Hypergraph key node detection method based on index model

Abstract

The invention discloses a hypergraph key node detection method based on an exponential model, which belongs to the field of hypergraph key node identification, and comprises the following steps: representing a hypergraph structure by a hyperedge set; converting the hyperedge set into a vertex set by using an s-line graph; the k Laplace energy is calculated; calculating a Laplace energy difference after the nodes are deleted; the Laplace energy centrality LC is calculated; calculating the shortest average distance; calculating the index centrality HEC of the hypergraph; calculating first 5% of nodes of the real world hypergraph data set; using an SIR model to test the performance of the method; drawing an infection diffusion curve; the HEC of the epilepsy areas and the non-epilepsy areas of the ten epilepsy patients is calculated; drawing an HEC contrast chart of the epileptic region and the non-epileptic region; counting an HEC average value and calculating a p value; and drawing a distribution box plot. According to the method, concepts of the index model and the s-line graph are combined, and the method can be used for identifying key nodes in the hypergraph. Compared with the prior art, the method has the advantages that the recognition accuracy and the calculation complexity are greatly improved, and the performance is better.

Description

Technical Field

[0001] The present invention relates to the field of identifying hypergraph key nodes, and in particular to a hypergraph key node detection method based on an exponential model. Background Art

[0002] Research has shown that identifying key nodes in a network is essential for studying network structure and function. Specifically, nodes with higher centrality are considered more important in a network because they have a greater impact on information propagation throughout the network and play a key role in maintaining network connectivity, promoting information dissemination, and ensuring the stability and functionality of the system. Compared with other nodes, the removal or failure of key nodes usually has a greater impact on the structure and performance of the network. Therefore, identifying these nodes is of great significance for enhancing network robustness, optimizing resource allocation, and preventing potential risks. For example, identifying key power stations or transmission lines in a power grid is essential to ensure the stable operation of the power system and cope with unexpected failures. By accurately locating these key nodes, power distribution can be optimized, the risk of power outages can be reduced, and the system's resilience can be enhanced. In addition, in the field of network security, identifying key server nodes can effectively prevent the spread of network attacks, promote timely protection measures, and improve the overall security and resilience of the network. However, most of the current mature network key node identification is aimed at ordinary graphs, focusing on the identification of key nodes in pairwise connected networks. With the development of network science, pairwise connections in ordinary graphs cannot reflect the complexity of interactions between nodes, so hypergraphs have been derived. Hypergraph is a network representation method that can characterize the connection between multiple nodes. It has been rapidly developed and studied due to its advantages of capturing the interactions between multiple nodes and reflecting the collective behavior of the network. At present, there are a few methods for identifying key nodes in hypergraphs, such as vector centrality, centrality identification based on von Neumann entropy, and centrality identification based on gravity model. The quantitative effects of these methods are not ideal and they cannot accurately identify the key nodes in the hypergraph. Summary of the invention

[0003] In order to solve the problem that the current method for identifying key nodes in a hypergraph has an unsatisfactory quantization effect and cannot accurately identify the key nodes in the hypergraph, the present invention aims to provide a hypergraph key node detection method based on an exponential model.

[0004] The technical solution adopted by the hypergraph key node detection method based on the exponential model of the present invention is: comprising the following steps:

[0005] S1. Collect six real-world hypergraph datasets and use hyperedge sets to represent the hypergraph structure.

[0006] S2, use the s-line graph structure to convert the hyperedge set into the vertex set of the ordinary graph, and convert the hypergraph into an ordinary graph;

[0007] S3, calculating the kth Laplace energy in the general graph;

[0008] S4, calculating the Laplace energy difference after deleting the node;

[0009] S5, calculate the Laplace energy centrality LC;

[0010] S6. Calculate the shortest average distance from a node to other nodes in a common graph;

[0011] S7. Calculate the exponential centrality HEC of the hypergraph;

[0012] S8, calculate the top 5% nodes in six real-world hypergraph datasets as hypergraph key nodes;

[0013] S9. Use the SIR epidemic model to verify the performance of the indicator;

[0014] S10. Draw the infection diffusion curve.

