Layering-based weighted network key node identification method

Abstract

The invention provides a weighting network key node identification method based on layering. The method comprises the following steps: step 1, establishing a multi-layer network initial model based on a layering network thought; 2, selecting two layers of networks with the maximum similar modularity gain after combination to perform iterative combination, and obtaining a multi-layer network model by taking the similar modularity gains of any two layers of networks are non-positive as a loop termination condition; step 3, considering betweenness indexes of each network node in the multi-layer network model, measuring importance of each network node in the multi-layer network model, and dividing the network nodes into interactive nodes, first-level non-interactive nodes and second-level non-interactive nodes; and 4, evaluating the importance of the interactive nodes, the primary non-interactive nodes and the secondary non-interactive nodes by adopting three betweenness indexes. According to the method, multi-layer network modeling is carried out on the premise that the global attributes of the key nodes of the complex network have limitation, and the recognition accuracy of the key nodes in the network nodes of the multi-layer network is ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of complex network recognition, and particularly relates to a method for identifying key nodes of a hierarchical weighted network. Background Art

[0002] In a network model, according to the different importance of nodes in the network, the nodes that play a crucial role in the network model are key nodes. Identifying and understanding key nodes is of great significance for the application of the network model. However, when using the network model to solve the key part identification, different network scales have limitations in the process of key node identification. On the one hand, the larger the network scale, the more factors the key identification algorithm needs to consider. On the other hand, in the existing key node identification algorithms, the node complexity does not increase in proportion to the network scale, generally increasing by a square multiple. Therefore, for a large-scale network model, it is crucial to give a key node identification algorithm with relatively low complexity but not low accuracy.

[0003] For a large-scale network model, the key nodes may not be particularly clear. Therefore, instead of giving an accurate key node identification algorithm, an idea that can solve the large-scale network problem is given.

[0004] Using hierarchical modeling for large-scale networks is a data preprocessing method. Based on the characteristics of specific problems as prior experience for layering, it can refine the solution to the problem. In the process of hierarchical modeling of large-scale networks, the most commonly used method is the community identification method, which divides the large-scale network into multiple layer networks through communities. Therefore, it is necessary to propose a similar community structure, and on this basis, adopt a network layering algorithm for modeling and then identify key nodes. Summary of the Invention

[0005] The purpose of the present invention is to solve the deficiency that the complexity of identifying key nodes in large-scale networks in the prior art is too high and difficult to achieve, and provide a method for identifying key nodes of a hierarchical weighted network, which performs multi-layer network modeling on the premise that the global attributes of key nodes in complex networks have limitations, and ensures the accuracy of identifying key nodes in each network node of the multi-layer network.

[0006] To achieve the above purpose, the technical solution provided by the present invention is:

[0007] A method for identifying key nodes of a hierarchical weighted network, comprising:

[0008] Step 1: Establish an initial model of a multi-layer network based on the idea of a hierarchical network;

[0009] Step 2: Select the two layer networks with the largest modularity gain after merging for iterative merging, and use the condition that the modularity gain of any two layer networks is non-positive as the loop termination condition to obtain a multi-layer network model;

[0010] Step 3: Consider the betweenness index of each network node in the multi-layer network model, measure the importance of each network node in the multi-layer network model, and divide the network nodes into interaction nodes, first-level non-interaction nodes, and second-level non-interaction nodes;

[0011] Step 4: Use three centrality indicators to evaluate the importance of the interaction nodes, the first-level non-interaction nodes, and the second-level non-interaction nodes; among them, the three centrality indicators include OD-betweenness, interaction betweenness, and block betweenness.

