Multi-layer complex network key node sorting identification method

Abstract

The invention discloses a multilayer complex network key node sorting identification method, and the specific process is as follows: generating an adjacent matrix # imgabs0 # of the whole network according to adjacent matrixes and inter-layer link relationships of different layers of a multilayer network, and calculating the degree value of each node in the multilayer network; the Bewenness value of each node in different layers in the multi-layer network is calculated respectively; carrying out normalization processing on the degree value and the Bewenness value of each node; fusing the degree value and the Beweenness value of each node to obtain an importance coefficient of each node; according to the obtained importance coefficient of each node, the nodes in the multi-layer network are arranged in a descending order; and identifying the key nodes by using a network disintegration algorithm. The method solves the problems that an existing key node identification method is large in error and low in efficiency for a multi-layer complex network. The method can be applied to control of basic networks such as electric power, communication, logistics and traffic, and has important application value in various fields such as infectious diseases and biomedical treatment.

Description

Technical Field

[0001] The present invention belongs to the field of network science and technology in complex systems, and specifically relates to a method for sorting and identifying key nodes in a multi-layer complex network. Background Art

[0002] Different nodes in the network have different degrees of influence on the topology and function of the network. The scale and structure of actual network systems are becoming increasingly complex, so it is very necessary to identify key nodes in the network: it can effectively improve the ability to monitor and maintain network operation, strengthen the stability of various networks in practical applications, and is critical to maintaining normal social order and protecting life and property safety. Therefore, the study of key nodes in the network has important theoretical significance and application value.

[0003] Traditional complex network analysis methods mostly focus on networks composed of a single type of connection between nodes, that is, single-layer networks. However, actual complex systems do not exist in isolation. With the development of information technology, the coupling connections between various systems are closer. Multi-layer networks can contain different types of elements and link relationships, which can more effectively represent the intricate interconnections and evolution mechanisms of complex systems in the real world, and help to achieve the prediction, analysis and control of real systems. The study of multi-layer networks has become an emerging field.

[0004] Compared with single-layer networks, multi-layer networks have more complex inter-layer connections, more complex interactions between nodes, and more fragile structures. In addition, with the rapid development of science and technology, the complexity and scale of networks have increased rapidly. Failure of a small number of nodes will cause large-scale failure of the entire network, and it becomes very difficult to find useful information from them. Although the analysis method for single-layer networks is relatively mature, it cannot be directly applied to multi-layer networks. It is still difficult to identify important nodes in multi-layer networks, and there has been no breakthrough in the research on the search method for key nodes.

[0005] The classification of methods currently proposed for identifying key nodes in multi-layer networks and the main problems they have are as follows: Based on the network structure: the layer centrality is proposed based on the probability of random walk jumping to neighbor nodes being affected by the layer where the node is located; the Multiplex PageRank method is proposed based on the interaction between nodes in different layers of the network, which is suitable for networks with fewer layers; the identification algorithm of key nodes in single-layer networks based on eigenvector centrality is extended to multi-layer networks, etc. Most of the above achievements are still committed to compounding and expanding the methods of measuring node importance in single-layer networks, ignoring the global properties of multi-layer networks. Therefore, this method has a large error in identifying key nodes in multi-layer networks.

[0006] Based on the propagation dynamics method: The SIR (Susceptible Infected Recovered Model) propagation model is extended on various types of dual networks, and a layer cross-propagation mode is adopted. On a two-layer network, the interaction of two pathogens propagated according to the SIR process (the two pathogens can have arbitrary inter-layer degree correlation), but the research process assumes that epidemics do not occur simultaneously. The SIS (Susceptible Infected Susceptible Model) model is used to dynamically study the malware propagation process on multiple networks. Most related studies are based on multi-layer network models with special assumptions, which have large errors in identifying key nodes of multi-layer systems in actual backgrounds.

[0007] Tensor-based methods: Use tensors to describe the topological structure of multi-layer interconnected networks. Based on the distribution of the same nodes in different layers, the random walk centrality, PageRank, eigenvector centrality, and betweenness centrality in single-layer networks are extended to multi-layer interconnected networks. Use fourth-order tensors to represent multi-layer networks, and propose a new method based on tensor decomposition to identify key nodes. However, the tensor method has a high computational complexity and is less efficient when dealing with large-scale problems. Summary of the invention

[0008] The purpose of the present invention is to provide a method for sorting and identifying key nodes in a multi-layer complex network, so as to solve the problem that the existing key node identification method has large errors and low efficiency for multi-layer complex networks.