[0015] A further improvement of the technical solution of the present invention is that in step S1, six real-world hypergraph data sets are collected, namely Vegas-bars, Madison-restaurants, Algebra-questions, High-school, House-committees and Senate-committees, and hyperedge sets are used to represent the hypergraph structure.

[0016] A further improvement of the technical solution of the present invention is that in step S3, the k-th Laplace energy in the general graph is calculated:

[0017]

[0018] Among them, μ i is the eigenvalue spectrum of the general graph Laplacian matrix L(G), and the Laplacian matrix is ​​defined as L(G) = D(G) - A(G).

[0019] A further improvement of the technical solution of the present invention is that: in step S4, by deleting the node v i Calculate the Laplace energy difference before and after deletion to evaluate the node v i Importance:

[0020]

[0021] Among them, G i =Gv i is a subgraph of graph G.

[0022] A further improvement of the technical solution of the present invention is that: in step S5, the node v is obtainedi The Laplace energy centrality LC:

[0023]

[0024] Among them, d i is the node v in the graph G i The degree, Δ i is the node containing the node v i The number of triangles.

[0025] A further improvement of the technical solution of the present invention is that in step S7, the calculated index is brought into the exponential model to obtain the exponential centrality of the s-line graph. Based on the cardinality of the hyperedge, the exponential centrality of the s-line graph is mapped to the centrality of the nodes in the hypergraph, and a smaller weight is assigned to the node with a larger cardinality. Therefore, the exponential centrality HEC of the node in the hypergraph is:

[0026]

[0027] Among them, LC(e j ) is the third Laplace energy centrality of the 1-line graph; is node e in the 1-line graph i The shortest distance to other nodes; Γ(v i ) is the node v i The set of associated hyperedges; α and β are linearly varying penalty factor functions, which normalize the exponential term and weight term of the exponential model to [0,1].

[0028] A further improvement of the technical solution of the present invention is that: in step S9, the top 5% of nodes are taken as the initial infected nodes I. In the first time step, the initial infected nodes will infect the susceptible nodes S in their hyperedges with an infection probability of β=0.1%; for each hyperedge containing the initial infected node, if the scores of the I and R nodes are greater than or equal to the threshold η=0.4, the infected node I in the hyperedge will infect the S nodes in its adjacent hyperedges with a probability of β in the next step, and this process will be repeated at each time step; in each time step, each I node in the hypergraph is restored to a node R with an independent probability of γ=1; the infection and recovery process will stop after five steps; such an infection process takes 100 Monte Carlo simulation experiments and takes the average value as the final infection process.

[0029] A further improvement of the technical solution of the present invention is that the HEC in step S7 is applied to the epilepsy data set to distinguish the epileptogenic area from the non-epileptic area, which specifically includes the following steps:

[0030] S7.1. Calculate the HEC of the epileptogenic and non-epileptogenic areas of ten epileptic patients;

[0031] S7.2. Draw a comparison map of HEC between the epileptogenic and non-epileptogenic areas;

[0032] S7.3, calculate the HEC mean and p-value;

[0033] S7.4. Draw a box plot of the distribution.

[0034] A further improvement of the technical solution of the present invention is that: in step S7.1, HEC is applied to the SEEG data of epileptic patients; the sampling channels corresponding to the brain regions in the data are regarded as nodes in a hypergraph, and the interactions between multiple nodes are regarded as hyperedges; the exponential centrality value of each channel is calculated to distinguish between epileptogenic areas and non-epileptic areas during epileptic seizures.

