Network alignment method based on intra-layer and inter-layer link optimization

Abstract

The invention discloses a network alignment method based on intra-layer and inter-layer link optimization. The method comprises the steps that a data set is acquired and preprocessed; performing consistency measurement, calculating consistency indexes of all known anchor nodes, and calculating a consistency value of the whole network based on the consistency indexes; on the basis of a consistency hypothesis, complementing lost links and potential links in a network layer by utilizing a relationship between different anchor node pairs, and optimizing an original adjacent matrix and node features by utilizing graph structure learning; calculating node embedding of the source network and the target network in different layers, and generating an initial hierarchical alignment matrix based on the embedding; according to the method, the topological structure of each layer of the network is learned and optimized by utilizing the graph structure, and more complete and consistent network representation is obtained; the consistency in network alignment is enhanced, and the effect of network alignment is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of graph machine learning, and specifically to a network alignment method based on optimizing intra-layer and inter-layer links. Background Technique

[0002] Traditional network alignment methods usually assume consistency, that is, the same entities in different networks should have similar features and topological connections. With the increase in network scale and data sparsity, the problem of missing intra-layer and inter-layer topological links becomes more prominent. For example, the edges between some intra-layer nodes are not discovered due to data loss or observation limitations, and the number of anchor node links between layers is also relatively scarce. These missing topological connections lead to the loss of network structure information, thereby affecting the structure-based network alignment effect. The consistency principle means that if a pair of cross-network nodes have consistent attributes and topological connections, then there may be an alignment between them.

[0003] However, in reality, the topological connections in the network are not always complete or consistent, especially in multi-layer networks, the phenomenon of missing intra-layer and inter-layer links is particularly common. For example, the connection patterns between nodes may be different, that is, the structures are inconsistent. In addition, the nodes in different network layers may have different attribute information, that is, the attributes are inconsistent. To address the issue of inconsistency in the network structure, existing technologies model multi-view structure representations in each graph, integrating node views, edge views, and subgraph views to reduce the impact of noise and inconsistency on the results, and by introducing perturbed data augmentation techniques and noise-adaptive loss functions, enabling the model to adapt to noise that violates the consistency constraints, or relaxing the strong consistency assumption through regularized meta-metric learning to mitigate the impact of inconsistency on alignment. However, by introducing regularization or multi-view techniques to relax the consistency constraints, although it can alleviate the inconsistency problem to a certain extent, it cannot fundamentally solve the problem of structural inconsistency, and lacks universality and wide applicability. Summary of the Invention

[0004] The purpose of the present invention is to provide a network alignment method based on optimizing intra-layer and inter-layer links to solve the problems presented in the above background technique.

[0005] To solve the above technical problems, the present invention provides the following technical solution: A network alignment method based on optimizing intra-layer and inter-layer links, including the following steps:

[0006] S1: Obtain a data set and perform preprocessing;

[0007] S2: Perform consistency measurement; the consistency measurement is to calculate the consistency index of all known anchor nodes and use the consistency index to calculate the consistency value of the entire network;

[0008] Preferably, the consistency in the consistency measure includes topological consistency and attribute consistency. The consistency measure in the topological consistency includes node degree consistency measure and curvature consistency measure. The attribute consistency is limited to networks with attributes and includes the consistency of node attributes.

[0009] Preferably, the node degree consistency measure is used to reflect the connectivity of a node in a network. The connectivity is specifically expressed as: when nodes v A and v a serve as a pair of anchor nodes, the node v A in the source network should be connected to nodes v B , v C and v D . The node v a in the target network should be connected to nodes v b , v c and v d . To maintain consistency, the node v a in the target network should also maintain the same connection pattern with nodes v b , v c and v d .

[0010] The node degree consistency measure is defined as follows: Define the degree consistency measure C1 to quantify the degree difference between the node v A in the source network and the node v a in the target network. Calculate the average value, maximum value, and minimum value of the degree differences of the known anchor nodes. The average value of the degree differences is expressed as:

[0011]

[0012] where |T| represents the number of known anchor links, deg(·) represents the degree of a node, and t s and t t represent nodes.

[0013] The maximum value and minimum value of the degree differences are expressed as follows:

[0014]

[0015] Normalize the average value, maximum value, and minimum value of the degree differences. The normalization formula is:

[0016]

[0017] to obtain the degree consistency measure C1.

