A method for identifying key nodes in complex networks based on improved dynamic sensitive centrality

Abstract

The present invention discloses a method for identifying key nodes in complex networks based on improved dynamic sensitive centrality, which relates to the technical field of complex network analysis and is used to identify key nodes with significant influence in aviation networks. Traditional methods mostly focus on analyzing the local or global impact of nodes, and do not fully consider the interactive impact between nodes. This method constructs a more accurate key node propagation impact assessment model by considering the node's own influence and the influence of neighboring nodes. Experimental results based on six actual aviation network data sets and the SIR propagation model show that IDS is significantly superior to existing traditional methods in node identification accuracy, especially on EU air‑2, where the Kendall correlation coefficient of IDS compared to DS centrality is improved by up to 105%. This method provides theoretical support for the stable operation of aviation networks and is suitable for identifying key nodes in different types of complex networks.

Description

Technical Field

[0001] The present invention relates to the technical field of complex network analysis, and in particular to a complex network key node identification method based on improved dynamic sensitive centrality. Background Art

[0002] The identification of key nodes in complex networks has attracted widespread attention in recent years. Complex networks have widespread applications in fields such as biology, society, and transportation, and the influence of nodes is crucial for network structure and function. Traditional node centrality metrics (such as degree centrality, closeness centrality, and eigenvector centrality) primarily consider the local or global influence of nodes, failing to fully capture the dynamic characteristics of networks and the interactions between nodes. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for identifying key nodes in complex networks based on improved dynamic sensitive centrality, in order to solve the above-mentioned problems. This method constructs a more accurate key node propagation impact assessment model by considering the influence of the node itself and the influence of neighboring nodes. Experimental results based on six actual aviation network data sets and the SIR propagation model show that IDS is significantly superior to existing traditional methods in node identification accuracy. In particular, on the EU aviation network, the Kendall correlation coefficient of IDS compared with DS centrality is improved by up to 105%. This method provides theoretical support for the stable operation of aviation networks and is suitable for identifying key nodes in different types of complex networks.

[0004] In order to achieve the above object, the present invention adopts the following technical solutions:

[0005] A complex network key node identification method based on improved dynamic sensitive centrality includes the following steps:

[0006] Step 1: Assume an undirected and unweighted network G = (N, E), where N is the node set and E is the edge set; the connection relationship between nodes is represented by the adjacency matrix A = {a ij} means; if two nodes are connected, then a ij = 1, otherwise 0; in an aviation network, airports represent nodes, and direct routes between airports represent connections in the network. The connections between different airports and routes form a complete network. The present invention introduces the relevant methods used in the subsequent steps, including the baseline methods DC, CC, and EC, as well as the improved methods DS and GSI.

[0007] Step 2: For the collected information, take degree centrality into account. Degree centrality measures the importance of a node by the number of neighboring nodes that the node is directly connected to. The specific model is as follows:

[0008]

[0009] Where n represents the number of nodes, Represents nodes i and a ij is the element in the i-th row and j-th column of the adjacency matrix A;

[0010] Step 3: Taking closeness centrality into account, determine the influential nodes through calculation; its expression is as follows:

[0011]

[0012] where d ij is the distance from node i to j;

[0013] Step 4: Eigenvector centrality describes the influence of a node in terms of the number and influence of its neighbors; the equation is expressed as:

[0014]

[0015] Where c = 1 / λ max is the proportionality constant, λ max Represents the adjacency matrix A and x=(x1,x2,...,x n ) T The eigenvector corresponding to the maximum eigenvalue of x j is the eigenvector centrality value of node j;

[0016] Step 5: Sort the nodes according to their degree and evaluate the influence of the nodes by their position attributes;

[0017] Step 6: Quantify the spread of node influence through the topological characteristics and dynamic characteristics of the network; the dynamic sensitive centrality DS expression is as follows:

[0018]