[0035] Due to the adoption of the above technical solution, the technical effects achieved by the present invention are as follows:

[0036] The present invention combines the exponential model and the s-line graph method, which can be used to quantify the centrality of key nodes in a hypergraph. First, the advantage of the exponential model in identifying the importance of nodes is mainly reflected in its sensitivity to network structure and its ability to capture dynamic characteristics. By considering the nonlinear relationship between nodes, the exponential model can effectively identify core nodes that play a key role in information dissemination or resource flow, which improves the accuracy of identification. In addition, the use of the s-line graph structure significantly reduces the computational cost of the hypergraph, making the calculation of HEC more efficient. Finally, the model also combines the Laplace energy centrality and shortest path distance methods to effectively integrate global and local information, thereby providing a more comprehensive assessment of node importance. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 A flowchart of a hypergraph key node detection method based on an exponential model of the present invention;

[0038] Figure 2 It is the diffusion curve of the top 5% nodes in the six real-world hypergraph datasets of the present invention in the SIR model;

[0039] Figure 3 This is a comparison diagram of the index centrality of the epileptogenic area and the non-epileptic area of ​​ten epilepsy patients of the present invention;

[0040] Figure 4 This is a box plot of significant differences between the epileptogenic area and the non-epileptic area after taking the average of the index centrality of ten epilepsy patients in the present invention. DETAILED DESCRIPTION

[0041] In order to make the purpose, technical solution and advantages of the present invention more clear, the present invention is further described in detail below in combination with specific implementation methods and with reference to the accompanying drawings. In the following description, the description of well-known structures and technologies is omitted to avoid unnecessary confusion of the concept of the present invention.

[0042] First reference Figure 1 It can be seen that the present invention comprises the following steps:

[0043] S1. Collect six real-world hypergraph datasets and use hyperedge sets to represent the hypergraph structure. The six real-world hypergraph datasets collected above are Vegas-bars, Madison-restaurants, Algebra-questions, High-school, House-committees, and Senate-committees, and hyperedge sets are used to represent the hypergraph structure.

[0044] S2. Use the s-line graph structure to convert the hyperedge set into the vertex set of the ordinary graph and convert the hypergraph into the ordinary graph.

[0045] S3, calculating the kth Laplace energy in the general graph;

[0046]

[0047] Among them, μ i is the eigenvalue spectrum of the general graph Laplacian matrix L(G), and the Laplacian matrix is ​​defined as L(G) = D(G) - A(G).

[0048] S4, by deleting node v i , calculate the Laplace energy difference before and after deletion to evaluate the node v i Importance:

[0049]

[0050] Among them, G i =Gv i is a subgraph of graph G.

[0051] S5. Get node v i The Laplace energy centrality LC:

[0052]

[0053] Among them, d i is the node v in the graph G i The degree, Δ i is the node containing the node v i The number of triangles.

[0054] S6. Calculate the node v in the general graph i The shortest average distance to other nodes.

[0055] S7. Calculate the exponential centrality HEC of the hypergraph. Bring the calculated index into the exponential model to obtain the exponential centrality of the s-line graph. Based on the cardinality of the hyperedges, map the exponential centrality of the s-line graph to the centrality of the nodes in the hypergraph, and assign smaller weights to nodes with larger cardinality. Therefore, the exponential centrality HEC of the nodes in the hypergraph is:

[0056]

[0057] Among them, LC(e j ) is the third Laplace energy centrality of the 1-line graph; is node e in the 1-line graph i The shortest distance to other nodes; Γ(v i ) is the node v i The set of associated hyperedges; α and β are linearly varying penalty factor functions, which normalize the exponential term and weight term of the exponential model to [0,1].

[0058] S7.1. Calculate the HEC of the epileptogenic and non-epileptic areas of ten epilepsy patients. To further verify the effectiveness of this method, the method was applied to SEEG data of epilepsy patients. The sampling channels corresponding to the brain regions in the data were regarded as nodes in the hypergraph, and the interactions between multiple nodes were regarded as hyperedges. The exponential centrality value of each channel was calculated to distinguish the epileptogenic and non-epileptic areas during epileptic seizures.

