Complex network high-order module and link prediction method based on low-order network reconstruction

Abstract

The invention discloses a complex network high-order module and link prediction method based on low-order network reconstruction, and the method comprises the steps: complementing a missing network structure through low-order link prediction, recognizing a potential high-order module in a prediction network, combining the topological features of nodes and edges, and discriminating a real high-order structure through a machine learning model. And finally, high-order network reconstruction is realized. According to the method, a low-order and high-order structure collaborative modeling mechanism is established, and the limitation that only binary interaction is processed traditionally is broken through; the method is suitable for various high-order network modeling tasks, does not depend on a fixed motif form, and has good universality and expandability.

Description

Technical Field

[0001] This invention relates to the field of link prediction technology, specifically to a high-order module and link prediction method for complex networks based on low-order network reconstruction. Background Technology

[0002] Existing link prediction research mainly focuses on improving the prediction accuracy of low-order networks (binary interactions), such as: efficient algorithms based on domain structure, parameterless methods based on common neighbor indexes, and network structure layering techniques using K-shell decomposition. Although these methods have made significant progress in low-order networks, they still have two fundamental limitations: (1) they cannot be directly applied to high-order network prediction, ignoring the structural characteristics of multi-node interaction modules; (2) they have not solved the transmission problem of "low-order missing leading to high-order distortion", and high-order networks are more sensitive to link missing. Although there have been attempts to predict high-order modules such as SimpleClosure and FSM algorithms, there are still three major technical pain points: ① they rely on specific triangular structure assumptions and are difficult to extend to multi-modules; ② the prediction effects of different high-order models such as hypergraphs and simple complexes vary significantly, and there is a lack of a unified framework; ③ they fail to achieve end-to-end collaboration between low-order prediction and high-order reconstruction, resulting in deviations between the prediction results and the real high-order modules (such as hyperlinks rather than combinations of multiple binary links). Summary of the Invention

[0003] Purpose of the invention: The purpose of this invention is to provide a method for predicting high-order modules and links in complex networks based on low-order network reconstruction, to construct a coupling mechanism between low-order link prediction and high-order module identification, and to solve the problem of the universality of high-order module representation.

[0004] Technical solution: The present invention provides a method for predicting high-order modules and links in complex networks based on low-order network reconstruction, comprising the following steps:

[0005] (1) Based on the network topology file and hyperlink file, extract the pairwise combinations of nodes in all hyperedges to generate the original low-order edge set and construct the low-order network graph;

[0006] (2) Based on hyperlink files, identify and record clearly defined real higher-order modules, including third-order modules and fourth-order modules, to form a set of higher-order modules, and decompose the higher-order modules into edge sets to generate a set of higher-order edges;

[0007] (3) On the low-order network graph, the motif matching algorithm is used to perform induced subgraph isomorphic matching to extract all closed triples and quadruples.

[0008] (4) Divide the original low-order edge set into training set and test set according to the proportion, and supplement isolated nodes to construct training network and test network; on the training network, extract the real high-order modules existing in hyperlinks as positive samples, and randomly select an equal amount of non-high-order structures from the triple or quadruple set of the training network as negative samples to construct a balanced binary classification sample set.

[0009] (5) On the training network, the link prediction algorithm is used to score the missing links, and the high-scoring edges are selected to fill in the missing links and construct the prediction low-order network; for the triples or quadruples in the balanced sample set, the structural features of the nodes and edges are extracted to form a structured training set and the classification model is trained.

[0010] (6) In the prediction of the low-order network, extract the new triples or quadruples as the test set input to the trained classification model and predict whether they are true high-order modules. Based on the prediction results, reconstruct the high-order network and evaluate the high-order link prediction and high-order module discrimination performance.

[0011] Furthermore, in step (1), the construction of the low-order network graph is as follows: extract all node pairs in the hyper-edges from the hyperlink file to ensure the clear definition of the high-order modules and avoid misjudging closed structures composed only of binary edges as high-order modules.

[0012] Furthermore, in step (3), the motif matching algorithm defines the triples as motifs. ,in, The motif consists of three nodes; This indicates that these three nodes are connected in pairs; a quadruple is defined as a complete subgraph of four nodes; the extracted triples or quadruples need to be sorted in ascending order by node number and deduplicated.