[0019] Where t is the time step, β is the propagation rate, and A is the adjacency matrix; L = (1, 1, ..., 1) T is an n×1 column vector, each component H=βA+(1-μ)I; in the expanded form, μ represents the recovery rate of the network, and I represents the identity matrix;

[0020] Step 7: Global structure influence (GSI) In this paper, the influence of nodes in the network is measured by the global and local influence of the nodes; the GSI of node i is defined as

[0021]

[0022] Where D i is the degree of node i, C(v i ), C(vj ) is the ks value of nodes i and j, dist(v i ,v j ) are the shortest paths from point i to adjacent node j, EC(v i ) is the eigenvector centrality value of node i;

[0023] Step 8: Calculate the node's own influence;

[0024] Step 9: Consider neighbor influence by combining the DS centrality of neighbor nodes; the distance between two nodes is inversely proportional to their influence on the network; the influence values ​​of all neighbor nodes within the specified range are accumulated to evaluate the neighbor influence value of the node;

[0025] Step 10: Based on the topological characteristics and dynamic characteristics of the network, determine the interaction influence of node i by considering the interaction influence between the node itself and its neighbor nodes; the expression of IDS is as follows:

[0026]

[0027] Step 11: Use the Kendall correlation coefficient to measure the correlation between the two sets of ranking data;

[0028] Step 12: Using an imprecise function, calculate the average propagation capacity of the top-pN nodes in the node ranking list generated by various key node mining algorithms, thereby achieving an effective evaluation of the performance of the key node mining algorithm; thereby determining the accuracy;

[0029] Step 13: In the process of obtaining the spread of nodes within the influence range, CCDF is introduced to represent the probability of the sorting result distribution:

[0030]

[0031] where n i Represents the number of nodes ranked r. When the number of nodes with different result values ​​in the ranking list is close to n, it indicates that this method can effectively distinguish the influence of each node.

[0032] In some embodiments, step 5: sorting nodes according to their degree and evaluating the influence of nodes by their positional attributes includes: K-shell decomposition (KS) as a coarse-grained method to sort nodes according to their degree and evaluate the influence of nodes by their positional attributes. The present invention also needs to draw on this method and delete nodes from the periphery to the core; when a node with degree 1 is deleted, a new node with degree 1 will appear in the remaining network; the deletion process continues until there are no nodes with degree 1 in the network; for each node deleted, ks=1, and this operation is repeated until there are no nodes in the network, at which point each node has an associated ks value.

[0033] In some embodiments, step 8: calculating the node's own influence includes: Inspired by the DS centrality and GSI methods, the present invention proposes a dynamic sensitive centrality (IDS) method that considers the interactive influence of the node itself and its neighboring nodes. In our proposed method, the node's own degree is first normalized. Then, the influence of the node's own DS centrality is introduced. The two parts are calculated by square root to obtain the node's own influence, which can be expressed as:

[0034]

[0035] Where D(i) is the degree of node i, N is the total number of network nodes, and DS(i) is the DS centrality value of node i.

[0036] In some embodiments, the formula used in step 9 is as follows:

[0037]

[0038] Where DS(j) represents the DS centrality value of the neighbor; j is the first-order neighbor of node i, and d(i,j) represents the shortest distance from node i to j.

[0039] In some embodiments, the step 11: using the Kendall correlation coefficient to measure the correlation between the two sets of ranking data includes the following formula:

[0040]

[0041] Where sgn(h) is the partition function; when h>0, sgn(h)=+1; when h<0, sgn(h)=-1, when h=0, sgn(h)=0; h i ,q i They represent the real propagation influence capability of node i and the propagation influence capability calculated by different key node mining algorithms; h j ,q jThey represent the real propagation influence capability of node j and the propagation influence capability calculated by different key node mining algorithms;

[0042] When the Kendall correlation coefficient is τ = 1, it indicates that the two sorted data are consistent; when τ = -1, it indicates that they are inconsistent; when τ = 0, it indicates that the two sequences are neither consistent nor inconsistent; when τ is closer to 1, the performance of the method is better, indicating that the method more accurately reflects the actual propagation process.