[0018] Preferably, the curvature consistency metric is used to reflect the structural characteristics of the local area where the node is located. The curvature consistency metric is as follows: Using the Ricci curvature difference at the node level as the evaluation criterion, calculate the Ricci curvature κ(i,j) of each edge in the network, aggregate the curvature information of the edge to each node, and take the average value:

[0019]

[0020] where ε represents the set of edges in the network, and κ i is the node-level curvature of node i;

[0021] Based on the node-level curvature difference, define the curvature consistency metric C2:

[0022]

[0023] where, and represent the node-level curvatures of nodes t s and t t ;

[0024] Use hyperbolic embedding to ensure the calculation of the attribute consistency of the anchor nodes. The hyperbolic space in hyperbolic embedding has a negative constant curvature. Embed the node attributes into the hyperbolic space and calculate the hyperbolic distance to measure the consistency of the node attributes. Specifically, the embeddings of nodes t s and t t in the hyperbolic space are h s and h t respectively, and the hyperbolic distance between them is given by the following formula:

[0025]

[0026] where c represents the curvature of the hyperbolic space, represents the Möbius addition in the hyperbolic space, and is defined as follows:

[0027]

[0028] Weightedly combine the topological consistency and attribute consistency metrics to obtain the final consistency metric C:

[0029] C = αC1 + βC2 + γC3

[0030] where α, β, and γ are hyperparameters that control the weights of different consistency metrics. Set γ = 0 when the input network has no node attributes.

[0031] S3: Based on the consistency assumption, use the relationships between different pairs of anchor nodes to complete the missing and potential links within the layer. Meanwhile, use graph structure learning to optimize the original adjacency matrix and node features;

[0032] Preferably, it specifically includes: setting (v A , v a ) and (v B , v b ) as two pairs of anchor nodes, where the uppercase letter subscript represents the nodes in the source network and the lowercase letter subscript represents the nodes in the target network; (v a , v b ) belongs to the edge set ε t of the target network. According to the consistency assumption, if there is an edge between v a and v b , and there are cross-layer anchor links between v A and v a , v B and v b , that is, (v a , v b ) ∈ ε, (v A , v a ) ∈ T, (v B , v b ) ∈ T, then it is speculated that there is also an edge between v A and v B in the edge set ε s between the source network and the target network, that is, (v A , v B ) ∈ ε s ; Check all pairs of anchor nodes to determine whether the pairs of anchor nodes satisfy the consistency assumption. When there is an edge between two known pairs of anchor nodes in one of the networks, check whether their corresponding nodes are still connected in the other network. If not, add an edge between these two nodes. The added edge is used to restore the connection pattern of the source network and the target network, make up for the information loss caused by the missing links in the network, and keep the consistency assumption valid; and the supplementation of the intra-layer edges is limited to between anchor nodes. During the process of supplementing intra-layer links, only the links that can be confirmed under the consistency assumption are allowed to be supplemented.

[0033] Preferably, the optimization of the original adjacency matrix and node features specifically includes: calculating the similarity between nodes using different kernel functions, constructing a similarity graph S, performing a sparsification operation, and retaining partial structural features of the original graph through residual links; the similarity graph is an important part of graph learning, used to reflect the similarity degree between nodes and provide a basis for edge weighting in the graph model. To achieve this goal, node embeddings are calculated using inner product kernels, cosine similarity kernels, and diffusion kernel functions; the sparsification operation is to sparsify it using the K-nearest neighbor algorithm. By traversing each node, the top K nodes most similar to the node are retained, and the edge weights of other nodes are set to zero. The mathematical expression is as follows:

[0034]

[0035] where top K(S i ) represents the top K minimum or maximum values of node i; processing the sparsified similarity matrix S K , for each pair of nodes (i, j), retain the maximum value in their similarity matrix. The mathematical expression of this process is as follows:

[0036]

[0037] to obtain a symmetric and sparse similarity matrix

[0038] S4: Calculate the node embeddings of the source network and the target network at different layers, generate an initial hierarchical alignment matrix based on the embeddings, identify stable nodes according to the alignment results, and adjust the embeddings according to the identification results.

[0039] Preferably, calculate the node embeddings of the source network and the target network at different layers, generate an initial hierarchical alignment matrix based on the node embeddings, identify stable nodes according to the defined node stability criteria, and use the current alignment results to identify the stable nodes and adjust the embeddings accordingly to amplify the influence of stable nodes in the alignment process.

[0040] Preferably, given a node v and its embeddings h (1) , …, h (k) at each layer of the network, where the network layers range from the first layer to the k-th layer, determine whether the node is a stable node. If node v is stable, then in the alignment matrices of all layers, the anchor node of node v is the same, and its corresponding alignment score is greater than a preset confidence factor λ, that is, for all layers 0 ≤ l ≤ k, when node v is stable, the following conditions are satisfied:

[0041]

[0042] where S (l)(v, u) represents the alignment score between node v and node u at the l-th layer. v' is the anchor node of node v. When the above formula holds, node v is considered stable, and nodes that do not satisfy the above formula are considered unstable nodes.

