Important node analysis method for improving K-shell von Noiemann entropy centrality

Abstract

The invention discloses an important node analysis method for improving K-shell von Noiemann entropy centrality, and particularly relates to the technical field of complex network analysis. According to the method, a traditional K-shell decomposition algorithm is improved, the von Noiemann entropy centrality and improved K-shell decomposition are combined, and more accurate and comprehensive analysis of important network nodes is achieved. According to the method, the problem of excessive coarse graining division existing in traditional K-shell decomposition is effectively solved, and compared with a traditional algorithm, the improved K-shell decomposition has higher distinction degree. In order to analyze key nodes in a complex network from a more comprehensive perspective, Von Noiemann entropy centrality and an improved algorithm are fused, and the capability of comprehensively analyzing the key nodes in the complex network from the perspective of a network structure and overall characteristics of the network is realized. The effectiveness of the algorithm is verified by using a plurality of common evaluation indexes, including monotonicity index, deliberate attack model and correlation coefficient analysis, and comparison with a plurality of classic algorithms in a plurality of real networks.

Description

Technical Field:

[0001] The present invention relates to the technical field of complex network analysis, and specifically to an important node analysis method for improving the K-shell von Neumann entropy centrality. Background Art:

[0002] With the rapid development of information technology, the complex network theory has shown great application potential in multiple fields, such as social networks, bioinformatics, and neuroscience. In complex network analysis, the evaluation of node importance is a key link to understand the network structure, function, and dynamic changes. However, the currently widely used K-shell decomposition algorithm has significant limitations in node importance evaluation. Although the K-shell decomposition algorithm can identify the core nodes in the network, its discrimination is low, and it cannot effectively distinguish the subtle differences between nodes with similar importance. Moreover, it overemphasizes the role of central nodes in complex networks. This overemphasis on central nodes may lead to a deviation in understanding the overall structure and function of the network and ignore the overall characteristics of complex networks. This is particularly insufficient in practical applications, especially in complex network research that requires accurate identification of key nodes. For example, in a complex simulated brain network, neuron populations in different brain regions participate in information processing at different frequencies and intensities, and the K-shell decomposition algorithm cannot accurately capture these differences, resulting in inaccurate evaluation of node importance. To address the above problems, an improved K-shell von Neumann entropy centrality algorithm (Improving K-shell Von Neumann entropy centrality, IKSVNEC) is proposed to solve the problems of small discrimination and overemphasis on central nodes while ignoring the overall network characteristics in K-shell decomposition, enabling the algorithm to more accurately and comprehensively distinguish important nodes in complex networks. Summary of the Invention:

[0003] The present invention aims to address the above problems and proposes an improved K-shell von Neumann entropy centrality algorithm to comprehensively and accurately evaluate the importance of complex network nodes. The main technical problems and objectives of the present invention include: (1) improving the discrimination of network node importance evaluation, being able to divide the network into more levels to distinguish the subtle differences between nodes; (2) accurately and comprehensively analyzing the key nodes in the network. The K-shell decomposition only distinguishes important nodes based on network structure characteristics, which is incomplete and inaccurate. The algorithm of the present invention should analyze key nodes in combination with the overall network characteristics. Through the implementation of the present invention, it is expected to provide new perspectives and methods for complex network research.

[0004] To achieve the above objectives, the technical solutions adopted by the present invention are as follows:

[0005] The present invention improves an important node analysis method for improving the K-shell von Neumann entropy centrality, and the specific steps are as follows:

[0006] Step 1: Read a complex network and use the concepts and methods of graphs to describe the complex network.

[0007] Step 2: Threshold the complex network to construct an unweighted and undirected network.

[0008] Step 3: Construct an adjacency matrix according to the thresholded result.

[0009] Step 4: Aiming at the problems of over-coarse-grained partitioning and little distinction between nodes existing in the K-shell decomposition, we adopt an improved K-shell decomposition, making the algorithm have a higher discrimination degree compared with the original algorithm.

[0010] Step 5: Aiming at the problem that the K-shell decomposition overemphasizes central nodes and ignores the overall characteristics of the network, the present invention combines the von Neumann entropy centrality with the improved K-shell decomposition, and proposes a node importance algorithm based on the improved K-shell von Neumann entropy centrality, enabling the algorithm to analyze the key nodes in the complex network from the perspectives of overall characteristics and structural characteristics.