[0013] Furthermore, in step (4), the ratio of the training set to the test set includes 5:5, 6:4, 7:3, 8:2 and 9:1, where the ratio represents the ratio of the number of edges in the training set to the number of edges in the test set; negative samples are randomly selected from the triplet or quadruplet set of the training network after excluding positive samples.

[0014] Furthermore, in step (5), the link prediction algorithm includes the Jaccard similarity coefficient, the resource allocation index RA, and the Katz index; among which,

[0015] The formula for calculating the Jaccard similarity coefficient is:

[0016]

[0017] in, and Representing nodes respectively and The set of neighbors;

[0018] The formula for calculating the resource allocation index RA is:

[0019]

[0020] in, Represents a node and common neighbors The degree;

[0021] The Katz index is calculated using the following formula:

[0022]

[0023] in, It is a decay factor used to control the impact of long paths on similarity. Indicates the length of the path. Represents a node and The length between is The number of paths.

[0024] Furthermore, in step (5), the structural features include: node degree, node betweenness centrality, node k-shell value, node clustering coefficient, and edge betweenness centrality; among which,

[0025] The degree of a node is defined as:

[0026]

[0027] in, Represents all nodes Connected nodes The set, The operation represents retrieving the number of elements in the set, thus obtaining the node. The degree;

[0028] Node betweenness centrality is defined as:

[0029] in, Indicates from node To the node The number of all shortest paths, This indicates the nodes passed through in these paths. The number of entries;

[0030] The node k-shell value is obtained through a shell decomposition algorithm;

[0031] The node clustering coefficient is defined as:

[0032]

[0033] in Represents a node The number of triangles involved, For nodes The degree, i.e., the number of neighbors. This indicates the maximum number of edges that can be formed between neighbors;

[0034] Edge betweenness centrality is defined as:

[0035]

[0036] in Indicates from node To the node The number of all shortest paths, This indicates that these paths contain edges. The number of paths.

[0037] Furthermore, in step (5), the classification model includes support vector machine (SVM), random forest, neural network, and logistic regression; after training, it is used to predict the probability of adding new higher-order modules.

[0038] Furthermore, in step (6), the acquisition of new triples or quadruples includes: extracting all triples or quadruples on the prediction low-order network and performing a difference set with the corresponding set of the training network; feature extraction is based only on node and edge structure features and does not include labels.

[0039] Furthermore, in step (6), the evaluation includes the performance of high-order link prediction and high-order module discrimination; both of which use precision and area under the curve (AUC) metrics. Precision is calculated by ranking the predicted probability to find the proportion of real samples in the test set, and AUC is calculated by the receiver operating characteristic (ROC) curve to measure the ability of the method to distinguish between real samples and non-real samples.

[0040] Beneficial Effects: Compared with existing technologies, this invention has the following significant advantages: This invention aims to achieve high-precision collaborative prediction of high-order modules and links in complex networks by utilizing low-order link prediction algorithms and high-order motif recognition machine learning models. Through the low-order link prediction algorithm, effective reconstruction of missing edges in the network is achieved, enhancing the integrity and connectivity of the topology; through the high-order motif recognition model, accurate identification of potential high-order modules is achieved, reconstructing the group collaboration relationships of the high-order network. Compared with existing technologies, this invention can better capture high-order module information and multi-node collaborative relationship characteristics in complex networks, and is particularly suitable for the recovery and reconstruction of systems with obvious group interaction patterns or high-order module dependencies. Attached Figure Description

[0041] Figure 1 This is a simplified flowchart illustrating the process of extracting triples from a low-order network according to the present invention.

[0042] Figure 2 The present invention obtains new triples by identifying triples in the prediction network and taking the difference between the triples set of the training network;

[0043] Figure 3 The four basic structural features of the network selected for this invention include the number of nodes, the number of low-order edges (i.e., the total number of connected edges in the network), the number of third-order module edges and the number of modules, and the number of fourth-order module edges and the number of modules. Detailed Implementation

[0044] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0045] This invention provides a method for predicting high-order modules and links in complex networks based on low-order network reconstruction, comprising the following steps:

[0046] (1) First, based on the topology file and hyperlink file, extract all node pairs (i.e., pairwise combinations) in the original hyperedges to construct a low-order edge set, and use this to construct a low-order network graph. ,in For a set of nodes, Let be the set of edges, such as Figure 1 As shown in the diagram. Then, combining the hyperlink files, the explicitly defined real high-order modules are identified and extracted. For example, a third-order module represents a hyperlink composed of three nodes (e.g., ABC), and a fourth-order module represents a hyperlink composed of four nodes (e.g., ABCD). The high-order modules are then broken down into edges and deduplicated to obtain a set of high-order edges. The red triangles in the diagram represent the real third-order modules marked in the hyperlink files. Based on this, all node information from the original network is supplemented to generate a complete high-order network, which is used for subsequent verification of high-order module identification and link prediction performance.

