maximum residual link entropy according to claim 1, characterized in that, Also includes: Based on the maximum selection of candidate perturbation edges using the scoring function; The target link edge set is obtained by iteratively adding from the candidate perturbation edges based on the objective of maximizing the normalized residual path entropy. The scoring function is expressed as follows: ； in, Represents the scoring function; This represents the shortest distance between two nodes; The number of endpoints of the node is represented by ; a and b represent the index labels of the node.

9. The inverse link disturbance protection method based on maximum residual link entropy according to claim 1, characterized in that, Specifically, it includes: Link prediction model For all node pairs The existence of links between them is predicted, and the prediction probability matrix is ​​finally obtained. ,in This refers to the set of sensitive links; For each pair of nodes The predicted information entropy is calculated separately to quantify the uncertainty of the model's prediction of the existence of the link.

10. A reverse link disturbance protection system based on maximum residual link entropy, characterized in that, include: The original network module is used to provide the original graph network and the set of sensitive links in the original graph network; A perturbation range module is constructed to define a perturbation range around the sensitive link set and to establish an m-hop subgraph within the perturbation range; The path entropy calculation module is used to calculate and find the set of target link edges that maximize the normalized residual path entropy within the range of the m-hop subgraph; The perturbation optimization module is used to perform targeted perturbation on the target link edge set to obtain an adversarial graph network.

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