[0009] The technical solution adopted by the present invention is a method for sorting and identifying key nodes in a multi-layer complex network, which is specifically implemented according to the following steps: Step 1: Generate the adjacency matrix of the entire network based on the adjacency matrix of different layers of the multi-layer network and the inter-layer link relationship , and calculate each node in the multilayer network The value of ; Step 2, ignoring the inter-layer links, calculate the Betwenness value of each node in each layer of the multi-layer network; Step 3: For each node’s degree and Betwenness value Perform normalization processing; Step 4: Fusion the degree value and Betweenness value of each node to obtain the importance coefficient of each node; Step 5, arranging the nodes in the multilayer network in descending order according to the importance coefficient of each node obtained in step 4; Step 6: Use the network collapse algorithm to identify key nodes. The present invention is also characterized in that: In step 1, each node in the multi-layer network The value of The expression is: (1) In formula (1), Represents the adjacency matrix of the overall network Middle Line Column elements.

[0010] The specific process of step 2 is: Step 2.1, for each node in the multi-layer network , select the source node and terminal node , use the depth-first search algorithm for unweighted graphs and call Dijkstra's algorithm for weighted graphs to calculate from arrive The number of shortest paths ; Step 2.2, for each node , calculate from the source node and terminal node The shortest path passes through the node The number of paths , calculate each node Betwenness value .

[0011] In step 2.2, each node Betwenness value The expression is: (2).

[0012] In step 3, the degree value of each node and Betwenness value The normalized expression is: (3) (4) In formula (3) and formula (4), Representation Node The degree value, Represents the maximum value among all node degrees. Representation Node Betweenness value, Indicates the maximum value among all nodes' Betweenness values.

[0013] In step 4, the expression of the importance coefficient of each node is: (5) In formula (5), represents the normalized degree value, Indicates the normalized Betweenness value.

[0014] The specific process of step 6 is: Step 6.1, according to the node arrangement order obtained in step 5, select Q (attack strength) nodes from the first node as the starting node, with the initial Q=1, as the attack target node; Step 6.2, calculate the critical index K of multi-layer network collapse: When K>2, the multi-layer network is not disintegrated. The attack target nodes and the nodes that lose the connection relationship with the network due to the removal of the attack target nodes are removed from the multi-layer network to obtain the remaining network. The ratio S(Q) of the size of the largest connected branch in the remaining network to the number of nodes when the multi-layer network is complete is calculated. S(Q) is used to measure the degree of damage to the multi-layer network. At this time, continue with step 6.3; When K≤2, the program ends and returns the Q value and its corresponding node set; at this time, the multi-layer network structure has disintegrated, S(Q) is 0, Q is the minimum attack strength, and the node corresponding to the minimum attack strength is the key node set of the multi-layer network; Step 6.3, according to the node arrangement order obtained in step 5, add one node at a time as the attack target node, Q = Q + 1, and repeat step 6.2.

[0015] In step 6.2, the expression of the critical index K of multi-layer network collapse is: (6) In formula (6), Representation Node The degree value, Represents the total number of nodes in the remaining network. The remaining network is the network after removing the attack target nodes from the multi-layer network and the nodes that lose the connection relationship with the multi-layer network due to the removal of the attack target nodes.

[0016] The beneficial effects of the present invention are: (1) In the method for ranking and identifying key nodes in a multi-layer complex network of the present invention, the node degree value considers more the neighbor attributes of the node, which is a local attribute of the network; the Betweenness value is from the perspective of the path, using the core degree of the "bridge" node in the network when traversing the network, which is a more global indicator, that is, using the complementary characteristics between different indicators; at the same time, considering the impact of the inter-layer link relationship on the entire network, when calculating the node degree value, the network is regarded as a whole, without distinguishing between inter-layer and intra-layer links, and when calculating the Betweenness value, the attribute differences of different layers of the network are distinguished and considered, so that the importance of the node can be more accurately measured to accurately identify the key nodes; (2) The method for sorting and identifying key nodes in a multi-layer complex network of the present invention is combined with a network disintegration method, dynamically considers the role of nodes in the network, and considers the mutual influence of combinations between nodes. The method of the present invention can find a smaller number of key node combinations. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 It is an overall implementation flow chart of the method for sorting and identifying key nodes in a multi-layer complex network of the present invention; Figure 2 is a schematic diagram of the structure of the interdependent network used in the experiment in the embodiment of the present invention; Figure 3 is the link relationship of the Internet network in the two-layer network used in the experiment in the embodiment of the present invention; Figure 4 is the link relationship of the bus line network in the two-layer network used in the experiment in the embodiment of the present invention; Figure 5 is the link correspondence between the two-layer networks used in the experiment in the embodiment of the present invention; Figure 6 4 is a convergence curve diagram of network collapse in an embodiment of the present invention. DETAILED DESCRIPTION