[0047] It should be noted that the higher-order modules predicted in this invention specifically refer to multi-node interaction relationships explicitly defined by hyperlinks, rather than closed structures composed of multiple binary edges. For example, although a triangular structure may be composed of edges AB, BC, and AC, it is not considered a true higher-order module if it is not defined as a higher-order structure in a hyperlink. This method differs from traditional methods that construct higher-order structures based on low-order combinations, and can more accurately capture the real multi-node interaction patterns in complex networks.

[0048] (2) In the established low-order network The above method uses motif matching to extract the triplet structure in the network. Taking a third-order module as an example, the target motif structure is defined as follows: ,in:

[0049] The motif consists of three nodes;

[0050] This indicates that these three nodes are connected in pairs, forming a closed triangular structure.

[0051] The definition of a fourth-order motif is similar, namely a complete subgraph consisting of four nodes. In this embodiment, the subsequent steps all take the identification and prediction of a third-order motif as an example. The processing flow of a fourth-order motif is the same and can be directly deduced.

[0052] By drawing With the phantom Perform induced subgraph isomorphism matching to obtain all subgraph embeddings identical to the triangular motif, i.e., extract all subgraph embeddings from the low-order network that satisfy the condition. triples Each triple is sorted in ascending order of node number and deduplicated, ultimately yielding a set of all unique triangular structures in the network. The extraction process for this triple is as follows: Figure 1 As shown, this provides a structural foundation for the discrimination and prediction of subsequent higher-order modules.

[0053] (3) Based on the low-order edge set, the network is divided into training and testing sets. The partition ratios selected in this invention include: 5:5, 6:4, 7:3, 8:2, and 9:1, where... This represents the ratio of the number of edges between the training set and the test set, i.e., links are randomly selected from the original low-order network. Partially build the test suite, retain A training set is partially constructed, and isolated nodes are added to it to build a network, providing an input graph for the low-order link prediction module.

[0054] After the training network is built, the corresponding real third-order modules in the training network are extracted using the set of real high-order modules as positive samples, and their number is set to be 1. The training set is then decomposed into edges to obtain the set of true high-order structure edges. Simultaneously, using a motif matching method, all triplet sets are enumerated from the training network, and the identified true high-order modules are removed, resulting in the set of non-high-order triplets. To construct a balanced binary classification training dataset, a number of triplets are randomly selected from this set. The triples are used as negative samples.

[0055] The final construction obtained by Positive samples and The training set, consisting of negative samples, is used for training the subsequent higher-order module discrimination model.

[0056] (4) Based on the constructed balanced triplet sample set (including real higher-order modules as positive samples and an equal number of non-higher-order structures as negative samples), such as Figure 3 As shown, this invention extracts structural features from the nodes and edges in each triplet and sets classification labels (positive samples are labeled 1, and negative samples are labeled 0). The selected features include: node degree, node betweenness centrality, node k-shell value, node clustering coefficient, and edge betweenness centrality. The definitions of each feature are as follows:

[0057] The degree of a node represents the number of locally connected nodes, that is, the number of edges that directly connect the node to other nodes. The formula is as follows:

[0058]

[0059] in Represents all nodes Connected nodes The set, The operation represents retrieving the number of elements in the set, thus obtaining the node. The degree;

[0060] Node betweenness centrality represents a node's "mediating ability" in a network, that is, the frequency with which a node appears in the shortest paths of all node pairs, as shown in the following formula:

[0061]

[0062] in Indicates from node To the node The number of all shortest paths, This indicates the nodes passed through in these paths. The number of entries;

[0063] K-shell value: Evaluates the core level of a node in a network using a shell decomposition algorithm. This algorithm continuously peels away nodes with degrees not exceeding a certain threshold from the network. The k-shell value of a node is determined by the stripping round in which it is removed. It reflects the node's centrality and propagation ability within the overall structure.

