A key node identification method based on network generalized energy and information entropy

Abstract

This invention discloses a key node identification method based on network generalized energy and information entropy, comprising the steps of: data standardization; node influence scoring; and node sorting, arranged in descending order according to the node scores. Compared with existing technologies, this invention creatively combines network energy and information entropy theories, considering both the local energy entropy attribute of network nodes and the energy entropy of the node's first-order and second-order neighbors, effectively improving the accuracy of key node identification; and by converting network generalized energy into network topology indicators, it reduces computational time complexity and improves the effectiveness of key node identification.

Description

Technical Field

[0001] The present invention relates to the field of computer networks, and in particular to a key node identification method based on network generalized energy and information entropy. Background Art

[0002] The key nodes of a complex network refer to some special nodes that can affect the structure and function of the network to a greater extent than other nodes in the network. In recent years, the research on node importance ranking has received more and more attention, not only because of its great theoretical research significance, but also because of its wide practical application value. Almost all complex systems (such as social, biological, information, technology, and transportation systems) can be naturally represented as networks. Among them, nodes represent various components of the system, and the edges between nodes represent the connection between the elements. The application of complex network theory and methods can help us better understand the characteristics of these complex systems and better predict and control them. The number of important nodes is generally very small, but their influence can quickly spread to most nodes in the network. For example, in a deliberate attack on a scale-free network, the attack on a small number of the most important nodes will cause the entire network to collapse; the microblogs sent by the most influential users on Weibo can quickly spread throughout the entire network; only 1% of companies control 40% of the global economy. It can be seen that important nodes have a huge impact on the structure and function of the network, and the ranking of node importance and the mining of important nodes are of great significance.

[0003] So far, scholars have proposed a series of classic methods to identify key nodes in complex networks, however, these existing methods have some defects to varying degrees. For example, degree centrality (DC), which only considers local information about network nodes; closeness centrality (CC) and betweenness centrality (BC), which focus on the shortest path and have high computational complexity, and their performance is poor if most information is not transmitted through the shortest path; eigenvector centrality (EC), which faces limitations in dealing with large-scale, asymmetric and heterogeneous networks; K-shell algorithm, which performs poorly in distinguishing nodes with the same degree value. Summary of the invention

[0004] In order to overcome the deficiencies of the prior art, the present invention proposes a key node identification method based on network generalized energy and information entropy.

[0005] The technical solution of the present invention is implemented as follows: a key node identification method based on network generalized energy and information entropy comprises the steps of

[0006] S1: Data standardization, abstracting the selected network data set into a network graph and calculating the adjacency matrix A(G), degree matrix D(G) and generalized matrix A α (G);

[0007] S2: Node influence scoring, including steps

[0008] S21: Calculate the generalized energy of the entire network. The calculation formula of the generalized energy of the entire network is: where d i is the network node v i The degree of , α∈[0,1] is the adjustable parameter of the generalized matrix;

[0009] S22: Calculate the generalized energy of network nodes, network node v i The generalized energy calculation formula is Where N i is node v i The neighbor set of d i is the network node v i The degree of , α∈[0,1] is the adjustable parameter of the generalized matrix;

[0010] S23: Calculate the first-order and second-order generalized energy of network nodes, network node v i The calculation formulas for the first-order generalized energy and the second-order generalized energy are and Where N i is node v i The set of neighbors of ;

[0011] S24: Construct the first-order and second-order generalized energy probability sets of network nodes based on information entropy. i The first-order generalized energy probability and second-order generalized energy probability formulas are and Where N i is node v i The set of neighbors of ;

[0012] S25: Calculate the first-order generalized energy entropy and second-order generalized energy entropy of network nodes, network nodes v i The first-order generalized energy entropy and second-order generalized energy entropy formulas are and Where N i is node v i The set of neighbors of ;

[0013] S26: Calculate the generalized energy entropy of network nodes, network nodes v i The generalized energy entropy is GEE(v i )=GEE1(vi )+λ i GEE2(v i ),in

[0014] S27: Calculate the propagation capacity of network nodes and convert node v i The GEE values ​​of the neighboring nodes are summed, that is, Where N i is node v i The set of neighbors of ;

[0015] S28: Define the generalized energy entropy centrality of a network node as Where N i is node v i The set of neighbors of ;

[0016] S3: Arrange the GEEC values ​​of each network node in the network obtained in step 2 in descending order, and use the GEEC sequence R = {R1, R2, ..., R n}, where r1 represents the value of the node with the largest GEEC value in the network, r n It represents the value of the node with the smallest GEEC value in the network.