[0017] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0018] Example 1 The present invention provides a method for sorting and identifying key nodes in a multi-layer complex network. Figure 1 As shown, the specific implementation steps are as follows: Step 1: Generate the adjacency matrix of the entire network based on the adjacency matrix of different layers of the multi-layer network and the inter-layer link relationship , and calculate each node in the multilayer network The value of ; Step 2, ignoring the inter-layer links, calculate the Betwenness value of each node in each layer of the multi-layer network; Step 3: For each node’s degree and Betwenness value Perform normalization processing; Step 4: Fusion the degree value and Betweenness value of each node to obtain the importance coefficient of each node; Step 5, arranging the nodes in the multilayer network in descending order according to the importance coefficient of each node obtained in step 4; Step 6: Use the network collapse algorithm to identify key nodes.

[0019] Example 2 Based on Example 1, in step 1, each node in the multi-layer network The value of The expression is: (1) In formula (1), Represents the adjacency matrix of the overall network Middle Line Column elements.

[0020] Example 3 Based on Example 2, the specific process of step 2 is: Step 2.1, for each node in the multi-layer network , select the source node and terminal node , use the depth-first search algorithm for unweighted graphs and call Dijkstra's algorithm for weighted graphs to calculate from arrive The number of shortest paths ; Step 2.2, for each node , calculate from the source node and terminal node The shortest path passes through the node The number of paths , calculate each node Betwenness value , the expression is: (2).

[0021] Example 4 Based on Example 3, in step 3, the degree value of each node and Betwenness value The normalized expression is: (3) (4) In formula (3) and formula (4), Representation Node The degree value, Represents the maximum value among all node degrees. Representation Node Betweenness value, Indicates the maximum value among all nodes' Betweenness values.

[0022] Example 5 The method for sorting and identifying key nodes in a multi-layer complex network of the present invention is specifically implemented according to the following steps: Step 1: Generate the adjacency matrix of the whole network by combining the adjacency matrices of different layers and the inter-layer link relationships of the multi-layer network , and calculate each node in the multilayer network The value of ; The specific operations are as follows: Taking a two-layer network as an example, the node numbers of the first layer network are , the adjacency matrix is ​​expressed as ; The node numbers of the second layer network are The adjacency matrix is ​​represented as ; then the adjacency matrix of the overall network is The expression is:

[0023] in, For one , placed in the adjacency matrix of the overall network The upper left corner; For one , placed in the adjacency matrix of the overall network 's lower right corner; and is a matrix generated by the inter-layer link relationship; if the node numbered 1 in the first layer has a link relationship with the node numbered m+1 in the second layer, then The first row and first column of (the first row and m+1 column of the adjacency matrix A of the entire network) is 1, otherwise it is 0, and so on to get the matrix , For one Similarly, we get the matrix , which is a Matrix of Consider the multi-layer network as a whole and calculate each node in the multi-layer network The value of ; Degree value It refers to the node Number of directly connected nodes: (1) In formula (1), Represents the adjacency matrix of the overall network Middle Line Column elements; Step 2, ignoring the inter-layer links, calculate the Betwenness value of each node in each layer of the multi-layer network; The specific process is: Step 2.1, for each node in the multi-layer network , select the source node and terminal node , use the depth-first search (BFS) algorithm for unweighted graphs and call Dijkstra's algorithm for weighted graphs to calculate arrive The number of shortest paths ; Step 2.2, for each node , calculate from the source node and terminal node The shortest path passes through the node The number of paths , calculate each node Betwenness value ; The expression is: (2) Step 3: For each node’s degree and Betwenness value Perform normalization processing; The expression is: (3) (4) In formula (3) and formula (4), Representation Node The degree value, Represents the maximum value among all node degrees. Representation Node Betweenness value, Indicates the maximum value among the Betweenness values ​​of all nodes; Step 4: Fusion the degree value and Betweenness value of each node to obtain the importance coefficient of each node; The expression is: (5) In formula (5), represents the normalized degree value, Represents the normalized Betweenness value; Step 5, arranging the nodes in the multilayer network in descending order according to the importance coefficient of each node obtained in step 4; Step 6, using the network disintegration algorithm to identify key nodes; The specific process is: Step 6.1, according to the node arrangement order obtained in step 5, select Q (attack strength) nodes from the first node as the starting node as the attack target node, and initially assume that Q=1, that is, take the node with the highest importance as the target attack node; Step 6.2, calculate the critical index K of multi-layer network collapse; The expression is: (6) in Representation Node The degree value, Represents the total number of nodes in the remaining network (the attack target nodes are removed from the network and the nodes that lose their connection with the network due to the removal of the attack target nodes).