[0059] S7.2. Based on the experimental results, draw a comparison chart of the index centrality of the epileptogenic areas and non-epileptic areas of ten epilepsy patients.

[0060] S7.3. Calculate the average of the exponential centrality of the epileptogenic zone and the non-epileptogenic zone during the epileptic seizures of ten patients, and calculate the p-value to evaluate the performance of this method in distinguishing the epileptogenic zone and the non-epileptogenic zone in the SEEG data of epilepsy patients.

[0061] S7.4. Draw a box plot of the significant difference between the epileptogenic area and the non-epileptic area after taking the average of the index centrality of ten epilepsy patients.

[0062] S8. Calculate the top 5% of nodes in six real-world hypergraph datasets as hypergraph key nodes; the larger the calculated value, the more important the node is in the hypergraph. According to the proposed method, calculate the top 5% of nodes in six real-world hypergraph datasets as hypergraph key nodes, paving the way for subsequent verification of the method performance.

[0063] S9. Use the SIR infectious disease model to verify the performance of the indicator; take the top 5% of nodes as the initial infected node I. In the first time step, the initial infected node will infect the susceptible node S in its hyperedge with an infection probability of β = 0.1%; for each hyperedge containing the initial infected node, if the scores of I and R nodes are greater than or equal to the threshold η = 0.4, then the infected node I in the hyperedge will infect the S nodes in its adjacent hyperedge with a probability of β in the next step, and this process will be repeated at each time step; in each time step, each I node in the hypergraph recovers to node R with an independent probability of γ = 1; the infection and recovery process will stop after five steps; such an infection process takes 100 Monte Carlo simulation experiments and takes the average value as the final infection process.

[0064] S10. Draw the infection diffusion curve.

[0065] Figure 2 The diffusion curves of the top 5% of nodes in the SIR model in six real-world hypergraph datasets are shown in the figure. As shown in the figure, HEC outperforms existing hypergraph key node identification methods in most datasets. Specifically, within five time steps, the impact of node infection is always high, which shows that HEC has a unique advantage in identifying key nodes.

[0066] Figure 3 The figure is a comparison of the index centrality of the epileptogenic area and the non-epileptic area of ​​ten epilepsy patients. The semi-violin plot in the figure shows the HEC distribution of the epileptogenic area and the non-epileptic area. The overall distribution of the epileptogenic area is greater than that of the non-epileptic area, which shows that this method can well distinguish the epileptogenic area from the non-epileptic area of ​​epilepsy patients. The black line in the figure shows the value distribution of the two areas more simply and clearly, further illustrating the effectiveness of the method.

[0067] Figure 4 The box plot of the significant difference between the epileptogenic area and the non-epileptic area after taking the average of the index centrality of ten epilepsy patients. In order to better illustrate the effectiveness and performance of this method, the HEC scores of ten epilepsy patients were averaged and a significance test was performed. The significance analysis showed that the results were significantly different (**** represents p<0.0001)

[0068] The method of the present invention is applied to six real-world hypergraph datasets and ten SEEG datasets of epilepsy patients. We conclude that the method can accurately identify key nodes in the hypergraph, which is conducive to a better understanding of network structure and function.

[0069] In the above embodiment, the present invention provides a hypergraph key node detection method based on an exponential model, which combines the exponential model and the s-line graph method, and can be used to quantify the centrality of key nodes in the hypergraph. First, the advantage of the exponential model in identifying the importance of nodes is mainly reflected in its sensitivity to network structure and its ability to capture dynamic characteristics. By considering the nonlinear relationship between nodes, the exponential model can effectively identify core nodes that play a key role in information dissemination or resource flow, which improves the accuracy of identification. In addition, the use of the s-line graph structure significantly reduces the computational cost of the hypergraph, making the calculation of HEC more efficient. Finally, the model also combines the Laplace energy centrality and shortest path distance methods to effectively integrate global and local information, thereby providing a more comprehensive assessment of node importance.