[0043] Compared with the prior art, the beneficial effects achieved by the present invention are as follows: The present invention aims at the method for complementing the intra-layer and inter-layer topological links, uses graph structure learning to optimize the topological structure of each layer of the network, so as to obtain a more complete and consistent network representation. The present invention enhances the consistency in network alignment by means of the original link information of the anchor nodes and the complemented topological structure, further improves the effect of network alignment, and in the process of representation learning, filters out reliable and information-rich anchor node pairs through the node alignment score, enriches the network representation by emphasizing the information of these nodes. Through extensive empirical verification, the technology of the present invention shows higher accuracy and remarkable generality. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] The drawings are used to provide a further understanding of the present invention, and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention, but do not constitute a limitation to the present invention.

[0045] In the drawings:

[0046] Figure 1 is a schematic flowchart of the method steps provided by the embodiment of the present invention;

[0047] Figure 2 is a schematic diagram of intra-layer edge complementation provided by the embodiment of the present invention;

[0048] Figure 3 is a schematic diagram of a pseudo-code example of emphasizing the stable anchor node part provided by the embodiment of the present invention;

[0049] Figure 4 is a schematic diagram of a pseudo-code example of an optimization algorithm based on intra-layer and inter-layer links provided by the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0050] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0051] The embodiments of the present invention are combined with Figures 1 - 4 , and specifically provide the following technical solutions: A network alignment method based on intra-layer and inter-layer link optimization, including the following steps:

[0052] S1: Obtain the dataset and preprocess it;

[0053] In this embodiment, 64 real-world datasets are selected for network alignment, including Douban online - Douban offline, ACM - DBLP, Flickr - Myspace, Flickr - Lastfm, Cora1 - Cora2, and Allmv - Imdb; the dataset sizes are different, presenting different characteristics, with sparse or dense edges; and due to the scarcity of anchor nodes in the real network, the dataset is divided into a 20% training set and an 80% test set.

[0054] S2: Perform consistency measurement, that is, calculate the consistency index of all known anchor nodes, and calculate the consistency value of the entire network based on the consistency index;

[0055] In this embodiment, in the network alignment task, the dataset usually contains two or more adjacency matrices, and some networks have node attributes. In this context, consistency is divided into topological consistency and attribute consistency (only for networks with attributes). Consistency measurement is mainly for known anchor nodes. By calculating the consistency index of all known anchor nodes and summing them, the consistency value of the entire network is finally obtained. Specifically, the goal of node consistency measurement is to evaluate the degree of difference between the topology and attributes of the same nodes across networks.

[0056] Among them, topological consistency: mainly focuses on the structural similarity between nodes in the network, especially the connection patterns of anchor nodes in different networks. An important aspect of consistency measurement is the degree consistency of nodes. The degree reflects the number of connections between a node and other nodes, and can reflect the position and importance of the node in the network.

[0057] Exemplarily, node degree consistency measurement: intuitively reflects the connectivity of nodes in the network. When nodes v A and v a serve as a pair of anchor nodes, ideally, node v A in the source network should be connected to nodes v B , v C and v D , while node v a in the target network should be connected to nodes v b , v c and v d . To maintain consistency, node v a in the target network should also be connected to nodes v b , v c and v d in the same connection pattern. Therefore, node v A in the source network and node va The degree difference should be as small as possible; to quantify this degree difference, a degree consistency metric C1 is defined, and the average, maximum, and minimum values of the degree differences of the known anchor nodes are calculated. The average degree difference is expressed as:

[0058]

[0059] where |T| represents the number of known anchor links, deg(·) represents the degree of a node, and t s and t t represent nodes.

[0060] The maximum value of the degree difference and the minimum value are expressed as follows to capture the extreme values of the degree difference:

[0061]

[0062] Finally, to eliminate the scale effect of the degree difference, these values need to be normalized to obtain the degree consistency metric:

[0063]

[0064] Exemplarily, the curvature consistency metric: In addition to the node degree, the local structural information of the edges is also crucial for evaluating the consistency of the node topology. The curvature of the edges can reflect the local structural characteristics where the nodes are located. For example, a tight structure such as a triangle will result in positive curvature, while a quadrilateral structure will show zero curvature, and a loose structure will produce negative curvature;

[0065] Therefore, in the topological consistency metric, the Ricci curvature difference at the node level is used as the evaluation criterion. Specifically, first, the Ricci curvature k(i,j) of each edge in the network is calculated, and then the curvature information of these edges is aggregated to each node and the average value is taken:

[0066]

[0067] where ε represents the set of edges in the network, and k i is the node-level curvature of node i.