[0024] When K>2, continue with step 6.3; at this time, the multi-layer network is considered to have not disintegrated, and the attack target nodes and the nodes that have lost their connection with the network due to the removal of the attack target nodes are removed from the multi-layer network to obtain the remaining network, and the ratio S(Q) of the size of the largest connected branch in the remaining network to the number of nodes when the multi-layer network is complete is calculated. S(Q) is used to measure the degree of damage to the multi-layer network, especially when the multi-layer network has not completely disintegrated. The larger the S(Q) value, the smaller the degree of damage to the multi-layer network. Conversely, it is considered that the multi-layer network has suffered relatively large damage; When K≤2, the program ends and returns the Q value and the corresponding node number. At this time, the multi-layer network only contains fragmented small fragments. At this time, it is considered that the multi-layer network structure has disintegrated. At this time, the default S(Q) is 0. At this time, Q is the minimum attack strength, that is, the minimum number of nodes for the disintegration of the multi-layer network. The nodes corresponding to the minimum attack strength are the key node set of the multi-layer network. Step 6.3: According to the node arrangement order obtained in step 5, add one node at a time as the attack target node, Q=Q+1, and repeat step 6.2 until the minimum number of nodes required to collapse the multi-layer network and its corresponding node set are found, which is the key node set of the multi-layer network; For example: Q=2, that is, the first two nodes with the highest importance are taken as the target attack nodes; Q=3, that is, the first three nodes with the highest importance are taken as the target attack nodes; and so on.

[0025] Example 6 The effect of the method of the present invention can be specifically illustrated by simulation experiments: 1. Experimental conditions The computer used in the experiment is equipped with a CPU i7-8700 processor, 8 GB RAM, Windows 10 system, and Matlab2017b. The experiment is aimed at a two-layer dependency network. Figure 2 As shown in the figure, the so-called two-layer dependency network consists of two interdependent sub-networks with the same size. The nodes are interdependent one-to-one. There is a cascade relationship in the interdependent network. When the nodes in a certain layer change, the impact may spread to many network layers. A node can only be considered valid when it is the largest connected branch of the network to which it belongs and the nodes it depends on also maintain normal functions.

[0026] Experimental data: Figure 3 and Figure 4 As shown, one layer is the Internet network, and the other layer is the bus line network, both of which are undirected and unweighted networks.

[0027] The general network information is shown in Table 1: Table 1

[0028] Where P is the number of nodes in the network, L1 is the number of edges in the Internet network, L2 is the number of edges in the bus line network, and T is the number of network layers. Two-layer network: The link relationship between the Internet network and the bus line network is as follows: Figure 5 As shown, for the sake of clarity, the link relationship is represented by a planar diagram.

[0029] 2. Experimental content First, calculate the degree value of each node in the two-layer network, and then calculate the Betweenness value of each node in the two-layer network respectively. Normalize all degree values ​​and Betweenness values, and then calculate the importance coefficient of each node. Then sort all nodes in descending order according to the importance coefficient. According to the order of arrangement, increase the number of attack nodes one by one, and calculate the ratio S(Q) of the number of nodes in the largest connected branch in the remaining network to the number of nodes when the multi-layer network is complete under different attack intensities; the calculation of the largest connected branch can be completed by depth-first search or breadth-first search. By judging Whether the value is less than or equal to 2, the minimum number of nodes and the corresponding node numbers that can completely disintegrate the two-layer network are obtained.

