A method and system for regulating the superdiffusion performance of a coupled double-layer network

Abstract

This application belongs to the technical field related to network hyperdiffusion design, and discloses a method and system for regulating the hyperdiffusion performance of a coupled double-layer network. The method includes: obtaining an original coupled double-layer network, where the minimum non-zero eigenvalue of network G1 is greater than the minimum non-zero eigenvalue of network G2, and alternately iteratively performing edge removal and edge addition until the diffusion degree of the two networks approaches a preset degree, and outputting the updated coupled double-layer network. Among them, the operation of performing edge removal includes: deleting edges with multiple edges and similar end nodes, and the operation of performing edge addition includes selecting nodes with longer paths and greater differences to add new connections. Through the above process, under the condition of ensuring that the total number of edges in the double-layer network remains basically unchanged, the in-layer topological structure of the network can be optimized, and the hyperdiffusion performance of the network can be enhanced to a large extent.

Description

Technical Field

[0001] This application belongs to the technical field related to network hyperdiffusion design. More specifically, it relates to a method and system for regulating the hyperdiffusion performance of a coupled double-layer network. Background Art

[0002] With the in-depth development of network science and application research, the "network of networks" that most widely reflects the real world, or the so-called "hypernetwork", can all be called multi-layer networks. In the real world, most nodes in complex networks have multiple functions, and are interconnected and interact with each other. Moreover, the multiple functions have qualitative (attribute) differences and cannot be superimposed, thus constituting a multi-layer network. Each layer of it is a single network with different functions or attributes, and there are complex relationships of interdependence and correlation between the layers.

[0003] Hyperdiffusion refers to the phenomenon in multi-layer networks that due to the coupling effect of inter-layer connections and the mutual influence of the structures of each single-layer network, the multi-layer network as a whole exhibits a stronger diffusion ability than any single-layer network when it exists independently. The hyperdiffusion phenomenon is unique to multi-layer networks, and it reveals the enhancement effect brought by the multi-layer structure during the diffusion process.

[0004] For example, the real-world transportation network usually consists of a subway network, a bus network, a light rail network, etc. The hyperdiffusion phenomenon means that the multi-level interconnection between different transportation modes can significantly improve the overall transportation efficiency.

[0005] For example, a smart grid usually consists of a physical layer and an information layer. The physical layer ensures the actual transmission and distribution of electricity; the information layer is responsible for collecting, transmitting, and processing data from the physical layer, providing data support for optimization decisions. By adjusting the interaction between the physical layer and the information layer to promote the generation of hyperdiffusion, the electricity supply process can be optimized to achieve efficient and reliable electricity supply.

[0006] The occurrence and performance of the hyperdiffusion phenomenon are significantly affected by the network topology structure, and the intra-layer topology structure plays a crucial role. By reasonably constructing and adjusting the intra-layer topology structure of the multi-layer network, the generation of the hyperdiffusion phenomenon can be promoted. Therefore, studying the influence of the intra-layer topology structure on the hyperdiffusion performance has practical application value. Currently, the research usually realizes the hyperdiffusion phenomenon in a double-layer network based on a single-layer network model with a special topology. However, in real applications, the original double-layer network form is usually given, and topological optimization is required to promote hyperdiffusion. At this time, the traditional method of promoting the hyperdiffusion phenomenon based on a special topology is not applicable.

[0007] Therefore, there is an urgent need to propose a method for regulating hyperdiffusion performance, which can regulate hyperdiffusion for different double-layer networks to adapt to practical applications. Summary of the Invention

[0008] In view of the above defects or improvement requirements of the prior art, the present application provides a method and system for regulating the superdiffusion performance of a coupled double-layer network, aiming to be able to regulate the superdiffusion of different double-layer networks to adapt to practical applications.

[0009] To achieve the above object, according to the first aspect of the present application, there is provided a method for regulating the superdiffusion performance of a coupled double-layer network, which includes:

[0010] Obtain an original coupled double-layer network, including networks G 1 and G 2 with different diffusion performances and the same nodes. The nodes of network G 1 are connected to the nodes of network G 2 one by one to achieve coupling. The smallest non-zero eigenvalue λ 1 (L 2 ) of network G 1 is greater than the smallest non-zero eigenvalue λ 2 (L 2 ) of network G 2 ;

[0011] Alternately iterate to perform edge removal and edge addition. During the alternate iteration, after each execution of at least one edge modification, re-compare λ 2 (L 1 ) and λ 2 (L 1 ). If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > k1, then continue to perform the alternate iteration; otherwise, the iteration ends and the updated coupled double-layer network is output. k1 is a set proximity threshold;