[0068] Next, based on the node-level curvature difference, a curvature consistency metric C2 is defined:

[0069]

[0070] where, and represent t s and t t the node-level curvatures of the nodes;

[0071] Attribute Consistency: Focus on the attribute similarity of the same entity in different networks. Ideally, the same node across networks should have similar attributes. However, due to the power-law distribution or scale-free property of most networks, embedding nodes into Euclidean space may lead to distortion of attribute representation, which makes the calculation of attribute consistency of anchor nodes unreliable.

[0072] To alleviate this problem, hyperbolic embedding is proposed as an effective solution. The hyperbolic space has a negative constant curvature, and its geometric structure results in less distortion for tree-like structure embeddings, making it suitable for representing the hierarchical relationships between nodes in the network. Therefore, node attributes are embedded into the hyperbolic space, and the hyperbolic distance is calculated to measure the consistency of node attributes.

[0073] Specifically, the embeddings of node t s and t t in the hyperbolic space are h s and h t , respectively, and the hyperbolic distance between them is given by:

[0074]

[0075] where c represents the curvature of the hyperbolic space, denotes the Möbius addition in the hyperbolic space, which is defined as follows:

[0076]

[0077] To comprehensively evaluate the node consistency across networks, the topological consistency and attribute consistency metrics are weighted and combined to obtain the final consistency metric C:

[0078] C = αC1 + βC2 + γC3

[0079] where α, β, and γ are hyperparameters that control the weights of different consistency metrics. If the input network has no node attributes, then γ = 0 is set.

[0080] Through this comprehensive consistency metric, the topological and attribute consistency of nodes in different networks can be accurately evaluated, thereby improving the accuracy and effectiveness of cross-network node alignment.

[0081] S3: Based on the consistency hypothesis, use the relationships between different pairs of anchor nodes to complement the missing and potential links within the layer, and at the same time, use graph structure learning to optimize the original adjacency matrix and node features;

[0082] In this embodiment, during the analysis and mapping of a multi-layer network, the completion of intra-layer links is an important task, especially when the anchor node pairs between the source network and the target network have been determined. In this method, from the perspective of the consistency hypothesis, the relationships between different anchor node pairs are studied to complete the missing and potential links in the network within the layer.

[0083] Specifically, let (v A , v a ) and (v B , v b ) be two pairs of anchor nodes, where the uppercase letter subscript represents a node in the source network and the lowercase letter subscript represents a node in the target network; (v a , v b ) belongs to the edge set ε t of the target network. According to the consistency hypothesis, if there is an edge between v a and v b , and there are cross-layer anchor links between v A and v a , v B and v b , that is, (v a , v b ) ∈ ε t , (v A , v a ) ∈ T, (v B , v b ) ∈ T, then it can be speculated that there may also be an edge between v A and v B in the edge set ε s between the source network and the target network, that is, (v A , v B ) ∈ ε s ; this speculation is based on the consistency hypothesis, believing that there is a mapping relationship and structural similarity between the anchor node pairs of the two networks, so as to ensure that the corresponding anchor nodes should have edge connectivity. Therefore, if the anchor node pairs between the source network and the target network are consistent, then the edges between them should also conform to the consistency principle.

[0084] Exemplarily, the task at this stage is to check all anchor node pairs to determine whether they satisfy the consistency hypothesis, that is, if there is an edge between two known anchor node pairs in one network, then check whether their corresponding nodes are still connected in the other network. If not, add an edge between these two nodes to restore the connection pattern between the source network and the target network. Adding an edge between the anchor node pairs can make up for the information loss caused by the missing links in the network and keep the consistency hypothesis valid.

[0085] Furthermore, it should be particularly emphasized that: the supplementation of intra-layer edges is limited to between anchor nodes. During the process of supplementing intra-layer links, only those links that can be confirmed under the consistency assumption are allowed to be supplemented. For the supplementation of edges between unknown node pairs, due to the lack of consistency assumption, such supplementation is considered infeasible. For example, in Figure 2 the node pair v A and v C should not be directly connected because there is a lack of clear mapping relationship and consistency guarantee between them which is prone to error accumulation. Without the consistency support between anchor node pairs, it is impossible to ensure the correctness and effectiveness of their connections. Therefore, all edge supplements must be strictly operated in accordance with the consistency of anchor node pairs to avoid introducing incorrect connections into the network; through this method, the missing links can be effectively supplemented between the source network and the target network, and the rationality of the network structure can be maintained. Especially through the connection supplementation between anchor node pairs, the accuracy of network mapping can be further improved, thereby providing a more reliable basis for subsequent multi-layer network analysis, information propagation research, and other related tasks. In addition, edge supplementation also helps to solve the problem of network incompleteness in practical applications, making the network model more comprehensive and enhancing the credibility of network analysis results.