[0011] Step 6: Finally, substitute the constructed binary adjacency matrix into the algorithm to obtain the calculated entropy centrality value, sort the newly obtained values, and obtain the node importance order.

[0012] The K-shell decomposition is an algorithm that can divide a complex network into multiple layers and identify core nodes. By gradually removing the nodes with the smallest degree in the network and updating the degree information of the nodes, the K-shell decomposition can divide the nodes into different layers, where the nodes with higher layer values usually have higher connectivity and importance. Therefore, the K-shell decomposition provides a method to understand the organizational structure of the network and find the core nodes with importance and influence. The specific steps are as follows:

[0013] Step 1: Delete the nodes with degree 1 and their connecting edges in the complex network, and find and delete the nodes with degree 1 and their connecting edges in the remaining nodes. Repeat the above operations until there are no nodes with degree 1 in the network. Record these deleted nodes and set their k s value to 1.

[0014] Step 2: In the newly generated complex network, remove the nodes with degree 2 and their connecting edges in the complex network, and find and delete the nodes with degree less than or equal to 2 and their connecting edges in the remaining nodes. Repeat the above operations until there are no nodes with degree less than or equal to 2 in the network. Record these deleted nodes and set their ks The value is 2.

[0015] Step 3: And so on, repeat the above operations until each node has its own value of k. s value.

[0016] The von Neumann entropy is an important concept in information theory. In the study of von Neumann entropy centrality, it is considered that the von Neumann entropy can reflect the regularity and complexity of the network and can effectively characterize the overall characteristics of the network. The change of the von Neumann entropy, that is, the von Neumann entropy centrality, can accurately present the impact of nodes on the network as a whole.

[0017] Before introducing the von Neumann entropy, we need to construct a Laplacian matrix L(G). When constructing L(G), we also need to use the matrix A(G), which is the adjacency matrix of the complex network G. The specific construction method is shown in Equation (1):

[0018] L(G) = D(G) - A(G) (1) where D(G) is the degree matrix, which is an n×n matrix, and its specific definition is shown in Equation (2):

[0019]

[0020] where d G (v i ) is the degree of node v i . Therefore, the Laplacian matrix can be written as:

[0021]

[0022] After normalization, we get:

[0023]

[0024] Performing spectral decomposition on it, we get L(G) = ΦΛΦ, where Λ = diag(λ 1 , λ 2 ,..., λ n ) is the diagonal matrix of eigenvalues. The normalized Laplacian matrix contains all the topological features of the network. For a matrix, the most important indicators are the eigenvalues, which are directly related to the topological properties of the network.

[0025] Therefore, substituting the eigenvalues into the von Neumann entropy formula, the von Neumann entropy is shown in Equation (5):

[0026]

[0027] where S(G) represents the entropy value of the von Neumann entropy, and node v iThe centrality can be defined as the change in the von Neumann entropy when removing the node and its connected edges from the network. Using C E (v i ) to represent the von Neumann entropy node centrality of node v i , we obtain:

[0028] C E (v i ) = S(G) - S(G / v i ) (6)

[0029] Beneficial effects:

[0030] In view of the deficiencies of the traditional K-shell decomposition method, the present invention proposes a node importance evaluation algorithm based on the improved K-shell von Neumann entropy centrality (IKSVNEC).

[0031] Compared with the prior art, this algorithm has the following outstanding advantages:

[0032] 1. More accurate node discrimination: By combining the improved K-shell decomposition, IKSVNEC achieves a precise evaluation of the importance of network nodes. Compared with the K-shell decomposition, IKSVNEC can better distinguish the importance levels of nodes in the network, and the division of nodes is more accurate, which helps to discover key nodes.

[0033] 2. More comprehensive network analysis: IKSVNEC not only retains the global structural perspective of the K-shell decomposition, but also introduces the consideration of the overall characteristics of the network through the calculation of the von Neumann entropy, effectively solving the problem that traditional methods overly focus on central nodes and ignore the overall characteristics of the network. This feature makes the algorithm more comprehensive and detailed when evaluating the importance of network nodes.