[0012] As a further limitation of the present invention, the Step 1 includes:

[0013] Step (11) Establish an initial multi-layer network model based on the hierarchical network idea. The initial multi-layer network model is composed of a triple (G, C, W), and the expression is:

[0014]

[0015] In formula (1), G represents a network set, G α represents the α-layer sub-network, α represents the α layer, L represents the number of network layers, X α represents the node set of the α-layer sub-network G α E represents the edge set of the α-layer sub-network G α represents the α-layer sub-network G α C represents the set of edges between the α-layer sub-network G α and the β-layer sub-network G β E αβ represents the α-layer sub-network G α and the β-layer sub-network G β The connecting edge, β represents the β layer, W represents the weight set, A represents the in-layer edge weight set, B represents the inter-layer edge weight set, represents the weight of the edge ;

[0016] Step (12) Treat each node in the initial multi-layer network model as a layer network, and the expression is:

[0017] M t ={V t 1 ,V t 2 ,...,V t N}=∪ α V tα Formula (2)

[0018] In formula (2), M t represents the division of the similar community structure at time t, represents the α-th layer sub-network at time t, V t α represents the node set of the α-th layer sub-network at time t, that is, the α-th similar community structure at time t, and the initial time t = 0.

[0019] As a further limitation of the present invention, the second step includes:

[0020] Step (21) Based on the initial model of the layer network, for each layer network attempts to merge with the adjacent layer network where: i ≠ j, i in the formula represents the i-th layer network, and j represents the j-th layer network;

[0021] If the layer network and the layer network can be merged into a new layer network, record the gain ΔSQ of the non-interacting node ratio in the new layer network. The gain ΔSQ of the non-interacting node ratio is the modularity gain, and the expression is:

[0022]

[0023] In formula (3), SQ represents the similar modularity, M t represents the division of the similar community structure at time t, represents the similar community structure after merging the layer network and , V t i represents the node set of the layer network , V t j represents the node set of the layer network ;

[0024] Step (22) Determine the multi-layer network model based on the layer network corresponding to the maximum similar modularity gain; specifically, if the modularity gain of any two layer networks is non-positive, then obtain the multi-layer network model and stop merging; otherwise, perform the next merge until the layer network L i and the layer network L j are found. Through the operation M k = M k-1 \ L max(i,j) , and L min(i,j) = L i ∪ L j to obtain the current similar community structure, and return to step (21).

[0025] As a further limitation of the present invention, step three includes:

[0026] Step (31) considers the importance of each network node in the multi-layer network model from the perspective of betweenness, and defines the centrality index ODBC; specifically, it calculates the shortest path length of the multi-layer network M and counts The shortest path length between nodes obtained by calculating through the LCDM method and node where, represents the number of paths passing through nodes and node in the shortest path between any two nodes and node of the multi-layer network M;

[0027] Step (32) calculates the OD-edge betweenness of each network node in the multi-layer network model. The expression is:

[0028]

[0029] In formula (4), represents the importance of the OD-edge betweenness in measuring the edge in all shortest paths, represents the number of shortest paths from node to node in the interaction network CG, represents the number of nodes passing through edge to node in CG;

[0030] Calculate the OD-node betweenness of all nodes in the multi-layer network model. The expression is:

[0031]

[0032] In formula (5), represents the importance of the OD-node betweenness in summarizing the associated edge in all shortest paths, represents the set of adjacent nodes of node in CG;

[0033] Step (33) calculates the block BC of the multi-layer network model; specifically, the node block betweenness BC expression is:

[0034] ​

[0035] In formula (6), represents the number of shortest paths between node and node represents the number of shortest paths that node passes through node and node ;

[0036] Step (34) calculates the block - betweenness Δ k of the first - level non - interacting nodes of the multi - layer network model, and the expression is:

[0037]

[0038] In formula (7), Δ k represents the importance of the multi - layer network and the non - interacting node in the shortest path, represents the equivalent triple represents the weight of the equivalent triple ; where:

[0039] The judgment condition for the equivalent triple is:

[0040] If and the weight of the edge α in the judgment network DG satisfies then is an equivalent triple;

[0041] In the judgment network DG α , G α is the layer network of the multi - layer network M, and the expression is G α =(X α , E α ), in the formula, X α represents the disjoint interacting nodes, and E α represents the edges in the network, α = 1, 2,..., L;

[0042] In the judgment network DG α , DG α is the judgment network of the layer network G α , and the expression is In the formula, X α represents the node set of each layer, represents the edge set in the judgment network, W represents the weight of the edge set in the judgment network, represents the interacting node set of the layer network G α ; Presentation layer network G α The non-interactive node set of

[0043] As a further limitation of the present invention, the fourth step includes:

[0044] Step (41) In the multi-layer network model of step three, the interactive nodes are sorted based on the size of the OD-betweenness centrality; specifically: if the network node of the multi-layer network model is an interactive node, the interactive nodes are directly sorted according to the size of the OD-betweenness centrality of the node;

[0045] Step (42) In the multi-layer network model of step three, the non-interactive nodes are divided into first-level non-interactive nodes and second-level non-interactive nodes, and then sorted based on the interaction betweenness Δ k and the block betweenness BC; specifically:

[0046] If the network node of the multi-layer network model is a non-interactive node, a decision network DG α is constructed according to the layer network G α , and if there are two nodes α in the network nodes of the decision network DG and and such that is an equivalent triple, then the non-interactive node is a first-level non-interactive node and is sorted according to the importance Δ k of the equivalent triple of the node; where represents the interactive node set of the layer network G α ;

[0047] Otherwise, the non-interactive node is a second-level non-interactive node and is sorted according to the size of the block betweenness BC.

[0048] The advantages of the present invention are:

[0049] The present invention performs multi-layer network modeling on the premise that the global attributes of the key nodes in the complex network have limitations, and ensures the accuracy of identifying key nodes among the network nodes of the multi-layer network.

[0050] The additional aspects and advantages of the present invention will be partially given in the following description, partially become obvious from the following description, or be understood through the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] The above and / or additional aspects and advantages of the present invention will become obvious and easy to understand from the description of the embodiments in conjunction with the following drawings, where:

[0052] Figure 1:Flow chart of a method for identifying key nodes in a hierarchical weighted network provided by the present invention;

[0053] Figure 2 :Diagrams of several similar community structure models of complex networks provided by the present invention;

[0054] Figure 3 :Schematic diagram of the double mapping relationship between a single-layer network and a multi-layer network provided by the present invention;

[0055] Figure 4 :Schematic diagram of the multi-layer network M and its interaction network provided by the present invention;

[0056] Figure 5 :Schematic diagram of the classification of six-layer network nodes of the multi-layer network M provided by the present invention;

[0057] Figure 6 :Schematic diagram of the sorting of betweenness centrality of three types of nodes (OD-betweenness centrality, interaction betweenness centrality, betweenness centrality) of the multi-layer network M provided by the present invention;

[0058] Figure 7 :Schematic diagram of the change in the sorting of nodes when adjusting the number of inter-layer edges provided by the present invention. Detailed implementation manners

[0059] The following details the embodiments of the present invention. The embodiments are exemplary and are intended to explain the present invention and should not be construed as limiting the present invention.

[0060] Please refer to Figure 1 , the embodiments of the present invention provide a method for identifying key nodes in a hierarchical weighted network, which can perform multi-layer modeling on the premise that the global attributes of key nodes in a complex network are limited, and ensure the accuracy of key node identification, and solve the problem that it is difficult to implement due to the excessive complexity of identifying key points in a large-scale complex network.

[0061] In order to verify the effectiveness of the method for identifying key nodes in a hierarchical weighted network proposed in the embodiments of the present invention, please refer to Figure 2 and Figure 4 , the embodiments of the present invention take a six-layer network (ER network, WS network) as the research object, conduct application research from a macroscopic perspective, and verify the accuracy of classifying network node identification into three categories. The multi-layer network M in the embodiments of the present invention is preferably a six-layer network M. If each layer network in the six-layer network M is an ER (WS) network, then the six-layer network M is denoted as M ER (M WS ).

[0062] Specifically, M in the embodiments of the present invention ER is a six-layer network, G 1 , G 2,...,G 6 is a random layer (ER) network. Please refer to Figure 5 , the edge connection probability of the random network in the embodiment of the present invention is p, and the number of interaction nodes in each layer network G α is The number of inter-layer edges is

[0063] The method for identifying key nodes in the weighted network according to the embodiment of the present invention includes the following steps 1 to 4:

[0064] Step 1: Establish an initial model of the layer network based on the idea of a hierarchical network.