[0043] In some embodiments, the step 12: using an imprecise function to calculate the average propagation capacity of the top-pN nodes in the node ranking list generated by various key node mining algorithms, thereby achieving an effective evaluation of the performance of the key node mining algorithm; thereby determining the accuracy; includes: using the formula

[0044] Among them, p is the proportion of selected nodes in the entire network, M d (p) and M s (p) represents the average propagation capability of top-pN nodes obtained using different methods and the average propagation capability of top-pN nodes obtained in the SIR propagation model; if M d (p) and M s (p) is closer, indicating that the average transmission capacity of the top-pN nodes selected by this method is more relevant to the transmission capacity of the top-pN nodes in actual situations, then ε(p) is closer to 0.

[0045] The beneficial effects of the complex network key node identification method based on improved dynamic sensitive centrality disclosed in this application may include but are not limited to:

[0046] This paper proposes a method for identifying key nodes in complex networks based on improved dynamic sensitive centrality. By integrating the interaction between the node itself and its neighboring nodes, it effectively identifies the key nodes in the aviation network. Based on multiple aviation network data sets and the SIR propagation model, experiments have verified that the IDS method significantly improves the accuracy of node identification. Compared with traditional methods, the IDS method performs better in multiple evaluation indicators (such as Kendall correlation coefficient, error function and complementary cumulative distribution function). In particular, in the EU aviation network, the correlation coefficient of this method is improved by up to 105% compared with DS centrality. This improved method not only provides theoretical support for the operational stability of the aviation network, but also provides an effective tool for dealing with sudden failures. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 The overall flow chart of the method provided by the present invention;

[0048] Figure 2 Topological diagrams of six aviation networks provided by the present invention;

[0049] Figure 3 The present invention provides an SIR model for six aeronautical networks to evaluate the shortest distances between first to fifth order neighbors and the propagation impact of nodes;

[0050] Figure 4 A schematic diagram of comparing the IDS method with six other methods using the Kendall coefficient in six aviation networks provided by the present invention;

[0051] Figure 5 Comparison of the accuracy of the imprecise functions obtained by the seven methods provided by the present invention in different aviation networks;

[0052] Figure 6 Schematic diagram of the comparison of seven methods using CCDF in six aviation networks provided by the present invention. DETAILED DESCRIPTION

[0053] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0054] On the contrary, this application covers any alternatives, modifications, equivalents, and solutions made within the spirit and scope of this application as defined by the claims. Furthermore, to facilitate a better understanding of this application, certain specific details are described in detail below in the detailed description of this application. Those skilled in the art will be able to fully understand this application without these details.

[0055] The following is a detailed description of a complex network key node identification method based on improved dynamic sensitive centrality involved in an embodiment of the present application. It should be noted that the following embodiments are only used to explain the present application and do not constitute a limitation of the present application.

[0056] like Figure 1-2 As shown, a complex network key node identification method based on improved dynamic sensitive centrality includes the following steps:

[0057] Step 1: Assume an undirected and unweighted network G = (N, E), where N is the node set and E is the edge set; the connection relationship between nodes is represented by the adjacency matrix A = {a ij} means; if two nodes are connected, then a ij= 1, otherwise 0; in an aviation network, airports represent nodes, and direct routes between airports represent connections in the network. The connections between different airports and routes form a complete network. The present invention introduces the relevant methods used in the subsequent steps, including the baseline methods DC, CC, and EC, as well as the improved methods DS and GSI.