[0064] The node clustering coefficient represents the tightness of connections between a node's neighbors, that is, the ratio between the actual number of edges connecting the node's neighbors and the maximum possible number of edges connecting the nodes. The calculation formula is:

[0065]

[0066] in Represents a node The number of triangles involved, For nodes The degree, i.e., the number of neighbors. This indicates the maximum number of edges that can be formed between neighbors (i.e., the maximum possible number of triangles).

[0067] Edge betweenness centrality represents the frequency with which an edge appears in the shortest paths between all pairs of nodes, reflecting the "bridging role" of the edge in connecting different communities or modules. The calculation formula is:

[0068]

[0069] in Indicates from node To the node The number of all shortest paths, This indicates that these paths contain edges. The number of paths.

[0070] Using the above metrics, features are extracted from the nodes and edges involved in each triplet, ultimately forming a complete feature vector. Classification labels are then added to create a balanced training sample set, providing input for subsequent training of machine learning models to recognize higher-order structures. For example... Figure 1 As shown, this step completes the construction of training set structure feature vectorization and label annotation.

[0071] (5) Based on the constructed training network, this invention employs three typical low-order link prediction algorithms—Jaccard Coefficient, Resource Allocation Index (RA), and Katz Index—to predict the missing potential links in the training network, thereby generating an ordered list of possible links. Subsequently, the top N edges with the highest scores (where N is the number of edges in the network, i.e., the number of missing edges) are selected first and added to the training network to construct a predicted complete low-order network for subsequent high-order structure reconstruction and recognition, such as... Figure 1 As shown.

[0072] The principles and calculation methods of the three link prediction algorithms are as follows:

[0073] Jaccard similarity coefficient: A similarity measure based on the overlap of the sets of common neighbors of nodes. Its calculation formula is as follows:

[0074]

[0075] in, and Representing nodes respectively and The set of neighbors. A higher score indicates that two nodes share more neighbors, and are more likely to form a potential link.

[0076] Resource Allocation Index (RA): Simulates the process of neighboring nodes "allocating resources" to the target node, emphasizing that lower-degree nodes provide a greater contribution. The calculation formula is:

[0077]

[0078] in Represents a node and common neighbors The degree of the graph. This method is particularly effective in sparse graphs and is suitable for identifying potential connections in small clusters.

[0079] Katz index: a global link prediction method that considers all possible links from... arrive The path is determined, and shorter paths are given higher weights. The calculation formula is:

[0080]

[0081] in It is a decay factor used to control the impact of long paths on similarity. Indicates the length of the path. Represents a node and The length between is The number of paths. The Katz index can capture indirect relationships and is an effective means of identifying weak connections.

[0082] (6) Based on the constructed predictive complete low-order network, the phantom recognition method is used to extract triples from the network to obtain the prediction network triple set. The difference operation is then performed between this set and the constructed training network triple set to obtain the set of triples added during the prediction process. This process is as follows: Figure 2 As shown, the gray triangles represent the new triplet structures generated through low-order link prediction.

[0083] Subsequently, using the defined structural feature extraction method for nodes and edges, features (excluding the label column) are calculated for each of the newly added triples, thereby constructing a test sample set for the corresponding link prediction algorithm. This test set will serve as input to the subsequent machine learning classification model to determine whether the newly added triples constitute a true higher-order module.

[0084] (7) The training and test sets were standardized, and the training set was used to train four machine learning models (neural network, random forest, support vector machine, and logistic regression). Then, the probability of newly formed triples being true third-order modules was predicted for the test set samples, and these predicted new third-order modules were marked in blue. Based on these predicted third-order modules, a higher-order network was reconstructed, and the prediction results were compared with the obtained higher-order network to evaluate the prediction results. The process is as follows: Figure 1 As shown.

[0085] (8) Based on the machine learning model, the predicted probability of the newly added triplet being a true third-order module is calculated. Each triplet is split into three edges, and the predicted probability is assigned as the weight of these three edges. At the same time, the training network triplet is also split into three pairs of edges, and a weight of 1 is assigned to each pair. Then, the predicted edge set and the training edge set are merged, and duplicate edges are deduplicated: when an edge appears repeatedly in multiple triplets, the edge with the larger predicted probability value is retained. The final edge set is consistent in size with the original low-order edge set from the previous step, forming the edge-weight set after prediction. Based on this, the performance evaluation of the third-order link and prediction is carried out, using two metrics: the third-order link prediction precision and the area under the curve (AUC).