[0070] The embodiments described above are merely descriptions of preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A hypergraph key node detection method based on an exponential model, characterized in that: The following steps are included: S1. Collect six real-world hypergraph datasets and use hyperedge sets to represent the hypergraph structure. S2, use the s-line graph structure to convert the hyperedge set into the vertex set of the ordinary graph, and convert the hypergraph into an ordinary graph; S3, calculating the kth Laplace energy in the general graph; S4, calculating the Laplace energy difference after deleting the node; S5, calculate the Laplace energy centrality LC; S6. Calculate the shortest average distance from a node to other nodes in a common graph; S7. Calculate the exponential centrality HEC of the hypergraph; S8, calculate the top 5% nodes in six real-world hypergraph datasets as hypergraph key nodes; S9. Use the SIR epidemic model to verify the performance of the indicator; S10. Draw the infection diffusion curve.

2. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: In step S1, six real-world hypergraph datasets are collected, namely Vegas-bars, Madison-restaurants, Algebra-questions, High-school, House-committees, and Senate-committees, and hyperedge sets are used to represent the hypergraph structure.

3. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: In step S3, the kth Laplace energy in the general graph is calculated: Among them, μ i is the eigenvalue spectrum of the general graph Laplacian matrix L(G), and the Laplacian matrix is ​​defined as L(G) = D(G) - A(G).

4. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: In step S4, by deleting node v i , calculate the Laplace energy difference before and after deletion to evaluate the node v i Importance: Among them, G i =Gv i is a subgraph of graph G.

5. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: In step S5, get node v i The Laplace energy centrality LC: Among them, d i is the node v in the graph G i The degree, Δ i is the node containing the node v i The number of triangles.

6. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: In step S7, the calculated index is brought into the exponential model to obtain the exponential centrality of the s-line graph. Based on the cardinality of the hyperedge, the exponential centrality of the s-line graph is mapped to the centrality of the nodes in the hypergraph, and smaller weights are assigned to nodes with larger cardinality. Therefore, the exponential centrality HEC of the nodes in the hypergraph is: Among them, LC(e j ) is the third Laplace energy centrality of the 1-line graph; is node e in the 1-line graph i The shortest distance to other nodes; Γ(v i ) is the node v i The set of associated hyperedges; α and β are linearly varying penalty factor functions, which normalize the exponential term and weight term of the exponential model to [0,1].

7. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: In step S9, the first 5% of nodes are taken as the initial infected nodes I. In the first time step, the initial infected node will infect the susceptible node S in its hyperedge with an infection probability of β = 0.1%; for each hyperedge containing the initial infected node, if the scores of I and R nodes are greater than or equal to the threshold η = 0.4, the infected node I in the hyperedge will infect the S nodes in its adjacent hyperedge with a probability of β in the next step, and this process will be repeated at each time step; in each time step, each I node in the hypergraph recovers to node R with an independent probability of γ = 1; the infection and recovery process will stop after five steps; such an infection process takes 100 Monte Carlo simulation experiments and takes the average value as the final infection process.

8. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 1, characterized in that: Applying the HEC in step S7 to the epilepsy data set to distinguish the epileptogenic area from the non-epileptic area specifically includes the following steps: S7.

1. Calculate the HEC of the epileptogenic and non-epileptogenic areas of ten epileptic patients; S7.

2. Draw a comparison map of HEC between the epileptogenic and non-epileptogenic areas; S7.3, calculate the HEC mean and p-value; S7.

4. Draw a box plot of the distribution.

9. The method for detecting key nodes of a hypergraph based on an exponential model according to claim 8, characterized in that: In step S7.1, HEC is applied to SEEG data of epileptic patients; the sampling channels corresponding to the brain regions in the data are regarded as nodes in the hypergraph, and the interactions between multiple nodes are regarded as hyperedges; The index centrality value was calculated for each channel and used to distinguish epileptogenic and non-epileptogenic areas during epileptic seizures.

Images