[0012] The operation of performing edge removal includes:

[0013] Find node pairs (i, j) that satisfy l ij (G 1 ) = l ij (G 2 ) = 1 and , where i ≠ j, and delete the edge between nodes i and j in network G 1 ; where l ij (X) represents whether there is an edge connection between nodes i and j in network X, and the value of 1 means there is an edge connection. represent the Fiedler components corresponding to node i and node j in the Fiedler vector of network X, respectively, represent the maximum component and the minimum component in the Fiedler vector of network X, respectively, where k <1> is a set proportionality coefficient less than 1;

[0014] The operations for adding an edge include:

[0015] Calculate the current network G 1 and the network G 2 the shortest network distances between the nodes of the network G after grid connection, form a network distance set, and determine the maximum value m in the network distance set max and the median m median ; find the node pair (i, j) that satisfies m ij ≥ m median , and , where i ≠ j, and add an edge between nodes i and j in network G 2 , where m ij is the shortest network distance between nodes i and j, and k <2> and k ′ are both set proportionality coefficients less than 1.

[0016] In some embodiments, the operations for removing an edge include:

[0017] Step S21: Randomly determine a node pair (i, j), where i ≠ j;

[0018] Step S22: Obtain the edge connection relationship l 1 between nodes i and j in network G ij (G 1 ) and the edge connection relationship l 1 between nodes i and j in network G ij (G 2 );

[0019] Step S23: Determine whether it satisfies l ij (G 1 ) = l ij (G 2 ) = 1. If not, update the node pair (i, j) and return to Step S22. If so, execute Step S24;

[0020] Step S24: Determine whether it satisfies If not, update the node pair (i, j) and return to Step S22. If so, set l ij (G 1 ) = 0 to complete the edge removal operation.

[0021] In some of these embodiments, the proportionality coefficient k <1> satisfies 0 < k <1> < 0.1.

[0022] In some of these embodiments, the operations added to the execution edge include:

[0023] Step S31: Randomly determine a node pair (i, j), where i ≠ j;

[0024] Step S32: Calculate the shortest network distances between the nodes of the current network G 1 and the network G 2 after they are interconnected to form a network distance set, and determine the maximum value m max and the median value m median ;

[0025] Step S33: Determine whether m ij ≥ m median . If not, update the node pair (i, j) and return to Step S32. If so, execute Step S34;

[0026] Step S34: Determine whether and . If not, update the node pair (i, j) and return to Step S32. If so, set l ij (G 2 ) = 1 to complete the edge addition operation.

[0027] In some of these embodiments, calculating the shortest network distances between the nodes of the current network G 1 and the network G 2 after they are interconnected to form a network distance set, and determining the maximum value m max and the median value m median , includes:

[0028] Generate the distance matrix M of network G, where the element in the i-th row and j-th column of the distance matrix is the network distance m ij between the node pair (i, j), record the maximum value m max in the distance matrix <, extract the non-diagonal elements in the upper triangular part or lower triangular part of the matrix and sort them monotonically by network distance, and determine the median m median .

[0029] In some of these embodiments, the proportionality coefficients k <2> and k ′ satisfy: 0.5 < k <2> < 1, 0.5 < k ′ < 1.

[0030] According to the second aspect of the present application, a superdiffusion performance regulation system for a coupled double-layer network is provided, which includes:

[0031] An acquisition unit for acquiring an original coupled double-layer network, including networks G and G with different diffusion performances and exactly the same nodes. The nodes of network G and the nodes of network G are connected in one-to-one correspondence to achieve coupling. The minimum non-zero eigenvalue λ(L) of network G is greater than the minimum non-zero eigenvalue λ(L) of network G; 1 and network G 2 , the nodes of network G 1 and the nodes of network G 2 are connected in one-to-one correspondence to achieve coupling. The minimum non-zero eigenvalue λ(L) of network G 1 is greater than the minimum non-zero eigenvalue λ(L) of network G 2 (L 1 ) is greater than the minimum non-zero eigenvalue λ 2 of network G 2 (L 2 );

[0032] An edge removal unit for performing an edge removal operation, including: finding node pairs (i, j) that satisfy l(G) = l(G) = 1 and, and deleting the edge between nodes i and j in network G; where l(X) represents whether there is an edge connection between nodes i and j in network X, and the value of 1 means there is an edge connection, respectively represent the Fiedler components corresponding to nodes i and j in the Fiedler vector of network X, respectively represent the maximum component and the minimum component in the Fiedler vector of network X, and k is a set proportionality coefficient less than 1; ij (G 1 ) = l ij (G 2 ) = 1 and of node pairs (i, j), i ≠ j, and deleting the edge between nodes i and j in network G 1 ; where l ij (X) represents whether there is an edge connection between nodes i and j in network X, and the value of 1 means there is an edge connection, respectively represent the Fiedler components corresponding to nodes i and j in the Fiedler vector of network X, respectively represent the maximum component and the minimum component in the Fiedler vector of network X, and k <1> is a set proportionality coefficient less than 1;