[0058] Step 2: For the collected information, take degree centrality into account. Degree centrality measures the importance of a node by the number of neighboring nodes that the node is directly connected to. The specific model is as follows:

[0059]

[0060] Where n represents the number of nodes, Represents nodes i and a ij is the element in the i-th row and j-th column of the adjacency matrix A;

[0061] Step 3: Taking closeness centrality into account, determine the influential nodes through calculation; its expression is as follows:

[0062]

[0063] where d ij is the distance from node i to j;

[0064] Step 4: Eigenvector centrality describes the influence of a node in terms of the number and influence of its neighbors; the equation is expressed as:

[0065]

[0066] Where c = 1 / λ max is the proportionality constant, λ max Represents the adjacency matrix A and x=(x1,x2,...,x n ) T The eigenvector corresponding to the maximum eigenvalue of x j is the eigenvector centrality value of node j;

[0067] Step 5: Sort the nodes by their degree and evaluate the influence of the nodes by their positional attributes; this includes: K-shell decomposition (KS) as a coarse-grained method to sort the nodes by their degree and evaluate the influence of the nodes by their positional attributes. The present invention also needs to draw on this method and delete nodes from the periphery to the core; when a node with degree 1 is deleted, a new node with degree 1 will appear in the remaining network; the deletion process continues until there are no nodes with degree 1 in the network; for each node deleted, ks = 1, and this operation is repeated until there are no nodes in the network, at which point each node has an associated ks value.

[0068] Step 6: Quantify the spread of node influence through the topological characteristics and dynamic characteristics of the network; the dynamic sensitive centrality DS expression is as follows:

[0069]

[0070] Where t is the time step, β is the propagation rate, and A is the adjacency matrix; L = (1, 1, ..., 1) T is an n×1 column vector, each component H=βA+(1-μ)I; in the expanded form, μ represents the recovery rate of the network, and I represents the identity matrix;

[0071] Step 7: Global structure influence (GSI) In this paper, the influence of nodes in the network is measured by the global and local influence of the nodes; the GSI of node i is defined as

[0072]

[0073] Where D i is the degree of node i, C(v i ), C(v j ) is the ks value of nodes i and j, dist(v i ,v j ) are the shortest paths from point i to adjacent node j, EC(v i ) is the eigenvector centrality value of node i;

[0074] Step 8: Calculate the node's own influence. Inspired by the DS centrality and GSI methods, this paper proposes an improved dynamic sensitive centrality (IDS) method that considers the interactive influence of the node itself and its neighboring nodes. In our proposed method, the node's degree is first normalized. Then, the influence of the node's own DS centrality is introduced. The two parts are calculated by square root to obtain the node's own influence, which can be expressed as:

[0075]

[0076] Where D(i) is the degree of node i, N is the total number of network nodes, and DS(i) is the DS centrality value of node i.

[0077] Step 9: Consider neighbor influence by combining the DS centrality of neighbor nodes. The distance between two nodes is inversely proportional to their influence on the network. The influence values ​​of all neighbor nodes within the specified range are accumulated to evaluate the neighbor influence value of the node. The formula used is as follows:

[0078]

[0079] Where DS(j) represents the DS centrality value of the neighbor; j is the first-order neighbor of node i, and d(i,j) represents the shortest distance from node i to j.

[0080] Step 10: Based on the topological characteristics and dynamic characteristics of the network, determine the interaction influence of node i by considering the interaction influence between the node itself and its neighbor nodes; the expression of IDS is as follows:

[0081]

[0082] Step 11: Use the Kendall correlation coefficient to measure the correlation between the two sets of ranking data; Use the Kendall correlation coefficient to measure the correlation between the two sets of ranking data; including: The formula is as follows:

[0083]

[0084] Where sgn(h) is the partition function; when h>0, sgn(h)=+1; when h<0, sgn(h)=-1, when h=0, sgn(h)=0; h i ,q i They represent the real propagation influence capability of node i and the propagation influence capability calculated by different key node mining algorithms; h j ,q j They represent the real propagation influence capability of node j and the propagation influence capability calculated by different key node mining algorithms; h i ,q i and h j ,q j There are three relationships between: i >h j ,q i >q j or h i <h j ,q i j ;h i >h j ,q i j or h i <h j ,q i >q j ;h i =h j ,q i =q j ​​, representing consistency, inconsistency, and consistency or inconsistency, respectively. In this experiment, we use the Kendall coefficient to analyze the correlation between the network ranking lists generated by various node ranking algorithms and the SIR model. If the resulting τ value is closer to 1, it means that the ranking list generated by the complex network key node mining algorithm is more likely to reflect the node ranking during the actual propagation process, indicating that the algorithm is more efficient.