[0065] Step 1 of the embodiment of the present invention includes:

[0066] Step (11) Initialize the layer network: Based on the idea of a hierarchical network, establish an initial model of a multi-layer network. The initial model of the multi-layer network is composed of a triple (G, C, W), and the expression is:

[0067]

[0068] In formula (1), G represents a network set, G α represents the α-th layer sub-network, α represents the α-th layer, L represents the number of network layers, X α represents the node set of the α-th layer sub-network G α E α represents the edge set of the α-th layer sub-network G α C represents the set of edges between the α-th layer sub-network G α and the β-th layer sub-network G β E αβ represents the connection edge between the α-th layer sub-network G α and the β-th layer sub-network G β , β represents the β-th layer, W represents the weight set, A represents the in-layer edge weight set, B represents the inter-layer edge weight set, represents the edge weight.

[0069] Step (12) Regard each node in the initial model of the multi-layer network as a layer network, and the expression is:

[0070] M t ={V t 1 ,V t 2 ,...,V t N}=∪ α V t α Formula (2)

[0071] In formula (2), M t represents the division of the similar community structure at time t, represents the sub-network of the α-th layer at time t, and V t α represents the node set of the sub-network of the α-th layer at time t, that is, the α-th similar community structure at time t. The initial time t = 0.

[0072] The node set X of each layer in the embodiment of the present invention α consists of two non-intersecting sets wherein is the interactive node set of layer G α and is the non-interactive node set. W = A ∪ B, where A is the set of edge weights within the layer and B is the set of edge weights between layers

[0073] Step 2: Select the two layer networks with the largest gain in the similarity modularity after merging for iterative merging, and use the condition that the gain in the similarity modularity between any two layer networks is non-positive as the loop termination condition to obtain a multi-layer network model.

[0074] Step 2 of the embodiment of the present invention includes:

[0075] Step (21) Based on the initial model of the layer network, for each layer network attempt to merge with the adjacent layer network where: i ≠ j, that is, different layer networks are attempted to be merged;

[0076] If the layer network and the layer network can be merged into a new layer network, record the gain ΔSQ of the non-interactive node ratio in the new layer network. The gain ΔSQ of the non-crossing node ratio is the gain in the modularity, and the expression is:

[0077]

[0078] In formula (3), SQ represents the similarity modularity, and M t represents the division of the similar community structure at time t, represents the similar community structure after merging the layer networks and , and V t i represents the node set of the layer network , and V t j represents the node set of the layer network .

[0079] In formula (3), represents the merged layer network and the subsequent similar community structure, L u represents the similar community to which node x u belongs. The similarity modularity SQ is used to measure the quality of the division of the similar community structure. Specifically, the similarity modularity SQ formula is:

[0080]

[0081] where SQ(M) represents the similarity modularity of the similar community structure, k u represents the degree of node x u , L v represents the similar community to which node x v belongs, and L u represents the similar community to which node x v belongs.

[0082] In step (22), merge the layer network with the largest gain in similarity modularity in the initial model of the multi-layer network, and then determine the multi-layer network model. Specifically, if the gain in modularity of any two layer networks is non-positive, the multi-layer network model is obtained and the merging stops; otherwise, the next merging is performed until the layer network corresponding to the maximum similarity modularity is found and the layer network By operating M k = M k-1 \G max(i,j) , and G min(i,j) = G i ∪G j the current similar community structure is obtained, and return to step (21).

[0083] Please continue to refer to Figure 2 , Figure 2 In, network (b) and network (c) are two similar community structures of network (a). The similarity modularity result of network (b) is SQ≈0.6667, and the modularity Q≈0.6561. The similarity modularity result of network (c) is SQ≈0.8333, and the modularity Q≈0.1768. For the division of network (b), there are fewer edges between different colors. Therefore Figure 2 (b) has a higher modularity, that is, Q (b) > Q (a) . For Figure 2 (c) division, the edges between different colors pass through fewer nodes. Therefore Figure 2 (c) has a higher similarity modularity, that is, SQ (b) > SQ (a) .

[0084] Step 3: Consider the betweenness index of each network node in the multi-layer network model, measure the importance of each network node in the multi-layer network model, and divide the network nodes into interactive nodes, primary non-interactive nodes, and secondary non-interactive nodes.