[0033] An edge addition unit for performing an edge addition operation, including: calculating the shortest network distance between the nodes of the current network G and the network G after merging to form a network distance set, determining the maximum value m and the median m in the network distance set; finding node pairs (i, j) that satisfy m ≥ m, and, and adding an edge between nodes i and j in network G, where m is the network distance between nodes i and j, and k and k are both set proportionality coefficients less than 1; 1 and network G 2 after merging the network G to form a network distance set, determining the maximum value m max and the median m median ; finding node pairs (i, j) that satisfy m ij ≥ m median , and of node pairs (i, j), i ≠ j, and adding an edge between nodes i and j in network G 2 ; where m ij is the network distance between nodes i and j, and k <2> and k ′ are both set proportionality coefficients less than 1;

[0034] A control unit for controlling the edge removal unit and the edge addition unit to alternately and iteratively perform edge removal and edge addition. During the alternating iteration, every time at least one edge modification is performed, λ is recompared. 2 (L 1 ) and λ 2 (L 1 ) are compared. If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > k1, then continue with the alternating iteration; otherwise, end the iteration and output the updated coupled bilayer network, where k1 is a set proximity threshold.

[0035] According to the third aspect of the present application, there is provided an electronic device including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the method described in any one of the above are implemented.

[0036] According to the fourth aspect of the present application, there is provided a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, the steps of the method described in any one of the above are implemented.

[0037] According to the first aspect of the present application, there is provided a computer program product including a computer program or instruction. When the computer program or instruction is executed by a processor, the steps of the method described in any one of the above are implemented.

[0038] Generally speaking, compared with the prior art through the above technical solutions conceived by the present application, the method for regulating the superdiffusion performance of the coupled bilayer network provided by the present application mainly has the following beneficial effects:

[0039] 1. The method for regulating the superdiffusion performance of the coupled bilayer network provided by the present application makes the diffusion performances of the two single-layer networks close by alternately and iteratively performing edge removal and edge addition. It is found that making the diffusion performances of each single-layer network close is more conducive to the emergence of the network superdiffusion phenomenon. Therefore, in the present invention, edges are removed from the single-layer network with stronger diffusion performance, and edges are added to the network with weaker diffusion performance. By alternately performing the processes of edge removal and addition, while ensuring that the total number of edges in the bilayer network remains unchanged, the intra-layer topological structure of the network is optimized, and the superdiffusion performance of the network is enhanced to a large extent.

[0040] 2. The method for regulating the superdiffusion performance of the coupled bilayer network provided by the present application sets two conditions when performing edge removal operations on the network G 1 with stronger diffusion ability, and can choose to perform operations on the network G2 Edges with multiple edges and edges with similar end nodes in it are deleted, thereby reducing the weakening effect of edge deletion operations on the diffusion ability of the network.

[0041] 3. For the method for regulating the super-diffusion performance of the coupled double-layer network provided by this application, when performing an edge addition operation on the network G with relatively weak diffusion ability 2 three conditions are set, and nodes with longer paths and greater differences can be selected to add new edges, thereby making the impact of adding edges on enhancing the diffusion of the network greater. Brief Description of the Drawings

[0042] Figure 1 is a flowchart of the steps of the method for regulating the super-diffusion performance of the coupled double-layer network in an embodiment of this application;

[0043] Figure 2 is a network schematic diagram of the original coupled double-layer network in an embodiment;

[0044] Figure 3 is the network G in an embodiment 1 network schematic diagram;

[0045] Figure 4 is the network G in an embodiment 2 network structure diagram;

[0046] Figure 5 is a diagram of the dynamic change process of the parameter ζ during the algorithm iteration process in an embodiment;

[0047] Figure 6 is a diagram of the dynamic change process of the parameter λ during the algorithm iteration process in an embodiment s ;

[0048] Figure 7 is a curve of the change of the minimum non-zero eigenvalue of each single-layer network and the double-layer network with the inter-layer coupling strength d after the algorithm iteration ends in an embodiment. Detailed Embodiments

[0049] In order to make the purpose, technical solutions and advantages of this application clearer, the following further elaborates on this application in combination with the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not used to limit this application. In addition, the technical features involved in the various embodiments of this application described below can be combined with each other as long as they do not conflict with each other.