[0085] When the Kendall correlation coefficient is τ = 1, it indicates that the two sorted data are consistent; when τ = -1, it indicates that they are inconsistent; when τ = 0, it indicates that the two sequences are neither consistent nor inconsistent; when τ is closer to 1, the performance of the method is better, indicating that the method more accurately reflects the actual propagation process.

[0086] Step 12: Using an imprecise function, calculate the average propagation capacity of the top-pN nodes in the node ranking list generated by various key node mining algorithms, thereby achieving an effective evaluation of the performance of the key node mining algorithm; thereby determining the accuracy; using the formula

[0087] Among them, p is the proportion of selected nodes in the entire network, M d (p) and M s (p) represents the average propagation capability of top-pN nodes obtained using different methods and the average propagation capability of top-pN nodes obtained in the SIR propagation model; if M d (p) and M s (p) is closer, indicating that the average transmission capacity of the top-pN nodes selected by this method is more relevant to the transmission capacity of the top-pN nodes in actual situations, then ε(p) is closer to 0.

[0088] Step 13: In the process of obtaining the spread of nodes within the influence range, CCDF is introduced to represent the probability of the sorting result distribution:

[0089]

[0090] where n i Represents the number of nodes ranked r. When the number of nodes with different result values ​​in the ranking list is close to n, it indicates that this method can effectively distinguish the influence of each node.

[0091] Table 1 shows the relevant topological properties of six aviation networks, introducing six real aviation networks: EU air-1, EU air-2, US air, CA, CZ, and MU. EU air-1 constitutes a comprehensive aviation network composed of five European airlines, and EU air-2 represents a comprehensive aviation network composed of 15 European airlines. US air represents the American Airlines network, CA is the Air China route network, CZ is the China Southern Airlines route network, and MU is the China Eastern Airlines route network. N represents the total number of nodes in the network, and E is the total number of edges in the network. <k>represents the average degree, β th represents the propagation threshold, γ represents the expansion threshold, C is the clustering coefficient, <d>represents the average distance, and D represents the network diameter.

[0092] Table 1 Related topological attributes

[0093]

[0094] Figure 2 A topological diagram of six aviation networks provided as an example of the present invention;

[0095] Topological graphs of six different aviation networks. Each node in the graph represents an airport, and an edge between two nodes represents a flight route. Each node in the graph represents an airport, and an edge between two nodes represents a flight route. These six aviation networks exhibit similar topological characteristics, exhibiting a layout characterized by dense centers and sparse edges. Observations show that a small number of nodes have a large number of connected edges, while most nodes have only a small number of edges. This indicates that nodes with many directly connected neighbors are more important than nodes with only a few neighbors.

[0096] Figure 3 An SIR model is provided through six aviation networks for an example of the invention. The shortest distances within the 1st, 2nd, 3rd, 4th and 5th order neighbors of the selected nodes in the IDS and the diffusion impact of the nodes are evaluated.

[0097] We set the propagation velocity β = 0.5, the time step t = 10, and the recovery rate μ = 1 for the propagation process of six different airline networks: EU Airlines 1, EU Airlines 2, American Airlines, CZ (China Southern Airlines), CA (Air China), and MU (China Eastern Airlines). The shortest distances within a node's first-, second-, third-, fourth-, and fifth-order neighbors were selected as variables. The Kendall correlation coefficient was used to compare the accuracy of the IDS under different conditions. The Kendall correlation coefficient performed best for the first-order neighbor range as the shortest distance, resulting in the highest IDS accuracy. When considering second-order neighbors, Kendall's τ significantly decreased. However, for third-, fourth-, and fifth-order neighbor ranges, Kendall's τ values ​​stabilized.