[0030] Comparative Example 1: Another node importance ranking index was selected: the inter-layer connection was regarded as equivalent to the ordinary intra-layer connection, and the degree value of the node was calculated for the entire interdependent network by formula (1). Then, the K-shell value of the node was calculated for different layers of the network, and the degree value and K-shell value of each node were normalized. The importance index of each node was calculated, and the nodes were sorted in descending order according to the importance index. Then, the minimum number of nodes and the corresponding node numbers for the complete disintegration of the two-layer network were obtained according to the network disintegration method.

[0031] The specific process is as follows: (1) The links between the two layers of the network are considered to be the same as the links within the layers, and the degree value of each node in the entire network is calculated using formula (1); (2) Calculate the K-Shell value of each node for different layers of the network. The specific calculation steps are as follows: S1: Remove the nodes with a current degree value of 1 from the network, and update the degree values ​​of the remaining nodes in the network until there are no more nodes with a degree value of 1 in the network. The removed network nodes are classified as 1-Shell layers and assigned a K-Shell value of 1. S2: Next, remove the nodes with a current degree value of 2 and update the degree values ​​of the remaining nodes in the network until there are no more nodes with a degree of 2 in the network. The removed nodes are classified as 2-Shell layers and assigned a K-Shell value of 2. S3: And so on, until all nodes in the network are assigned K-Shell values. The node layer with the largest K-Shell value is the core layer of the network, indicating that the node is more important.

[0032] (3) Normalize the degree and K-Shell value of each node respectively; (4) The node importance coefficient of the multi-layer network is obtained by combining the two indicators (the normalized degree value plus the normalized K-Shell value).

[0033] (5) Arrange the nodes in descending order according to their importance coefficients.

[0034] (6) Using the network disruption strategy to find the key nodes of the network, this process is the same as step 6 of the present invention.

[0035] Comparative Example 2: In order to illustrate that steps (1) and (2) of the present invention are necessary and indispensable, in Example 2, we first regard the inter-layer connection as equivalent to the ordinary intra-layer connection, calculate the K-Shell value of the node for the entire interdependent network, and then calculate the degree value of the node for the two-layer network respectively. The K-Shell value and degree value obtained for each node are normalized, and the importance index of each node is calculated. The nodes are arranged in descending order according to the importance index. The minimum number of nodes and the corresponding node numbers for the complete disintegration of the two-layer network are obtained according to the network disintegration method.

[0036] The specific process is as follows: (1) The links between the two layers of the network are considered to be the same as the links within the layers, and the K-Shell value of each node in the entire network is calculated. The specific steps are similar to step (2) in comparative example 1.

[0037] (2) Calculate the degree of each node in different layers of the network.

[0038] (3) Normalize the K-Shell value and degree value of each node respectively; (4) Combining the two indicators (the sum of the normalized K-Shell value and the normalized degree value), the node importance coefficient of the multi-layer network is obtained.

[0039] (5) Arrange the nodes in descending order according to their importance coefficients.

[0040] (6) Using the network disruption strategy to find the key nodes of the network, this process is the same as step 6 of the present invention.

[0041] The minimum set of nodes that causes the network to collapse is the key node set in the multi-layer network. Different node importance indicators directly affect the number of minimum key node sets found in the end. The smaller the number, the more important the single node found, that is, this sorting and identification method is more reasonable.

[0042] 3. Experimental results Using the node importance calculation method of the multi-layer network, each node has a certain importance coefficient. The nodes are arranged in descending order according to their importance. The attack nodes are added in order from large to small importance to find the minimum set of nodes required to collapse the network. This set is the key node set of the network.

[0043] The horizontal axis is the attack strength Q, and the vertical axis is the S(Q) value. The convergence diagram between the attack strength and the attack effect is as follows: Figure 6 To test the effectiveness of the method, the S(Q) value of the network after the attack is observed. It can be seen that the convergence curve of the method of the present invention converges faster. When the S(Q) value returns to zero, it means that the multi-layer network is completely collapsed under the current attack intensity. The broken line reflects the convergence speed of each method.

[0044] The experimental data are shown in Table 2: Table 2

[0045] The network is attacked in sequence according to the importance index of the nodes. In comparative example 1, when the number of attacked nodes reaches 7, the network completely collapses, and the key nodes found are numbered as 1, 8, 2, 7, 3, 4, and 5. In comparative example 2, when the number of attacked nodes reaches 6, the network completely collapses, and the key nodes found are numbered as 1, 3, 8, 23, 2, and 48. Using the method of the present invention, when 3 nodes are attacked, the network completely collapses, and the key nodes found are node 1, node 48, and node 49.