[0034] Specific performance indicators and quantitative effects:

[0035] In the present invention, multiple groups of commonly used performance evaluation indicators for node importance algorithms are used to verify the algorithm effect, including monotonicity indicators, attack models, and correlation coefficients, and compared with multiple groups of traditional algorithms. In the experiment of the monotonicity indicator, the IKSVNEC algorithm shows the highest node discrimination. In the experiment of the attack model, the deletion of the important nodes calculated by the IKSVNEC algorithm causes greater damage to the network connectivity compared with other algorithms, proving that the nodes calculated by it are more important than those calculated by other algorithms. In the experiment of the correlation coefficient, the important nodes calculated by the IKSVNEC algorithm are most similar to the important nodes simulated by the SIR model. Except for being slightly lower than the PageRank algorithm in the Zebra network, the correlation coefficients are basically above 0.8 in other networks. Description of the drawings:

[0036] Figure 1 is the network topology structure used in the embodiments of the present invention. (a) Zebra network (b) Continental United States connection network (c) Weaver bird network (d) Schematic diagram of power network.

[0037] Figure 2 is a schematic diagram of the experimental results of the deliberate attack models of various algorithms in the embodiments of the present invention in different networks. Figure 3 is a schematic diagram of the experimental results of the deliberate attack models of various algorithms in the embodiments of the present invention in different networks. Specific implementation manner:

[0038] The embodiments of the present invention will be described in detail below. The examples of the embodiments are shown in the accompanying drawings and tables. The verification of the effectiveness of the algorithms of the present invention will be described below with reference to the accompanying drawings and tables. Examples of the algorithms in multiple real complex networks are shown in the accompanying drawings and tables, and evaluation indicators such as monotonicity indicators, deliberate attack models, and correlation coefficients are used to compare with various traditional algorithms to verify the effectiveness of the algorithms of the present invention.

[0039] The specific method is as follows:

[0040] Step 1: Define a complex network in the form of a graph in graph theory. The complex network can be represented by points and edges, denoted as: G=(v, ε). Among them, v={1, 2, 3, 4,..., N} represents N nodes, represents the edge set.

[0041] Step 2: In the present invention, the case of undirected complex networks is mainly considered. Most of the complex networks we read in life contain weights. In order to more clearly understand the structure and correlation of complex networks, it is necessary to determine the threshold of the network edge weights and binarize the complex network, that is, when the relationship strength between nodes is greater than the threshold, the edge weight is set to 1, otherwise it is 0.

[0042] Step 3: According to the results after thresholding, construct an adjacency matrix. A reasonable threshold is very important, which can filter out the edges with weak connectivity in the network. Reasonable thresholding can reduce the complexity of the network and better distinguish the network regions with large weight differences.

[0043] Step 4: Improve the counting rule of K-shell decomposition so that the improved K-shell decomposition has higher discrimination than K-shell decomposition. The specific steps are as follows:

[0044] Step 1: Delete the nodes and edges with a node degree of 1 in the complex network, record these deleted nodes, and set their k s value to 1.

[0045] Step 2: Repeat the operation in Step 1 in the updated complex network. Each time the complex network is updated, the k s value is increased by 1, and record these deleted nodes and the k s values until there are no nodes with degree 1 in the updated complex network.

[0046] Step 3: Delete the nodes with degree 2 and their connecting edges in the updated complex network. The k s value is increased by 1, and record these deleted nodes and the k s values.

[0047] Step 4: Continue to update the complex network. Delete the nodes with degree less than or equal to 2 and their connecting edges in the updated complex network. The k s value is increased by 1, and record these deleted nodes and the k s values. Repeat the above operations in the updated complex network until there are no nodes with degree less than or equal to 2 in the updated complex network.

[0048] Step 5: And so on, repeat the above operations until each node has its own k s value.

[0049] Step 5: Combine the von Neumann entropy centrality with the improved K-shell decomposition, and analyze the key nodes in the complex network from the perspective of the overall characteristics and structure of the network. The specific method is as follows:

[0050] Step 1: For a complex network G, we use the improved K-shell decomposition algorithm to calculate and record the k s value of each node in the complex network G.

[0051] Step 2: Before calculating the entropy value of the network, we need to construct the adjacency matrix A(G) of the complex network G, convert it into a Laplacian matrix, and normalize it to obtain the normalized Laplacian matrix.