[0086] In this embodiment, the model also utilizes the idea of graph structure learning to optimize the original adjacency matrix and node features. First, the similarity between nodes is calculated using different kernel functions, and then a similarity graph S is constructed and sparsified. Finally, some structural features of the original graph are retained through residual links. Next, the details of the entire process will be introduced specifically.

[0087] Kernel functions: The similarity graph is an important part of graph learning. It can reflect the similarity degree between nodes and provide a basis for edge weighting in the graph model. To achieve this goal, three commonly used kernel functions are selected to calculate node embeddings, namely the inner product kernel, the cosine similarity kernel, and the diffusion kernel.

[0088] First, the similarity S between node i and node j is defined by the following formula ij , where k(·,·) represents the kernel function:

[0089] S ij = k(h i , h j )

[0090] The following will discuss in detail three different kernel functions, each of which is suitable for different similarity measurement methods:

[0091] (1) Inner product kernel: The inner product kernel is a common and computationally simple similarity metric method. It evaluates the similarity between two node embedding vectors by calculating their inner product. The inner product kernel assumes that if the embedding vectors of two nodes are closer, then the similarity between them is higher. Therefore, the larger the value of the inner product, the closer the connection between the nodes. The calculation formula of this kernel function is:

[0092]

[0093] where σ is a Sigmoid function used to normalize the edge weights. The Sigmoid function maps the inner product value to the range [0, 1], thus ensuring that the edge weights in the similarity graph are probability values.

[0094] (2) Cosine similarity kernel: The cosine similarity kernel is based on the principle of cosine similarity and evaluates the similarity by calculating the cosine of the angle between node embedding vectors. Different from the inner product kernel, the cosine similarity kernel pays more attention to the directionality of the vectors rather than their magnitudes. Therefore, the smaller the angle between two embedding vectors, the greater their similarity. The calculation formula of the cosine similarity kernel is:

[0095] S ij =cos(h i ⊙w,h j ⊙w)

[0096] where ⊙ represents the element-wise multiplication operation, and w is a trainable parameter. This method allows for further adjustment of the similarity calculation by learning the weight w, enhancing the expressive power of the model.

[0097] (3) Diffusion kernel: The diffusion kernel is a method for calculating node similarity based on the diffusion process of the graph. It constructs a transition matrix T through the personalized PageRank algorithm and uses this transition matrix to propagate the similarity of nodes. The diffusion kernel takes into account the multi-hop relationships between nodes, that is, the similarity of a node is not only affected by its direct neighbors but also by distant nodes. The calculation formula of the diffusion kernel is:

[0098]

[0099] where, θ k is the weighting coefficient that controls the intensity of diffusion, and T k represents the transition matrix in the k-th round. Through this method, the similarity of nodes can be calculated more carefully according to the graph structure, especially suitable for graphs with complex connection patterns.

[0100] Exemplarily, since the similarity matrix S calculated through the above kernel function is usually very dense, resulting in high storage and computational complexity, the K-nearest neighbor (KNN) algorithm is adopted to sparsify it. The goal of KNN sparsification is to retain the most relevant neighbors of each node, avoid the interference of irrelevant nodes, thereby enhancing the confidence of the edges in the similarity graph and reducing the introduction of noise.

[0101] Specifically, traverse each node, retain the top K nodes most similar to this node, and set the edge weights of other nodes to zero. The mathematical expression is as follows:

[0102]

[0103] where top K(S i ) represents the top K minimum or maximum values of node i.

[0104] Furthermore, to ensure that the finally obtained similarity graph is symmetric, the sparsified similarity matrix S K is further processed. Specifically, for each pair of nodes (i, j), retain the maximum value in their similarity matrix, thereby ensuring that is symmetric. The mathematical expression of this process is as follows:

[0105]

[0106] Through this processing, a symmetric and sparse similarity matrix is obtained whose storage and computational efficiency are significantly optimized.

[0107] Residual connection: Even through the above methods, it is still difficult to ensure that all important topological structures and connection information in the original graph are completely retained. However, the structure of the original graph usually contains rich topological prior knowledge, and this information may play a crucial role in the final learning results in practical applications. Therefore, how to retain these topological priors and reduce noise in the learning process of the similarity graph has become an important challenge.

[0108] In this embodiment, to solve this problem, the present invention proposes a way of residual connection to effectively incorporate the information of the original adjacency matrix into the finally obtained adjacency matrix. Specifically, by combining the adjacency matrix obtained from similarity graph learning with the original adjacency matrix, it is possible to avoid losing important topological information in the original graph while maintaining the advantages brought by the similarity graph. Using the weighted summation method, the final adjacency matrix A * is the weighted sum of the similarity graph learning adjacency matrix and the original adjacency matrix A, that is:

[0109]

[0110] Among them, α is a hyperparameter that controls the balance between the similarity graph learning adjacency matrix and the original adjacency matrix. By adjusting the value of α, the proportion of similarity information and original topological information in the final adjacency matrix can be flexibly controlled.