[0050] Embodiment 1

[0051] As Figure 1The figure shows a flowchart of the steps for regulating the superdiffusion performance of a coupled double-layer network in an embodiment. The following introduces the relevant steps thereof.

[0052] Step S1: Obtain an original coupled double-layer network, which includes networks G with different diffusion performances and exactly the same nodes 1 and network G 2 , and the minimum non-zero eigenvalue λ 1 (L 2 ) of network G 1 is greater than the minimum non-zero eigenvalue λ 2 (L 2 ) of network G 2 .

[0053] As Figure 2 shown is a network schematic diagram of an original coupled double-layer network in an embodiment, which includes networks G with different diffusion performances and exactly the same nodes 1 and network G 2 , and the nodes of network G 1 are connected to the nodes of network G 2 one by one to achieve coupling. The minimum non-zero eigenvalue λ 1 (L 2 ) of network G 1 is greater than the minimum non-zero eigenvalue λ 2 (L 2 ) of network G 2 .

[0054] Taking the transportation network as an example, network G 1 can be the subway network, which includes stations A to F. Network G 2 can be the bus network, which also includes stations A to F. The six stations of the subway network are connected to the six stations of the bus network one by one. The same nodes have different attributes in network G 1 and network G 2 . Station A has the attribute of a subway station in network G 1 and the attribute of a bus node in network G 2 .

[0055] Taking the smart grid as an example again, network G 1 can be the physical layer network, which includes grid nodes 1 to 100. Network G 2 can be the information layer network, which also includes grid nodes 1 to 100. The 100 grid nodes of the physical layer network are connected to the 100 grid nodes of the information layer network one by one. The same nodes have different attributes in network G 1 and network G 2 . Grid node 1 has the attribute of a physical layer grid node in network G 1The attributes in it are physical layer nodes, which are used to achieve the actual transmission and distribution of electric power. In network G 2 The attributes in it are information layer nodes, which are used to collect, transmit and process data from the physical layer, providing data support for optimization decisions. The two layers cooperate closely to jointly form the overall framework of the smart grid, achieving efficient, reliable and environmentally friendly power supply.

[0056] In practical applications, each layer of the network is very complex, and the number of nodes in each layer reaches up to hundreds.

[0057] Among them, network G 1 and network G 2 have different diffusion performances, that is, their minimum non-zero eigenvalues are different. Here, the one with the larger minimum non-zero eigenvalue is defined as network G 1 , and its minimum non-zero eigenvalue is λ 2 (L 1 ). The one with the smaller minimum non-zero eigenvalue is defined as network G 2 , and its minimum non-zero eigenvalue is λ 2 (L 2 ).

[0058] For a definite double-layer network, the Laplacian matrix, the minimum non-zero eigenvalue λ 2 , the Fiedler vector and the distance matrix M of each layer of the network and the network after the merger of all network nodes can be calculated, and each parameter can be obtained through existing methods.

[0059] In the coupled double-layer network, both network G 1 and network G 2 are unweighted undirected networks with N nodes. Define the Laplacian matrix of network G 1 as L 1 , and the Laplacian matrix of network G 2 as L 2 . The element in the i-th row and j-th column of the Laplacian matrix indicates whether there is an edge connection between node i and node j. If there is, the value is 1; if not, the value is 0. The Supra-Laplacian matrix L of the coupled double-layer network is described as:

[0060]

[0061] Among them, d represents the inter-layer coupling strength, and I N is the identity matrix.

[0062] The diffusion performance of the network is usually described by the minimum non-zero eigenvalue λ 2 of the Laplacian matrix. Network G 1The minimum non-zero eigenvalue λ 2 is λ 2 (L 1 ), for the network G 2 the minimum non-zero eigenvalue λ 2 is λ 2 (L 2 ). For the double-layer network, the minimum non-zero eigenvalue λ 2 is In the double-layer network composed of G 1 and G 2 , if the inequality is satisfied, it is said that superdiffusion occurs in the double-layer network. will increase with the increase of the inter-layer coupling strength d and finally converge to λ s , where λ s is the minimum non-zero eigenvalue corresponding to the matrix , which is usually used to describe the superdiffusion performance.

[0063] The Fiedler vector is the eigenvector corresponding to λ 2 . Its eigen-components imply the network structure characteristics. For example, the positive and negative of the eigen-components can be used to mine the network community structure, and the similarity of the eigen-components can quantify the similarity of the connection situation and position of the corresponding nodes in the network and other structural characteristics.