[0052] Step 3: Using the normalized Laplacian matrix for spectral decomposition, we can obtain the eigenvalues of the network. Substitute these eigenvalues into the formula of the von Neumann entropy to calculate the entropy value of the network.

[0053] Step 4: To further explore the importance of each node in the network structure, we can calculate the change in the von Neumann entropy when removing each node and its connected edges. In this step, we can use the k s value of each node recorded before, and multiply it by the change value of the von Neumann entropy of the corresponding node obtained to get a new centrality calculation method.

[0054] Step 6: Substitute the unweighted undirected adjacency matrix into the algorithm, sort the calculated values, and obtain a new order of node importance.

[0055] The following combines Figure 1 Figures 2 to 2 and Tables 1 to 3 to describe in detail the validation of the effectiveness of the algorithm of the present invention.

[0056] As Figure 1 shown, it is the network topology structure used in the present invention. The network topology structure comes from a public dataset. The complex network in the dataset is already an unweighted undirected complex network and does not require operations in Steps 1 to 3. Figure 1 It shows the topology structures of four real networks, namely: Zebra network, USA network, Weaver network, and Power network. Table 1 shows the relevant information of several networks.

[0057] As shown in Table 2, Table 2 shows the monotonicity indicators of various algorithms in different networks. In order to evaluate the discriminative power of different algorithms in node importance ranking, the monotonicity indicator is used for quantitative measurement. This indicator measures the evaluation effect of nodes by calculating the proportion of nodes with the same ranking index. This method can not only effectively judge the resolution of different algorithms, but also provide useful information for comprehensively understanding the node propagation potential. When each node has a unique importance ranking value, it means that the algorithm can clearly distinguish the importance of each node. When M(R) = 0, it means that the importance of nodes cannot be distinguished and the importance of each node is the same; when M(R) = 1, it means that the importance of each node can be clearly shown and each node has a different node importance. From the results in the table, it can be seen that the algorithm of the present invention shows good results in several networks. In the USA network and the Weaver network, the monotonicity indicator of this algorithm reaches 1, which means that the algorithm has complete monotonicity, has a high resolution, and can accurately distinguish the importance of each node. It also shows better performance than other algorithms in other networks, verifying the excellent performance of the algorithm.

[0058] Table 1 Relevant information of Zebra network, USA network, Weaver network, and Power network

[0059]

[0060]

[0061] Table 2 Monotonicity indicators of various algorithms in different networks

[0062]

[0063] As Figure 2As shown, the figure presents the experimental results of various algorithms for deliberately attacking models in different networks. The deliberately attacking model is a common evaluation metric for testing the performance of node importance algorithms. The basic principle of the algorithm is as follows: Use the node importance algorithm to calculate the node importance ranking of a complex network, and delete the node with the highest node importance ranking and its connected edges in the complex network. Since the nodes and edges in the complex network are deleted, the topological structure of the complex network changes, and some parameters corresponding to the complex network also change. Studying the changes in the parameters corresponding to this complex network can measure the performance of the node importance algorithm. However, due to the change in the topological structure of the complex network, the node importance ranking of some nodes may change. Therefore, after each deliberate attack and calculation of the required network parameters, it is necessary to recalculate the node importance ranking using the node importance algorithm as the basis for the next deliberate attack.

[0064] During the process of deliberate attack, along with the deletion of nodes and connected edges, we use the network efficiency decline rate ε to quantitatively measure the damage to the network caused by node removal. As can be seen from the figure, the network efficiency decline rate ε of the IKSVNEC algorithm shows a faster rising speed compared to other algorithms, that is, the network efficiency changes faster, indicating that the deletion of the important nodes calculated by the IKSVNEC algorithm causes greater damage to the network compared to other algorithms, and these nodes are more important. The IKSVNEC algorithm is significantly superior to other algorithms in small networks such as the Zebra network, USA network, and Weaver network, showing good results. It also basically maintains a good level in relatively larger networks such as the Power network. In summary, the IKSVNEC algorithm proposed in this paper has good ability to distinguish important nodes and shows better performance than other algorithms in medium and small networks with practical significance.