[0111] The finally obtained adjacency matrix combines the structures of the similarity graph and the original graph. However, this matrix contains edge weights, which are not always necessary for many downstream tasks. To remove these edge weights and ensure an unweighted adjacency matrix, a threshold is set for the edge weights. Only when the edge weight is greater than this threshold will the edge be retained to form the final adjacency matrix. In this way, an adjacency matrix that better meets the requirements of downstream tasks can be obtained, and this matrix achieves a balance in capturing similarity information between nodes and maintaining the topological prior of the original graph structure.

[0112] The adjacency matrix A obtained by this process * can be used as the input in subsequent graph learning tasks to help achieve goals such as graph network alignment and learning. Finally, by using this weighted summation method, not only is the noise effectively reduced, but also the topological prior information of the original graph is fully utilized, providing a more reliable input for graph analysis.

[0113] S4: Calculate the node embeddings of the source network and the target network at different layers, generate an initial hierarchical alignment matrix based on the embeddings, identify stable nodes using the alignment results, and adjust the embeddings according to the identification results.

[0114] In this embodiment, the interference of noise in the network alignment task often leads to the accumulation of matching errors, thereby reducing the alignment accuracy. To alleviate this problem, an iterative optimization method based on stability refinement is designed to enhance the influence of cross-layer stable anchor nodes on the final alignment result. Before generating the final alignment matrix, this method gradually optimizes the node embeddings, emphasizes the matching relationships of stable nodes, and weakens the error propagation caused by noise. Specifically, the model first calculates the node embeddings of the source network and the target network at different layers and generates an initial hierarchical alignment matrix based on these embeddings. On this basis, according to the defined node stability criteria, stable nodes are identified. Subsequently, these stable nodes are identified using the current alignment results, and the embeddings are adjusted accordingly to amplify the influence of stable nodes during the alignment process.

[0115] Exemplarily, embedding stability is the core of detecting and screening stable nodes. Specifically, given a node v and its embeddings h (1) ,…,h (k), the purpose is to determine whether the node is a stable node. If node v is stable, then in the alignment matrices at all levels, the anchor node of node v should be consistent, and its corresponding alignment score should be greater than a preset confidence factor λ. In other words, for all levels 0 ≤ l ≤ k, if node v is stable, then the following conditions hold:

[0116]

[0117] where S (l) (v, u) represents the alignment score between node v and node u at the l-th level, and v' is the anchor node of node v. If the above formula holds, then node v is considered stable. In contrast to stable nodes, nodes that do not satisfy the above formula are considered unstable nodes. For these nodes, the embedding differences may not be entirely caused by noise and may be affected by other factors. Therefore, during the alignment process, it is necessary to further identify these unstable nodes and filter them out from the refinement process to improve the accuracy of the alignment result.

[0118] Exemplarily, taking the aggregation process of a graph neural network as an example, the refinement process adopts a weighted propagation mechanism, making stable nodes have a greater influence in the embedding propagation, while unstable nodes have a relatively smaller influence. Specifically, for a certain node v, the aggregation process of its embedding is as follows:

[0119]

[0120] where represents the set of embeddings of neighbor node t, represents the degree normalization factor of node t, and α(v) represents the influence factor of node v. Intuitively, if node v is stable, then its influence factor should be strengthened to a greater extent to affect the alignment optimization process:

[0121] α(v) = β × α(v)

[0122] where β > 1 is an accumulation constant to ensure that the influence of stable nodes gradually increases. Finally, we apply this mechanism to the update rule of GCN, making the embedding propagation pay more attention to stable nodes:

[0123]

[0124] where is the normalized degree matrix, and Q is a diagonal matrix satisfying Q(v, v) = α(v). This strategy ensures that stable nodes occupy a greater weight in the information propagation process, thereby reducing the interference of noise.

[0125] Specifically, during the iterative optimization process, the model identifies stable nodes based on the current alignment matrix and updates the influence factors of these nodes to strengthen their contributions to the alignment. At the same time, the model uses the adjusted influence factors to optimize the embeddings of the nodes and recalculates the alignment matrix based on this. To ensure that the optimization process of the matching results develops in the direction of stable convergence, the model adopts a greedy strategy and selects the optimal alignment matrix according to the cumulative change of the alignment score. If the current alignment score exceeds the historical optimal value, the final alignment result is updated. Through this iterative refinement process, the method of the present invention can effectively reduce the interference of noise, improve the stability and accuracy of the alignment matrix, and finally obtain a more robust cross-network alignment result.