[0085] Step 3 of the embodiment of the present invention includes:

[0086] Step (31): Consider the importance of each network node in the multi-layer network model from the perspective of betweenness, and define the OD-node betweenness index ODBC; specifically, calculate the shortest path length of the multi-layer network M and count Calculate the shortest path length between nodes obtained by the LCDM method and node wherein, represents the number of paths passing through node and node in the shortest path between any two nodes and node of the multi-layer network M. Specifically, the Dijkstra algorithm is used to calculate these distance matrices to obtain the shortest path length between nodes and node .

[0087] Step (32): Calculate the OD-edge betweenness of each network node in the multi-layer network model. The expression is:

[0088]

[0089] In formula (4), represents that the OD-edge betweenness measures the importance of edge in all shortest paths, denoted as represents the number of shortest paths from node to node in the interactive network CG, represents the number of nodes passing through edge from node to node

[0090] Calculate the OD-node betweenness of all nodes in the multi-layer network model. The expression is:

[0091]

[0092] In formula (5), the OD-node betweenness measures the importance of node in the interactive nodes in all shortest paths, denoted as Represents the set of adjacent nodes of the node in the CG .

[0093] Step (33) calculates the block BC of the multi-layer network model; specifically, the node block betweenness BC expression is:

[0094]

[0095] In formula (6), represents the number of shortest paths between node and node . represents the number of shortest paths that node passes through node and node .

[0096] Step (34) calculates the block betweenness (equivalent triple importance) Δ k of the first-level non-interactive nodes of the multi-layer network model, and the expression is:

[0097]

[0098] In formula (7), Δ k represents the importance degree in the shortest path from node to node in the interactive network, represents the equivalent triple represents the equivalent triple ; where:

[0099] The judgment condition for the equivalent triple is:

[0100] If and the weight of the edge α in the judgment network DG satisfies then is an equivalent triple;

[0101] In the judgment network DG α , G α is the layer network of the multi-layer network M, and the expression is G α =(X α , E α ), in the formula, X α represents non-overlapping interactive nodes, and E α represents the edges in the network, α = 1, 2,..., L;

[0102] In the judgment network DG α , DG α is the layer network Gα The decision-making network, with the expression In the formula, X α represents the node set of each layer, represents the edge set in the decision-making network, W represents the weight of the edge set in the decision-making network, represents the interaction node set of the layer network G α ; represents the non-interaction node set of the layer network G α .

[0103] Step 4: Use three centrality metrics to evaluate interaction nodes, primary non-interaction nodes, and secondary non-interaction nodes.

[0104] Step 4 of the embodiment of the present invention includes:

[0105] Step (41) In the multi-layer network model of the embodiment of the present invention, the interaction nodes are sorted based on the size of ODBC (OD-betweenness centrality); specifically: for the network nodes in the multi-layer network model, if they are interaction nodes, the interaction nodes are directly sorted according to the size of the ODBC of the nodes;

[0106] Step (42) In the multi-layer network model of the embodiment of the present invention, the non-interaction nodes are divided into primary non-interaction nodes and secondary non-interaction nodes, and then sorted based on the interaction betweenness Δ k and the block betweenness BC; specifically:

[0107] For the network nodes in the multi-layer network model, if they are non-interaction nodes, a decision-making network DG α is constructed according to the layer network G α . If there are two nodes α in the network nodes of the decision-making network DG such that is an equivalent triple, then the non-interaction node is a primary non-interaction node and is sorted according to the importance Δ k of the node's equivalent triple; where represents the interaction node set of the layer network G α ; otherwise, the non-interaction node is a secondary non-interaction node and is sorted according to the size of the centrality metric BC.

[0108] In the embodiment of the present invention, for the six-layer network M ER , the connection probability of the random layer network G 1 , G 2 ,..., G 6 is P = 0.2. Figure 6 In the network shown in k , the nodes are sorted according to OD-betweenness centrality, interaction betweenness Δ Figure 7The increased number of inter-layer edges shown and the change of node sorting. Figure 7 In each sub-graph in, the horizontal axis represents the network M ER The sorting Rank M obtained by the above sorting method ER , Figure 7 In each sub-graph in, the vertical axis represents in the six-layer network M ER The mapping network proj(M ER ). Among them, according to the sorting result of the classical betweenness of nodes, the red nodes represent the sorting result of interactive nodes, the blue nodes represent the sorting result of first-level non-interactive nodes, and the orange nodes represent the sorting result of second-level non-interactive nodes.