[0086] The specific process for calculating the precision of third-order link prediction is as follows:

[0087] First, read the set of edges composed of all real third-order modules in [1] as the overall set of real edges;

[0088] Then, the known set of third-order module edges in the training set is removed to obtain the set of true third-order module edges corresponding to the test set, and its number of edges is denoted as . Using the edge-weight set constructed in the previous step, sort it from highest to lowest weight;

[0089] Before sequential traversal When predicting edges, if an edge does not belong to the set of real edges in the training set but belongs to the set of real edges in the test set, it is considered a correct third-order link prediction and the count is accumulated.

[0090] Finally, divide by the number of hits. The precision of the third-order link prediction is calculated and used to measure the recovery performance of the proposed method at the higher-order edge level.

[0091] The specific process for calculating the area under the prediction curve of a third-order link is as follows:

[0092] Each edge in the predicted edge set is labeled with a true label: if the edge belongs to the true 3rd-order edge set in the test set, it is labeled as a positive sample (1); otherwise, it is labeled as a negative sample (0). Simultaneously, the prediction score of the edge is used as the prediction probability of the corresponding sample. Subsequently, the True Positive Rate (TPR, representing the proportion of true positive examples correctly predicted as positive) and False Positive Rate (FPR, representing the proportion of true negative examples incorrectly predicted as positive) are calculated at different thresholds, and the Receiver Operating Characteristics (ROC) curve is plotted with FPR on the horizontal axis and TPR on the vertical axis. The area under the ROC curve is calculated to obtain the AUC value. The closer the AUC is to 1, the better the method's performance in distinguishing between true and false 3rd-order links.

[0093] Finally, the performance of the present invention in the structure discrimination task of the third-order module prediction is evaluated. The precision and area under the curve are still used as two indicators to quantify the structure discrimination ability of the method.

[0094] The specific calculation process for the third-order module prediction precision is as follows:

[0095] First, obtain the predicted probability that the newly added triples output by the trained machine learning model are true third-order modules;

[0096] Then, all real 3rd-order modules are read, and the sets of real 3rd-order modules known from the training network are removed to obtain the set of real 3rd-order modules in the test network, i.e., the missing real 3rd-order modules, the number of which is denoted as . ;

[0097] Subsequently, the newly added triples are sorted from highest to lowest predicted probability, and the top-scoring triples are selected. The set of three triples is the most likely candidate set of third-order modules;

[0098] If any of these candidate triples exists in the set of real third-order modules in the test network, it is considered a correct prediction, and the number of hits is accumulated.

[0099] Finally, the number of hits divided by The precision index of the third-order module prediction is obtained and used to evaluate the accuracy performance of the method in third-order module recognition.

[0100] The specific process for calculating the area under the predicted curve using the third-order module is as follows:

[0101] First, calculate the number of triples in the fully connected network consisting of all nodes in the network, and denote the total number as . ;

[0102] Then, subtract the set of triples generated after low-order link prediction from this set to obtain the set of triples that are not considered to form triples during link prediction, and denote its number as . ;

[0103] For all newly predicted triples, label them based on whether they appear in the real third-order module set of the test network: if they exist, they are positive samples (label 1); otherwise, they are negative samples (label 0). Simultaneously, count the number of correctly predicted real higher-order triples. The number of prediction errors To ensure the integrity of the negative sample distribution, further supplementary construction is needed. A negative sample triple with a probability of 0;

[0104] Based on this, the true labels and predicted scores of all predicted samples are input into the ROC curve constructor, and a curve is plotted with FPR as the horizontal axis and TPR as the vertical axis. The area under the curve is calculated to obtain the AUC value. The closer the AUC is to 1, the better the method's performance in distinguishing between true and false third-order modules.

[0105] (11) The fourth-order link and module prediction is implemented using the same method as described above, that is, the high-order structure composed of four nodes is used as the prediction target, and the process is consistent with that of the third-order prediction, including low-order link prediction, motif identification, structural feature extraction, classification model training and prediction. In terms of performance evaluation, the calculation methods of the precision and AUC of the fourth-order module are consistent with those of the third-order module. The discriminative ability and prediction effect of the method are measured by comparing with the real fourth-order module of the test network. Finally, the prediction results of the third-order and fourth-order modules are summarized in a unified manner.