[0126] It should be noted that in this article, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the term "comprising", "including" or any other variation thereof is intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements not only includes those elements, but also includes other elements not expressly listed, or also includes elements inherent to such process, method, article or device.

[0127] Finally, it should be noted that the above are only the preferred embodiments of the present invention and are not used to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A network alignment method based on optimizing intra-layer and inter-layer links, characterized in that: It includes the following steps: S1: Obtain a dataset and perform preprocessing; S2: Perform consistency measurement; The consistency measurement is to calculate the consistency index of all known anchor nodes and use the consistency index to calculate the consistency value of the entire network; S3: Based on the consistency hypothesis, use the relationships between different pairs of anchor nodes to complete the missing and potential links within the layer in the network. At the same time, use graph structure learning to optimize the original adjacency matrix and node features; S4: Calculate the node embeddings of the source network and the target network at different layers, generate an initial hierarchical alignment matrix according to the embeddings, identify stable nodes based on the alignment results, and adjust the embeddings according to the identification results.

2. The network alignment method based on optimizing the intra-layer and inter-layer links according to claim 1, wherein: The consistency in the consistency measurement includes topological consistency and attribute consistency. The consistency measurement in the topological consistency includes node degree consistency measurement and curvature consistency measurement. The attribute consistency is limited to networks with attributes and includes the consistency of node attributes.

3. The network alignment method based on optimizing the intra-layer and inter-layer links according to claim 2, wherein: The node degree consistency metric is used to reflect the connectivity of a node in a network. The connectivity is specifically expressed as follows: When nodes v A and v a serve as a pair of anchor nodes, node v A in the source network should be connected to nodes v B , v C and v D . Node v a in the target network should be connected to nodes v b , v c and v d . To maintain consistency, node v a in the target network should also have the same connection pattern with nodes v b , v c and v d . The node degree consistency metric is defined as follows: Define the degree consistency metric C1 to quantify the degree difference between node v in the source network A and node v in the target network a . Calculate the average value, maximum value, and minimum value of the degree differences of the known anchor nodes. The average value of the degree differences is expressed as: where |T| represents the number of known anchor links, deg(·) represents the degree of a node, and t s and t t represent nodes; The maximum value of the degree difference and the minimum value are shown as follows: Normalize the average value, maximum value, and minimum value of the degree difference. The normalization formula is: Obtain the degree consistency measurement C1.

4. The network alignment method based on the optimization of intra-layer and inter-layer links according to claim 2, characterized in that: The curvature consistency measurement is used to reflect the local structural characteristics where the node is located. The curvature consistency measurement is: Use the Ricci curvature difference at the node level as the evaluation criterion, calculate the Ricci curvature κ(i,j) of each edge in the network, aggregate the curvature information of the edges to each node, and take the average value: where ε represents the set of edges in the network, and κ i is the node-level curvature of node i; Based on the curvature difference at the node level, define the curvature consistency measurement C2: Among them, and represent the nodal level curvatures of t s and t t nodes; Guarantee the attribute consistency calculation of anchor nodes by using hyperbolic embedding. The hyperbolic space in hyperbolic embedding has a negative constant curvature. Embed node attributes into the hyperbolic space, calculate the hyperbolic distance, and measure the consistency of node attributes. Specifically, node t s and t t 's embeddings in the hyperbolic space are h s and h t respectively, and the hyperbolic distance between them is given by the following formula: where \(c\) represents the curvature of the hyperbolic space, denotes the Möbius addition in the hyperbolic space and is defined as follows: Weightedly combine the topological consistency and attribute consistency measurements to obtain the final consistency measurement C: C = αC1 + βC2 + γC3 where α, β, and γ are hyperparameters that control the weights of different consistency measurements. Set γ = 0 when the input network has no node attributes.

5. A network alignment method based on optimizing intra-layer and inter-layer links according to claim 1, characterized in that: The specific process of using the relationships between different pairs of anchor nodes to complete the missing and potential links within the layer includes: Set (v A , v a ) and (v B , v b ) as two pairs of anchor nodes, where the uppercase letter subscript represents the node in the source network, and the lowercase letter subscript represents the node in the target network; (v a , v b ) belongs to the edge set ε t , According to the consistency assumption, if there is an edge between v a and v b , and there are cross-layer anchor links between v A and v a , v B and v b , that is, (v a , v b ) ∈ ε t , (v A , v a ) ∈ T, (v B , v b ) ∈ T, then it is speculated that there is also an edge between v A and v B in the edge set ε s between the source network and the target network, that is, (v A , v B ) ∈ ε s ; Check all pairs of anchor nodes to determine whether the pair of anchor nodes satisfies the consistency hypothesis. When there is an edge between two known pairs of anchor nodes in one of the networks, check whether their corresponding nodes are still connected in the other network. If not, add an edge between these two nodes. The added edge is used to restore the connection pattern between the source network and the target network, make up for the information loss caused by the missing links in the network, and keep the consistency hypothesis valid; Moreover, the addition of edges within the layer is limited to between anchor nodes. During the process of supplementing intra-layer links, only the links that can be confirmed under the consistency hypothesis are allowed to be supplemented.