[0065] To better measure the performance of the algorithm in this paper, the Kendall Tau correlation coefficient τ is used to measure the correlation between the node importance rankings of different algorithms and the node importance rankings simulated by the SIR model. This is also a commonly used method to measure the performance of node importance algorithms. The Kendall Tau correlation coefficient τ is one of the statistical methods used to measure the correlation between two variables. It is used to measure the degree of consistency between the ranks of two variables. The value range of the Kendall Tau correlation coefficient τ is between -1 and 1. When the Kendall Tau correlation coefficient τ is closer to 1, it indicates that the algorithm sorting result is closer to the node importance ranking simulated by the SIR model. Table 3 shows the correlation coefficient τ between the node importance rankings obtained by the algorithm proposed in this paper and other methods and the importance rankings simulated by the SIR model. The optimal values of the corresponding networks are indicated in bold in the table. It can be seen from Table 3 that the IKSVNEC algorithm is only slightly lower than the PageRank algorithm in the Zebra network, and the correlation coefficients in the network are basically above 0.8. In other networks, it shows the highest similarity, and the performance of the IKSVNEC algorithm is better than that of the other three algorithms.

[0066] Table 3 Correlation coefficient τ of various algorithms in different networks

[0067]

Claims

1. An important node analysis method for improving K-shell von Neumann entropy centrality, characterized in that: The following steps are involved: Step 1: Read a complex network and describe it using graph theory concepts and methods; Step 2: Thresholding the complex network to construct an unweighted and undirected network; Step 3: Construct an adjacency matrix based on the thresholded results; Step 4: Improve the K-shell decomposition algorithm. By improving the counting rule of K-shell decomposition, the algorithm has higher node discrimination than the traditional K-shell decomposition algorithm, and calculates the hierarchical value of each node according to the complex network; Step 5: construct the Laplace matrix of the complex network, perform normalization and spectral decomposition to obtain the eigenvalues ​​of the network, and substitute them into the von Neumann entropy centrality formula; Step 6: Use the von Neumann entropy centrality formula to calculate and obtain the entropy value of the entire network, and further calculate the change in von Neumann entropy when each node and its edges in the network are removed, so as to obtain the von Neumann entropy centrality value of each node; Step 7: Combine the von Neumann entropy centrality with the improved K-shell decomposition to obtain the improved K-shell von Neumann entropy centrality, and calculate the improved K-shell von Neumann entropy centrality value of each node; Step 8: Sort the calculated improved K-shell von Neumann entropy centrality values ​​of each node to obtain the importance order of the nodes.

2. The method according to claim 1, characterized in that The specific steps of improving the K-shell decomposition algorithm include: Step 1: Delete nodes and edges with a node degree of 1 in the complex network, record these deleted nodes, and set their k s The value is 1; Step 2: Repeat Step 1 in the updated complex network. Each time the complex network is updated, k s The value increases by 1, recording the deleted nodes and the nodes’ k s value until there is no node with degree 1 in the updated complex network; Step 3: Delete nodes and edges with a node degree of 2 in the updated complex network, k s The value increases by 1, recording the deleted nodes and the nodes’ k s value; Step 4: Continue to update the complex network, delete the nodes and edges with node degrees less than or equal to 2 in the updated complex network, k s The value increases by 1, recording the deleted nodes and the nodes’ k s value, repeat the above operation in the updated complex network until there is no node with degree less than or equal to 2 in the updated complex network; Step 5: Repeat the above steps until each node has its own k s value.

3. The method according to claim 1, characterized in that The method for obtaining the complex network characteristic value includes: Step 1: Use the adjacency matrix and degree matrix of the complex network to construct the Laplace matrix according to the formula; Step 2: Normalize the Laplace matrix and perform spectral decomposition to obtain eigenvalues.

4. The method according to claim 1, characterized in that: The von Neumann entropy centrality value is determined by calculating the change in network entropy after removing the node. The specific calculation formula is: the importance of the node is measured by using the change in network entropy before and after the node is removed.

5. The method according to claim 1, characterized in that The improved K-shell von Neumann entropy centrality combines the von Neumann entropy centrality value of each node with the hierarchical value k of each node obtained by the improved K-shell decomposition. s Multiplying together gives the improved K-shell von Neumann entropy centrality value.

6. The method according to claim 1, characterized in that The method can improve the discrimination of network node importance assessment and divide the network into more levels, so as to more accurately distinguish the subtle differences of nodes, and analyze key nodes in combination with the overall characteristics of the network.

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