[0017] Furthermore, the key node identification method based on network generalized energy and information entropy also includes step S4: accuracy analysis, including the steps

[0018] S41: Use the SIR propagation model to measure the actual propagation capability of network nodes and generate a sequence of SIR models σ = {σ1,σ2,...,σ n};

[0019] S42: Obtain the node sequence R obtained by different measurement methods, and use the Kendall correlation coefficient to represent the correlation between the sequence σ and the sequence R. The Kendall correlation coefficient is defined as , where n c is the number of identical pairs of elements in the two sequences, n d is the number of inconsistent pairs of elements in the two sequences, and n is the size of the two pairs of elements;

[0020] S43: The ability of each centrality measure to distinguish node propagation efficiency is tested through the monotonicity function. Monotonicity (M) is defined as: Where n is the number of elements in the sequence R, N r is the number of elements in the rth level;

[0021] S44: Using ranking similarity function Characterizes the correlation of the top L rankings of sequences σ and R, where σ(L) and R(L) represent the top L rankings of σ and R, respectively;

[0022] S45: Result analysis.

[0023] Furthermore, in S42 , a node sequence R obtained by 6 measurement methods is obtained.

[0024] The beneficial effect of the present invention lies in that, compared with the prior art, the present invention combines network energy and information entropy theory. On the one hand, it takes into account the local energy entropy properties of network nodes and the energy entropy of the first-order neighbors and second-order neighbors of the nodes, which can effectively improve the recognition accuracy of key nodes; on the other hand, by converting the network generalized energy into network topology indicators, the time complexity of calculation is reduced, thereby improving the recognition effectiveness of key nodes. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 It is a flow chart of an embodiment of a key node identification method based on network generalized energy and information entropy of the present invention;

[0026] Figure 2 It is a schematic diagram of the key node identification method based on network generalized energy and information entropy;

[0027] Figure 3 It is a Kendall correlation coefficient comparison diagram of the key node identification accuracy obtained by the method of the present invention and six centrality measurement methods DC, BC, CC, CLD, CNC, and LEC in the Jazz network;

[0028] Figure 4 It is a comparison chart of the F(L) of the key node identification accuracy obtained by the method of the present invention and six centrality measurement methods DC, BC, CC, CLD, CNC, and LEC on the USAir network;

[0029] Figure 5 This is a monotonicity comparison table of key node identification accuracy obtained by using the method of the present invention and six centrality measurement methods DC, BC, CC, CLD, CNC, and LEC on nine real networks. DETAILED DESCRIPTION

[0030] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0031] See also Figure 1 and Figure 2 The key node identification method based on network generalized energy and information entropy provided by the present invention specifically includes:

[0032] Step 1: Data standardization: abstract the selected network data set into a network graph and calculate the adjacency matrix, degree matrix and generalized matrix;

[0033] S11: Convert the network structure into processable data, adjacency matrix A(G), degree matrix D(G) and generalized matrix A α (G) to achieve data standardization. The nodes in the network G represent entities, and the connection between entities is abstracted as an edge. If the node and its edge relationship are represented by the adjacency matrix A = (a ij ) n×n It is represented by, where n represents the number of nodes in the network, a ij Represents the relationship between any two nodes i and j in the network. Degree matrix D(G) = diag(d1, d2,…, d n ), where d i Represents node v i The degree of node v i The number of neighbor nodes. Generalized matrix D α (G) = αD(G) + (1-α)A(G), where α is the adjustment parameter of the adjacency matrix and the degree matrix.

[0034] Step 2: Node scoring. Generalized energy entropy centrality (GEEC) is proposed by combining generalized energy theory and information entropy theory. The network topology is quantified and the GEEC value of all nodes is calculated, i.e., the node influence score.

[0035] S21: Calculate the generalized energy of the entire network; the generalized energy calculation formula of the network is where μ i is the eigenvalue corresponding to the generalized matrix of the network. We transform the calculation of generalized energy into the calculation of network topology indicators through the knowledge of matrix theory and graph theory, which significantly reduces the time complexity of the calculation, that is, where d i is the network node v i The degree of , α∈[0,1] is the adjustable parameter of the generalized matrix.

[0036] S22: Derive the generalized energy formula of the network node according to step S21; the generalized energy of the node can be reflected by the change of the generalized energy of the network after deleting the node, that is, Then node v i The generalized energy calculation formula is Where N i is node v i The set of neighbors.