[0109] The Kendall correlation coefficient τ is used to measure the correlation between the classical betweenness sorting of the mapping network and the similar betweenness sorting of the multi-layer network. The expression is as follows:

[0110]

[0111] In the formula, C represents the number of consistent pairs in Rank1 and Rank2, and D represents the number of divergent pairs in Rank1 and Rank2. If the relative relationship of the sorting of the two variables is consistent, it is a consistent pair, otherwise it is a divergent pair.

[0112] From Figure 7 it can be seen that in the sorting of the three types of nodes, namely interactive nodes, first-level non-interactive nodes and second-level non-interactive nodes in the embodiments of the present invention, when the number of inter-layer edges is increased, the sorting of interactive nodes is close to the sorting result of the classical betweenness, and the Kendall correlation coefficient is close to 1. It shows the correctness and universality of classifying nodes into three categories from a macroscopic perspective, and the interactive node class can be directly used as the key class in the network.

[0113] The embodiments of the present invention provide theoretical and technical guidance for real networks such as traffic networks, and effectively solve the problems that the key point recognition method of large-scale complex networks is complex or the accuracy rate cannot meet the requirements and it is difficult to realize practical applications. The embodiments of the present invention first establish an initial model of the layer network, consider the modularity gain of merging each layer network, and obtain a multi-layer network model with the constraint that the gain of any two layer networks is non-positive; secondly, consider the betweenness index of network nodes, measure the importance of network nodes in the multi-layer network, divide the network nodes into interactive nodes, first-level non-interactive nodes and second-level non-interactive nodes, and propose three centrality indexes, namely OD-point betweenness, interactive betweenness and block betweenness, to evaluate the three types of nodes respectively; finally, conduct a case analysis with a six-layer network to verify the accuracy and universality of the network node recognition of this method from a macroscopic perspective, and prove that the embodiments of the present invention can perform multi-layer network modeling on the premise that the global attributes of key nodes in complex networks have limitations, and ensure the accuracy of key node recognition.

[0114] As described above, it is only the specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention can easily think of various equivalent modifications or substitutions, and these modifications or substitutions should all be covered within the protection scope of the present invention.

Claims

1. A method for identifying key nodes in a weighted network based on layering, characterized in that: include: Step 1: Establish a multi-layer network initial model based on the layered network concept; Step 2: Select two layers of networks with the largest similar modularity gain after merging, and perform iterative merging. The similar modularity gain of any two layers of networks is non-positive as the loop termination condition to obtain a multi-layer network model. Step 3: Considering the betweenness index of each network node in the multi-layer network model, measuring the importance of each network node in the multi-layer network model, and dividing the network nodes into interactive nodes, primary non-interactive nodes and secondary non-interactive nodes; Step 4: Use three betweenness indices to evaluate the importance of the interactive nodes, the first-level non-interactive nodes, and the second-level non-interactive nodes; wherein the three centrality indices include OD-point betweenness, interaction betweenness, and block betweenness.

2. The method for identifying key nodes of a weighted network based on layering according to claim 1, characterized in that: The step one comprises: Step (11) establishes a multi-layer network initial model based on the layered network concept. The multi-layer network initial model consists of a triple (G, C, W) and is expressed as: In formula (1), G represents the network set, G α represents the αth layer subnetwork, α represents the αth layer, L represents the number of network layers, X α Represents the αth layer subnetwork G α The node set, E α Represents the αth layer subnetwork G α The edge set of C represents the αth layer subnetwork G α With the β-th layer sub-network G β The set of edges between αβ Represents the αth layer subnetwork G α With the β-th layer sub-network G β , β represents the β layer, W represents the weight set, A represents the edge weight set within the layer, and B represents the edge weight set between layers. Represents edge The weight of Step (12) regards each node in the multi-layer network initial model as a layer network, and the expression is: In formula (2), M t represents the similar community structure division at time t, represents the αth layer subnetwork at time t, Represents the node set of the αth layer sub-network at time t.