[0098] Figure 4 An example of the invention provides a schematic diagram of comparing the IDS method with six methods using the Kendall correlation coefficient in six aviation networks;

[0099] Six methods, DC, CC, EC, KS, DS, and GSI, are used to compare with IDS. The time step is set to t = 10, the recovery rate μ = 1, the diffusion rate β = (0, 0.1], and the red dotted line indicates the diffusion threshold β th .like Figure 3 As shown in the figure, IDS performs exceptionally well in the four air networks EU air-1, US air, CA, and CZ. In most cases, IDS obtains Kendall τ that is closest to 1 compared to other methods. Furthermore, IDS also demonstrates superior performance in the EU air-2 and MU networks. EC, DS, and GSI perform relatively well, but DC, CC, and KS perform slightly worse. In the EU air-2 network, IDS significantly improves results compared to DS centrality, with a maximum improvement of 105% when β = 0.1. Across different air networks, IDS achieves higher accuracy than other methods, indicating that this method can more accurately identify potential nodes in the network.

[0100] Figure 5 For the purpose of an example of the invention, the accuracy comparison of the imprecise functions obtained by seven methods in different aeronautical networks is provided.

[0101] At the same time, we use the imprecision function ε(p) to further evaluate the accuracy of the method. During the experiment, the propagation rate and time step are set to β = 0.05, t = 10 and μ = 1, and the p value is 0.1 to 0.3. Figure 4 As can be seen, IDS generally achieves lower imprecision function values ​​than the other methods. Across various P values, the results obtained by our method are consistently close to or equal to 0. In particular, in the EUair-2 aggregated aeronautical network, IDS outperforms all other methods. DC and CC exhibit poor performance across these six networks, while the KS method exhibits the greatest fluctuation in imprecision function values ​​among the seven methods. This demonstrates that IDS can effectively and accurately predict the top 10% to 30% of nodes in an aeronautical network, demonstrating significant influence.

[0102] Figure 6 The present invention provides an example of the comparison of seven CCDF methods in six aviation networks.

[0103] Finally, CCDF is used to further verify the superiority of this method. Figure 5 As shown in the figure, we set β = 0.05, μ = 1, and t = 10. KS, due to its poor ability to identify key nodes in the network, decays the fastest and finally decays to 0, followed by DC and CC. Compared with the other four methods (EC, DS, GSI, and IDS), these three methods decay faster. Among DS, GSI, and IDS, IDS shows a slower decay rate and outperforms the other methods. By comparing with other methods, the effectiveness of this method in evaluating the propagation influence of each node in the network is demonstrated. This method can more accurately identify key nodes and avoid the problem of multiple nodes receiving the same centrality value in the same ranking.

[0104] In the experimental verification process, we introduced six different methods to compare with IDS: four baseline methods (DC, CC, EC and KS) and two improved methods (DS and GSI). We used the actual data of airlines to construct the topology graphs of six different aviation networks. Figure 1 As shown in the figure, each node represents an airport, and an edge between two nodes represents an air route. These six aviation networks exhibit similar topological characteristics, exhibiting a layout characterized by dense centers and sparse edges. Observation reveals that a small number of nodes have a large number of connected edges, while most nodes have only a small number of edges. This indicates that nodes with many directly connected neighbors are more important than nodes with only a few.