[0037] S23: Calculate the first-order and second-order generalized energy of network nodes; the first-order refers to the node's neighbor set, and the second-order refers to the node's neighbor's neighbor set. i The generalized energy is E i To simplify, the node v i The calculation formulas for the first-order generalized energy and the second-order generalized energy are and Where N i is node v i The set of neighbors.

[0038] S24: Construct the first-order and second-order generalized energy probability sets of network nodes based on information entropy; node v i The first-order generalized energy probability and second-order generalized energy probability formulas are and Where N i is node v i The set of neighbors.

[0039] S25: Calculate the first-order generalized energy entropy and second-order generalized energy entropy of network nodes; according to the definition of information entropy, the formulas of first-order generalized energy entropy and second-order generalized energy entropy are respectively and Where N i is node v i The set of neighbors.

[0040] S26: Calculate the generalized energy entropy by combining the first-order and second-order generalized energy entropy information of the node; Considering the node v i The propagation effect of is mainly determined by the first-order and second-order neighbors, and the contribution of the first-order neighbors is greater than that of the second-order neighbors. Therefore, the coefficient of the first-order generalized energy entropy is 1 by default, and the coefficient of the second-order generalized energy entropy is λ i Represents. Then the node v i The generalized energy entropy is expressed as GEE(v i )=GEE1(v i )+λ i GEE2(v i ),in

[0041] S27: Calculate the propagation capacity of network nodes through the neighbor information of nodes, and i The GEE values ​​of the neighboring nodes are summed, that is, Where N i is node v i The set of neighbors.

[0042] S28: The generalized energy entropy centrality of the final network node is defined as Where N i is node vi The set of neighbors.

[0043] S29: The importance score of each node in the network can be calculated by the main method GEEC of the present invention. The higher the score, the more critical the node.

[0044] Step 3: Arrange the GEEC values ​​of each node in the network obtained in step 2 in descending order, and use the GEEC sequence R = {R1, R2, ..., R n}, where r1 represents the value of the node with the largest GEEC value in the network, r n It represents the value of the node with the smallest GEEC value in the network;

[0045] Step 4, accuracy analysis, uses the SIR model to evaluate the actual propagation capability of the node, and then compares and analyzes the accuracy and effectiveness of the present invention.

[0046] S41: The SIR propagation model is used to measure the real propagation capability of a node. In the SIR model, the states of network nodes are divided into three categories: susceptible (S), infected (I), and recovered (R). Initially, it is assumed that there is only one node v i In the infected state, other nodes are in the susceptible state. The infected node infects the neighboring nodes in the susceptible state with probability β. Similarly, the infected node changes to the recovery state with probability γ. This process is repeated until there are no nodes in the infected state in the network. At this time, the number of nodes in the recovery state in the network is considered to be node v i Finally, the real propagation capacity of each node in the network is obtained, and a sequence of SIR model σ={σ1,σ2,...,σ n}.

[0047] S42: Kendall correlation coefficient: After calculating the node propagation capability and generating the sequence σ in step S41, the node sequence R obtained by different measurement methods is obtained. The higher the correlation between the two lists, the higher the accuracy of the corresponding method in evaluating the node expansion capability. The Kendall correlation coefficient can be used to represent the correlation between the two lists.

[0048] Assume {(σ1,R1),(σ2,R2),...,(σ n ,R n )} is a set of ordered pairs consisting of the sequence σ and the sequence R. For any pair of elements (σ i ,R i ) and (σ j ,R j ), when σ i >σ j And R i>R j or i <σ j And R i <R j When , the two pairs of elements are considered to be consistent; when σ i >σ j And R i <R j or i >σ j And R i <R j When , the two pairs of elements are considered inconsistent; when σ i =σ j or R i =R j , then the two pairs of elements are considered neither consistent nor inconsistent.

[0049] The definition of Kendall's correlation coefficient is as follows:

[0050] Among them, n c is the number of identical pairs of elements in the two sequences, n d is the number of inconsistent pairs of elements in the two sequences, and n is the size of the two pairs of elements.

[0051] S43: The monotonicity function is used to test the ability of each centrality measure to distinguish the node propagation efficiency. The definition of monotonicity (M) is as follows:

[0052]

[0053] Where n is the number of elements in the sequence R, N r is the number of elements in the rth level. The value of M is always a number in the range [0,1]. The larger the value of M, the stronger the ability of the metric to distinguish nodes in the network.