Claims

1. A method for predicting high-order modules and links in complex networks based on low-order network reconstruction, characterized in that, Includes the following steps: (1) Based on the network topology file and hyperlink file, extract the pairwise combinations of nodes in all hyperedges to generate the original low-order edge set and construct the low-order network graph. (2) Based on hyperlink files, identify and record clearly defined real high-order modules, including third-order modules and fourth-order modules, to form a set of high-order modules, and decompose the high-order modules into edge sets to generate a set of high-order edges; (3) On the low-order network graph, the motif matching algorithm is used to perform induced subgraph isomorphic matching to extract all closed triplet and quadruple sets; (4) Divide the original low-order edge set into training set and test set according to the proportion, and supplement isolated nodes to build training network and test network; on the training network, extract the real high-order modules existing in hyperlinks as positive samples, and randomly select an equal amount of non-high-order structures from the triple or quadruple set of the training network as negative samples to build a balanced binary classification sample set. (5) On the training network, the link prediction algorithm is used to score the missing links, and the high-scoring edges are selected to fill in the missing links and construct the prediction low-order network. For triples or quadruples in the balanced sample set, the structural features of nodes and edges are extracted to form a structured training set and the classification model is trained. (6) In the prediction of the low-order network, extract the new triples or quadruples as the test set input to the trained classification model and predict whether they are true high-order modules. Based on the prediction results, reconstruct the high-order network and evaluate the high-order link prediction and high-order module discrimination performance.

2. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (1), the construction of the low-order network graph is as follows: extract all node pairs in the hyper-edges from the hyperlink file to ensure the clear definition of the high-order modules and avoid misjudging closed structures composed only of binary edges as high-order modules.

3. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (3), the motif matching algorithm defines the triple as motif M = (V m E m ), where V m ={0,1,2} indicates that the module consists of three nodes; E m ={(0,1),(0,2),(1,2)} indicates that these three nodes are connected in pairs; a quadruple is defined as a complete subgraph of four nodes; the extracted triples or quadruples need to be sorted in ascending order by node number and deduplicated.

4. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (4), the ratio of the training set to the test set includes 5:5, 6:4, 7:3, 8:2 and 9:1, where the ratio represents the ratio of the number of edges in the training set to the number of edges in the test set; negative samples are randomly selected from the triple or quadruple set of the training network after excluding positive samples.

5. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (5), the link prediction algorithm includes the Jaccard similarity coefficient, the resource allocation index RA, and the Katz index; among which, The formula for calculating the Jaccard similarity coefficient is: Where Γ(u) and Γ(v) represent the neighbor sets of nodes u and v, respectively; The formula for calculating the resource allocation index RA is: Among them, C D (z) represents the degree of z, the common neighbor of nodes u and v; The Katz index is calculated using the following formula: Where β is the decay factor, used to control the impact of long paths on similarity, and l represents the path length. This represents the number of paths of length l between nodes u and v.

6. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (5), the structural features include: node degree, node betweenness centrality, node k-shell value, node clustering coefficient, and edge betweenness centrality; among which, The degree of a node is defined as: C D (v)=|{u∈V|(v,u∈E)}| Where {u∈v|(v,u∈E)} represents the set of all nodes u connected to node v, and the |·| operation represents taking the number of elements in the set, thus obtaining the degree of node v; Node betweenness centrality is defined as: Where, σ st σ represents the number of all shortest paths from node s to node t. st (v) represents the number of paths that pass through node v; The node k-shell value is obtained through a shell decomposition algorithm; The node clustering coefficient is defined as: Where T(v) represents the number of triangles that node v participates in, and C D (v) represents the degree of node v, i.e., the number of its neighbors. This indicates the maximum number of edges that can be formed between neighbors; Edge betweenness centrality is defined as: Where σ st σ represents the number of all shortest paths from node s to node t. st (e) represents the number of paths that contain edge e.

7. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (5), the classification model includes support vector machine (SVM), random forest, neural network and logistic regression; after training, it is used to predict the probability of adding new higher-order modules.

8. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (6), the acquisition of new triples or quadruples includes: extracting all triples or quadruples on the prediction low-order network and performing a difference set with the corresponding set of the training network; feature extraction is based only on node and edge structure features and does not include labels.

9. The method for predicting high-order modules and links in complex networks based on low-order network reconstruction according to claim 1, characterized in that, In step (6), the evaluation includes the performance of high-order link prediction and high-order module discrimination; both of which use precision and area under the curve (AUC) as indicators. Precision is calculated by ranking the predicted probability to find the proportion of real samples in the test set, and AUC is calculated by the receiver operating characteristic (ROC) curve to measure the ability of the method to distinguish between real samples and non-real samples.

Images