[0105] The study of key node identification in complex networks can provide valuable insights into understanding and analyzing the structure and propagation characteristics of networks. This is of great significance for optimizing network performance. To identify key nodes, this paper proposes an improved method, IDS, based on network topology and dynamic characteristics, which takes into account the influence of both itself and neighboring nodes. First, the self-influence of a node is calculated using its own degree centrality and DS centrality. Second, the first-order neighbor influence of a node is considered, and the interaction influence of the node is determined by combining these two aspects. To verify the accuracy and reliability of the proposed method, three evaluation indicators are used: Kendall correlation coefficient, imprecision function, and complementary cumulative distribution function. (1) The intrusion detection method proposed in this paper outperforms the other six methods by the Kendall correlation coefficient. In the EU air-2 network, when β = 0.1, the Kendall correlation coefficient is increased by 105% compared with DS centrality. (2) In six different air networks, the value of IDS is lower than that of other methods using the imprecise function. Especially in the EU air-2 network, the fluctuation range of the IDS detection result value is smaller than that of other methods and is closer to 0. (3) In the CCDF evaluation, the results of IDS decreased more slowly than those of the other methods. In all six networks, the slope of the CCDF line followed the diagonal line. Therefore, IDS was able to more effectively distinguish nodes with different influence orders. All the above evaluation indicators show that the performance of this method is better than that of other methods. This intrusion detection system can provide a reference for accurately identifying key nodes in the network.

[0106] Due to the continuous increase in global air traffic and the growing number of routes, the complexity of network topologies is also increasing. When unforeseen factors cause delays or closures at certain airports, the normal operation of the aviation network may be affected. In theory, an intrusion detection system based on the diffusion dynamics model can be used to identify important airports in the aviation network. Identifying important nodes in the aviation network helps to accurately optimize the layout of routes and airports, improving the network's resilience and anti-interference capabilities. In addition to aviation networks, it can also be extended to email, electricity, protein, social, and various other networks to identify basic nodes. With the development of society, complex networks have evolved from single-layer networks to multi-layer networks, weighted networks, and temporal networks. Therefore, future research will focus on these networks to further expand the applicability of intrusion detection systems.

[0107] The above description is only a preferred embodiment of the present invention and is used to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.< / d> < / k>

Claims

1. A complex network key node identification method based on improved dynamic sensitive centrality, characterized by: The following steps are involved: Step 1: Assume an undirected and unweighted network G = (N, E), where N is the node set and E is the edge set; the connection relationship between nodes is represented by the adjacency matrix A = {a ij } means; if two nodes are connected, then a ij =1, otherwise 0; Step 2: For the collected information, take degree centrality into account. Degree centrality measures the importance of a node by the number of neighboring nodes that the node is directly connected to. The specific model is as follows: Where n represents the number of nodes, represents the number of edges of node i, a ij is the element in the i-th row and j-th column of the adjacency matrix A; Step 3: Taking closeness centrality into account, determine the influential nodes through calculation; Its expression is as follows: where d ij is the distance from node i to j; Step 4: Eigenvector centrality describes the number and influence of a node in its neighbors; the equation is expressed as: Where c = 1 / λ max is the proportionality constant, λ max Represents the eigenvector x=(x1,x2,...,x n ) T The corresponding maximum eigenvalue; x j is the eigenvector centrality value of node j; Step 5: Sort the nodes according to their degree and evaluate the influence of the nodes by their position attributes; Step 6: Quantify the spread of node influence through the topological characteristics and dynamic characteristics of the network; the dynamic sensitive centrality DS expression is as follows: Where t is the time step, β is the propagation rate, and A is the adjacency matrix; L = (1, 1, ..., 1) T is an n×1 column vector, each component H=βA+(1-μ)I; in the expanded form, μ represents the recovery rate of the network, and I represents the identity matrix; Step 7: Calculate GSI to measure the influence of nodes in the network by their global and local influence; the GSI of node i is defined as Where D i is the degree of node i, C(i), C(j) are the ks values ​​of nodes i and j, d(i, j) are the shortest paths from point i to the adjacent node j, and EC(i) is the eigenvector centrality value of node i; Step 8: Calculate the node's own influence; Step 9: Consider neighbor influence by combining the DS centrality of neighbor nodes; the distance between two nodes is inversely proportional to their influence on the network; the influence values ​​of all neighbor nodes within the specified range are accumulated to evaluate the neighbor influence value of the node; Step 10: Based on the topological characteristics and dynamic characteristics of the network, determine the interaction influence of node i by considering the interaction influence between the node itself and its neighbor nodes; the expression of IDS is as follows: IDS(i)=SI(i)*NI(i); where SI(i) represents the node’s own influence and NI(i) represents the node’s neighbor influence; Step 11: Use the Kendall correlation coefficient to measure the correlation between the two sets of ranking data; Step 12: Using an imprecise function, calculate the average propagation capacity of the top-pN nodes in the node ranking list generated by various key node mining algorithms, thereby achieving an effective evaluation of the performance of the key node mining algorithm; thereby determining the accuracy; Step 13: In the process of obtaining the spread of nodes within the influence range, CCDF is introduced to represent the probability of the sorting result distribution: where n i Represents the number of nodes ranked m. When the number of nodes with different result values ​​in the ranking list is close to n, it indicates that this method can effectively distinguish the influence of each node.