[0054] S44: As we all know, nodes with greater influence have higher rankings, but the Kendall correlation coefficient takes into account all rankings in the sequence σ and R. In order to illustrate the ranking accuracy of key nodes, the ranking similarity function is used. Characterizes the correlation of the top L rankings of sequences σ and R, where σ(L) and R(L) represent the top L rankings of σ and R, respectively.

[0055] S45: See Figure 3 , Figure 4 and Figure 5 , the GEEC sequence obtained in step 2, the sequence R = {R1, R2, ..., R n} and the sequence σ of the SIR model of step S41 = {σ1,σ2,...,σ n}Comparison shows that the GEEC method has the highest correlation with the SIR model, that is, the Kendall correlation coefficient is the largest. Furthermore, the monotonicity (M) value of GEEC is closest to 1 among the 9 actual networks, indicating that the GEEC method has the strongest ability to identify network nodes. The ranking analysis of the key nodes of the network shows that the GEEC method is most correlated with the actual propagation ability of the key nodes, that is, the method can accurately identify the key nodes in the network, which is of great significance for the evolution and control of the network.

[0056] The above is a preferred embodiment of the present invention. It should be pointed out that a person skilled in the art can make several improvements and modifications without departing from the principle of the present invention. These improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A key node identification method based on network generalized energy and information entropy, characterized in that: Included steps S1: Data standardization, abstracting the selected network data set into a network graph and calculating the adjacency matrix A(G), degree matrix D(G) and generalized matrix A α (G); S2: Node influence scoring, including steps S21: Calculate the generalized energy of the entire network. The calculation formula of the generalized energy of the entire network is: where d i is the network node v i The degree of , α∈[0,1] is the adjustable parameter of the generalized matrix; S22: Calculate the generalized energy of network nodes, network nodes v i The generalized energy calculation formula is Where N i is node v i The neighbor set of d i is the network node v i The degree of , α∈[0,1] is the adjustable parameter of the generalized matrix; S23: Calculate the first-order and second-order generalized energy of network nodes, network node v i The calculation formulas for the first-order generalized energy and the second-order generalized energy are and Where N i is node v i The set of neighbors of ; S24: Construct the first-order and second-order generalized energy probability sets of network nodes based on information entropy. i The first-order generalized energy probability and second-order generalized energy probability formulas are and Where N i is node v i The set of neighbors of ; S25: Calculate the first-order generalized energy entropy and second-order generalized energy entropy of network nodes, network nodes v i The first-order generalized energy entropy and second-order generalized energy entropy formulas are and Where N i is node v i The set of neighbors of ; S26: Calculate the generalized energy entropy of network nodes, network node v i The generalized energy entropy is GEE(v i )=GEE1(v i )+λ i GEE2(v i ),in S27: Calculate the propagation capacity of network nodes and convert node v i The GEE values ​​of the neighboring nodes are summed up, that is, Where N i is node v i The set of neighbors of ; S28: Define the generalized energy entropy centrality of a network node as Where N i is node v i The set of neighbors of ; S3: Arrange the GEEC values ​​of each network node in the network obtained in step 2 in descending order, and use the GEEC sequence R = {R1, R2, ..., R n }, where r1 represents the value of the node with the largest GEEC value in the network, r n It represents the value of the node with the smallest GEEC value in the network.

2. The key node identification method based on network generalized energy and information entropy according to claim 1 is characterized in that: The method for identifying key nodes based on network generalized energy and information entropy also includes step S4: accuracy analysis, including the steps S41: Use the SIR propagation model to measure the actual propagation capability of network nodes and generate a sequence of SIR models σ = {σ1,σ2,...,σ n }; S42: Obtain the node sequence R obtained by different measurement methods, and use the Kendall correlation coefficient to represent the correlation between the sequence σ and the sequence R. The Kendall correlation coefficient is defined as Among them, n c is the number of identical pairs of elements in the two sequences, n d is the number of inconsistent pairs of elements in the two sequences, and n is the size of the two pairs of elements; S43: The ability of each centrality measure to distinguish node propagation efficiency is tested through the monotonicity function. Monotonicity (M) is defined as: Where n is the number of elements in the sequence R, N r is the number of elements in the rth level; S44: Using ranking similarity function Characterizes the correlation of the top L rankings of sequences σ and R, where σ(L) and R(L) represent the top L rankings of σ and R, respectively; S45: Result analysis.

3. The key node identification method based on network generalized energy and information entropy as claimed in claim 2 is characterized in that: In S42, a node sequence R obtained by six measurement methods is obtained.

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