[0046] It can be seen from the experimental data that the degree centrality of the nodes is first calculated for the entire interdependent network, and then the Betweenness centrality of the nodes is calculated for the two-layer network respectively, and the importance coefficient of the nodes is ranked. The network collapse method is used to identify the key node set. In this case, fewer nodes need to be attacked for network collapse. The minimum node set that causes the network to collapse is the key node set in the multi-layer network. The fewer the number of key nodes, the more important the single node found is, and the more critical the nodes found by this method are to the entire network structure and performance, that is, the method of the present invention is more accurate in measuring the importance of nodes.

[0047] The identification of key nodes in multi-layer networks is conducive to better monitoring and maintenance of network operations. It is crucial to the control of basic networks related to people's livelihood, such as national power networks, communication networks, and transportation networks. It also has important application value in infectious diseases, biomedicine and other fields.

Claims

1. A method for sorting and identifying key nodes in a multi-layer complex network, characterized in that: Follow the steps below to implement it: Step 1: Generate the adjacency matrix of the entire network based on the adjacency matrix of different layers of the multi-layer network and the inter-layer link relationship , and calculate each node in the multilayer network The value of ; Step 2, ignore the inter-layer links and calculate the Betwenness value of each node in different layers of the network respectively; Step 3: For each node’s degree and Betwenness value Perform normalization processing; Step 4: Fusion the degree value and Betweenness value of each node to obtain the importance coefficient of each node; Step 5, arranging the nodes in the multilayer network in descending order according to the importance coefficient of each node obtained in step 4; Step 6: Use the network collapse algorithm to identify key nodes.

2. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 1, characterized in that: In step 1, each node in the multi-layer network The value of The expression is: （1） In formula (1), Represents the adjacency matrix of the overall network Middle Line Column elements.

3. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 2 is characterized in that: The specific process of step 2 is: Step 2.1, for each node in the multi-layer network , select the source node and terminal node , use the depth-first search algorithm for unweighted graphs and call Dijkstra's algorithm for weighted graphs to calculate arrive The number of shortest paths ; Step 2.2, for each node , calculate from the source node and terminal node The shortest path passes through the node The number of paths , calculate each node Betwenness value .

4. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 3 is characterized in that: In step 2.2, each node Betwenness value The expression is: （2）。 5. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 4, characterized in that: In step 3, the degree value of each node and Betwenness value The normalized expression is: （3） （4） In formula (3) and formula (4), Representation Node The degree value, Represents the maximum value among all node degrees. Representation Node Betweenness value, Indicates the maximum value among all nodes' Betweenness values.

6. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 5, characterized in that: In step 4, the expression of the importance coefficient of each node is: （5） In formula (5), represents the normalized degree value, Indicates the normalized Betweenness value.

7. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 6, characterized in that: The specific process of step 6 is: Step 6.1, according to the node arrangement order obtained in step 5, select Q nodes from the first node as the starting node, with the initial Q=1, as the attack target node; Step 6.2, calculate the critical index K of multi-layer network collapse; When K>2, the multilayer network is not disintegrated. The attack target nodes and the nodes that lose the connection relationship with the network due to the removal of the attack target nodes are removed from the multilayer network to obtain the remaining network. The ratio S(Q) of the size of the largest connected branch in the remaining network to the number of nodes when the multilayer network is complete is calculated. S(Q) is used to measure the degree of damage to the multilayer network. At this time, continue with step 6.3; When K≤2, the program ends and returns the Q value and its corresponding node set; at this time, the multi-layer network structure has disintegrated, S(Q) is 0, Q is the minimum attack strength, and the node corresponding to the minimum attack strength is the key node set of the multi-layer network; Step 6.3, according to the node arrangement order obtained in step 5, add one node at a time as the attack target node, and repeat step 6.

2.

8. The method for sorting and identifying key nodes in a multi-layer complex network according to claim 7, characterized in that: In step 6.2, the expression of the critical index K of multi-layer network collapse is: （6） In formula (6), Representation Node The degree value, Represents the total number of nodes in the remaining network. The remaining network is the network after removing the attack target nodes from the multi-layer network and the nodes that lose the connection relationship with the multi-layer network due to the removal of the attack target nodes.

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