2. The method for identifying key nodes in complex networks based on improved dynamic sensitive centrality according to claim 1, characterized in that: Step 5: Sort the nodes according to their degree and evaluate the influence of the nodes by their location attributes; including: deleting nodes from the periphery to the core; deleting nodes with a degree of 1, and new nodes with a degree of 1 will appear in the remaining network; the deletion process continues until there are no nodes with a degree of 1 in the network; for each node deleted, ks=1, and this operation is repeated until there are no nodes in the network, at which point each node has an associated ks value.

3. The method for identifying key nodes in complex networks based on improved dynamic sensitive centrality according to claim 2, characterized in that: The step 8: calculating the node's own influence includes: normalizing the node's own degree; introducing the influence of the node's own DS centrality; and calculating the square root of the two parts to obtain the node's own influence, which is expressed as: Where D(i) is the degree of node i, K is the total number of network nodes, and DS(i) is the DS centrality value of node i.

4. The method for identifying key nodes in complex networks based on improved dynamic sensitive centrality according to claim 3, characterized in that: The formula used in step 9 is as follows: Where DS(j) represents the DS centrality value of the neighbor; j is the first-order neighbor of node i, and d(i,j) represents the shortest distance from node i to j.

5. The method for identifying key nodes in complex networks based on improved dynamic sensitive centrality according to claim 4, characterized in that: Step 11: using the Kendall correlation coefficient to measure the correlation between the two sets of ranking data; including: the formula is as follows: Where sgn(h) is the partition function; when h>0, sgn(h)=+1; when h<0, sgn(h)=-1, when h=0, sgn(h)=0; h i ,q i They represent the real propagation influence capability of node i and the propagation influence capability calculated by different key node mining algorithms; h j ,q j They represent the real propagation influence capability of node j and the propagation influence capability calculated by different key node mining algorithms; When the Kendall correlation coefficient is τ = 1, it indicates that the two sorted data are consistent; when τ = -1, it indicates that they are inconsistent; when τ = 0, it indicates that the two sequences are neither consistent nor inconsistent; when τ is closer to 1, the performance of the method is better, indicating that the method more accurately reflects the actual propagation process.

6. The method for identifying key nodes in complex networks based on improved dynamic sensitive centrality according to claim 5, characterized in that: Step 12: using an imprecise function to calculate the average propagation capabilities of the top-pN nodes in the node ranking list generated by various key node mining algorithms, thereby achieving an effective evaluation of the performance of the key node mining algorithm; To determine the accuracy; including: using the formula Among them, p is the proportion of selected nodes in the entire network, M d (p) and M s (p) represents the average propagation capability of top-pN nodes obtained using different methods and the average propagation capability of top-pN nodes obtained in the SIR propagation model; if M d (p) and M s (p) is closer, indicating that the average transmission capacity of the top-pN nodes selected by this method is more relevant to the transmission capacity of the top-pN nodes in actual situations, then ε(p) is closer to 0.

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