6. A network alignment method based on optimizing intra-layer and inter-layer links according to claim 1, characterized in that: The specific process of optimizing the original adjacency matrix and node features includes: Calculate the similarity between nodes using different kernel functions, construct a similarity graph S, perform a sparsification operation, and retain part of the structural characteristics of the original graph through residual links; The similarity graph is an important part of graph learning, which is used to reflect the similarity degree between nodes and provide a basis for edge weighting in the graph model. To achieve this goal, calculate the node embeddings using inner product kernels, cosine similarity kernels, and diffusion kernel functions; The sparsification operation is to sparsify it using the K-nearest neighbor algorithm. By traversing each node, the top K nodes most similar to the node are retained, and the edge weights of other nodes are set to zero. The mathematical expression is as follows: Among them, top K(S i ) represents the first K minimum or maximum values of node i; Processing the sparsified similarity matrix S K , for each pair of nodes (i, j), retain the maximum value in their similarity matrix. The mathematical expression for this process is as follows: Obtain a symmetric and sparse similarity matrix 7. A network alignment method based on optimizing intra-layer and inter-layer links according to claim 6, characterized in that: The retention of partial structural features of the original graph through residual links specifically includes: Effectively incorporate the information of the original adjacency matrix into the finally obtained adjacency matrix. By combining the adjacency matrix obtained from similarity graph learning with the original adjacency matrix, and using a weighted summation method, let the final adjacency matrix A * be the weighted sum of the adjacency matrix for similarity graph learning and the original adjacency matrix A, that is: Among them, α is a hyperparameter used to control the balance between the similarity graph learning adjacency matrix and the original adjacency matrix. By adjusting the value of α, the proportion of similarity information and original topological information in the final adjacency matrix is controlled. At the same time, a weight threshold for the edges is set. Only when the weight of an edge is greater than this threshold will the edge be retained to form the final adjacency matrix A * .

8. A network alignment method based on optimizing intra-layer and inter-layer links according to claim 1, characterized in that: Step S4 specifically includes: calculating the node embeddings of the source network and the target network at different layers, generating an initial hierarchical alignment matrix based on the node embeddings, identifying stable nodes according to the defined node stability criteria, and using the current alignment result to identify the stable nodes, and adjusting the embeddings accordingly to amplify the influence of the stable nodes in the alignment process.

9. A network alignment method based on optimizing intra-layer and inter-layer links according to claim 8, characterized in that: The stable node recognition includes: Given a node v and its embeddings h (1) ,…,h (k) in each layer of the network, where the network layers range from the first layer to the k-th layer, determine whether the node is a stable node. If the node v is stable, then in the alignment matrices of all layers, the anchor node of the node v is consistent, and its corresponding alignment score is greater than a preset confidence factor λ. That is, for all layers 0 ≤ l ≤ k, when the node v is stable, the following conditions are satisfied: Among them, S (l) (v, u) represents the alignment score between node v and node u at the l-th layer. v' is the anchor node of node v. When the above formula holds, node v is considered stable, and nodes that do not satisfy the above formula are considered unstable nodes.

10. A network alignment method based on optimizing intra-layer and inter-layer links according to claim 9, characterized in that: The identification of the stable nodes using the current alignment result specifically includes: further identifying unstable nodes and filtering them out from the refinement process. The refinement process is as follows: for node v, the aggregation process of its embedding is as follows: Among them, represents the embedding set of neighbor node t, represents the degree normalization factor of node t, α(v) represents the influence factor of node v. When node v is stable, its influence factor is strengthened to influence the alignment optimization process to a greater extent: α(v) = β × α(v) where β > 1 is an accumulation constant used to ensure that the influence of stable nodes gradually increases. Applying the mechanism to the update rule of GCN enables the propagation of embeddings to pay more attention to stable nodes: Among them, is the normalized degree matrix, and Q is a diagonal matrix that satisfies Q(v, v) = α(v), and stable nodes occupy a greater weight in the information propagation process; Finally, stable nodes are identified based on the current alignment matrix, the influence factors of these nodes are updated, and the embeddings of the nodes are optimized using the adjusted influence factors. On this basis, the alignment matrix is recalculated, and the greedy strategy is simultaneously adopted to select the optimal alignment matrix according to the cumulative change of the alignment score. When the current alignment score exceeds the historical optimal value, the final alignment result is updated, and the iterative optimization of the refinement process is carried out in this way.

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