3. The method for identifying key nodes of a weighted network based on layering according to claim 1, characterized in that: The second step comprises: Step (21) is based on the initial model of the layer network, Try to connect to the adjacent layer network Merge, where: i≠j; If we can make the layer network and layer network Merge into a new layer network, record the gain ΔSQ of the proportion of non-interactive nodes in the new layer network, the gain ΔSQ of the proportion of non-interactive nodes is the modularity gain, and the expression is: In formula (3), SQ represents similar modularity, M t represents the similar community structure division at time t, Represents the merging layer network and The subsequent community structure Presentation Layer Network The node set of Presentation Layer Network The node set of Step (22) determines the multi-layer network model based on the layer network with the largest similar modularity gain; specifically, if the modularity gains of any two layer networks are non-positive, the multi-layer network model is obtained and the merging is stopped; otherwise, the next merging is performed until the layer network L corresponding to the largest similar modularity is found. i and layer network L j , by operating M k =M k-1 \L max(i,j) , and L min(i,j) =L i ∪L j Get the current similar community structure and return to step (21).

4. The method for identifying key nodes of a weighted network based on layering according to claim 1, characterized in that: The step three comprises: Step (31) considers the importance of each network node in the multi-layer network model from the perspective of betweenness and defines the centrality index ODBC; specifically, calculates the shortest path length of the multi-layer network M and counts Nodes are obtained by calculating the LCDM method and nodes The shortest path length between Represents any two nodes in a multilayer network M and nodes The shortest path between nodes and nodes The number of paths; Step (32) calculates the OD-edge betweenness of each network node in the multi-layer network model, and the expression is: In formula (4), OD-edge betweenness measures the edges in a multilayer network M. In all The importance of the shortest path between Represents a node in the interactive network CG To Node The number of shortest paths, Indicates the edge in CG Node To Node The number of shortest paths; Calculate the OD-point betweenness of all nodes in the multi-layer network model, the expression is: In formula (5), Represents the OD-point betweenness summary node Related Edges In all The importance of the shortest path between Represents a node in CG The set of neighbors of ; Step (33) calculates the block BC of the multi-layer network model; specifically, the expression of the node block betweenness BC is: In formula (6), Representation Node and nodes The number of shortest paths between Representation Node Passing Node and nodes The number of shortest paths; Step (34) calculates the block betweenness Δ of the first-level non-interactive nodes of the multi-layer network model k , the expression is: In formula (7), Δ k Represents multi-layer networks and non-interacting nodes The importance of the shortest path, Represents equivalent triples Represents equivalent triples The weight of ; where: Equivalent triples The judgment conditions are: like And in the determination network DG α The edge The weights satisfy but is an equivalent triple; Determination network DG α In, G α is the layer network of the multi-layer network M, expressed as G α =(X α ,E α ), where X α represents disjoint interaction nodes, E α represents the edge in the network, α=1,2,...,L; Determination network DG α In, DG α is the layer network G α The decision network of Where, X α represents the node set of each layer, represents the edge set in the decision network, W represents the weight of the edge set in the decision network, Presentation Layer Network G α The set of interaction nodes, Presentation Layer Network G α A set of non-interacting nodes.

5. The method for identifying key nodes of a weighted network based on layering according to claim 1, characterized in that: The fourth step comprises: Step (41) In the multi-layer network model of step 3, the interactive nodes are sorted based on the size of the OD-point betweenness ODBC; specifically: if the network nodes of the multi-layer network model are interactive nodes, the interactive nodes are sorted directly according to the size of the OD-point betweenness of the nodes; Step (42) In the multi-layer network model of step 3, non-interactive nodes are divided into primary non-interactive nodes and secondary non-interactive nodes, and then based on the interaction betweenness Δ k And block betweenness BC is sorted; specifically: If the network nodes of the multi-layer network model are non-interactive nodes, then according to the layer network G α Construct decision network DG α , determine the network DG α If there are two nodes in the network node and and Make is an equivalent triple, then the non-interactive node is a first-level non-interactive node, according to the importance of the equivalent triplet of the node Δ k Sort by size; among them, Presentation Layer Network G α The set of interaction nodes; Otherwise, non-interactive nodes They are secondary non-interactive nodes and are sorted according to the size of their block betweenness BC.

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