[0064] Define the network G 1 =(V(G 1 ), E(G 1 )), the network G 2 =(V(G 2 ), E(G 2 )). V is the node set and E is the edge set. Then V(G 1 ) = V(G 2 ). After the networks G 1 and G 2 are interconnected, the network G = G 1 ∪G 2 =(V(G 1 ), E(G 1 )∪E(G 2 ))=(V(G 2 ), E(G 1 )∪E(G 2 ).

[0065] The Fiedler vector of the network G 1 is denoted as and are its maximum and minimum components respectively.

[0066] For the network G 2The Fiedler vector is denoted as and are its maximum component and minimum component respectively.

[0067] The Fiedler vector of network G is denoted as and are its maximum component and minimum component respectively.

[0068] The element m in the i-th row and j-th column of the distance matrix M of the network ij represents the value of the shortest network path between node i and node j in the network, that is, the minimum number of connected edges required to move from node i to node j, namely the shortest network distance. For example, if there is a connecting edge between node 1 and node 2, then the minimum number of connected edges required to move from node 1 to node 2 is 1. Suppose there is a connecting edge between node 1 and node 2, a connecting edge between node 2 and node 3, and no connecting edge between node 1 and node 3, then the minimum number of connected edges required to move from node 1 to node 3 is 2.

[0069] Since the diffusion capabilities of network G 1 and network G 2 in the original coupled bilayer network are different, the minimum non-zero eigenvalue λ 1 of network G 2 (L 1 ) is greater than the minimum non-zero eigenvalue λ 2 of network G 2 (L 2 ), that is, the diffusion capability of network G 1 is greater than that of network G 2 . It has been found through research that making the diffusion performances of each single-layer network closer is more conducive to the emergence of the network super-diffusion phenomenon. Therefore, in the present invention, edges are removed from the single-layer network with stronger diffusion performance, and edges are added to the network with weaker diffusion performance. By alternately performing the processes of edge removal and addition, while ensuring that the total number of connected edges in the bilayer network remains unchanged, the intra-layer topological structure of the network is optimized, and the super-diffusion performance and degree of the network are enhanced to a large extent. The specific steps are as follows.

[0070] Step S2: Perform an edge removal operation on network G 1 and then re-compare λ 2 (L 1 ) and λ 2 (L 1 ). If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L2 ) If k1, then jump to step S3; otherwise, the iteration ends and the updated coupled bilayer network is output.

[0071] Among them, the edge removal operation includes: finding node pairs (i, j) that satisfy l ij (G 1 ) = l ij (G 2 ) = 1 and , i ≠ j, and deleting the edge between nodes i and j in network G 1 . l ij (G 1 ) is the element in the i-th row and j-th column of the Laplacian matrix L of network G 1 , indicating whether there is an edge connection between nodes i and j in network G 1 ; l 1 (G ij ) is the element in the i-th row and j-th column of the Laplacian matrix L of network G 2 , indicating whether there is an edge connection between nodes i and j in network G 2 . 2 are the Fiedler components of nodes i and j in network G 2 respectively, and k are the Fiedler components of nodes i and j in network G 1 respectively, and k <1> is a set proportionality coefficient less than 1.

[0072] In the present invention, the edges selected for deletion need to meet two conditions:

[0073] The first condition is l ij (G 1 ) = l ij (G 2 ) = 1. This condition means that reconnected edges in the bilayer network are preferentially selected for deletion. A larger network diameter is not conducive to improving the diffusion performance. The diffusion performance of the bilayer network is positively correlated with the diffusion performance of the union network G 1 ∪ G 2 . Therefore, the edges selected for deletion in the G 1 layer should have less impact on the diffusion performance of the union network G 1 ∪ G 2 . In the present invention, selecting to remove reconnected edges does not affect the topological structure of the union network G 1 ∪ G 2 . Therefore, it does not increase the diameter of the union network G 1 ∪ G 2 , making the edge deletion operation have less impact on weakening the diffusion of the union network G 1 ∪ G 2 ;

[0074] The second condition is This condition uses the coefficient k <1> to quantify the magnitude of the difference between nodes. The greater the difference between two nodes i and j, the lower the similarity, and thus the greater the difference in the corresponding eigen-components in the Fiedler vector. Conversely, the smaller the difference between two nodes i and j, the higher the similarity, and thus the smaller the difference in the corresponding eigen-components in the Fiedler vector. By adjusting this coefficient, nodes with a similarity higher than a certain level can be selected, and deleting the edges between similar nodes can also reduce the impact of edge deletion on the diffusion ability of the network.

[0075] In one embodiment, the coefficient k <1> satisfies 0 < k <1> < 0.1. Within this range, nodes with extremely high similarity are found and the edge between them is deleted, so that the impact of edge deletion on the diffusion ability of the network is minimized.

[0076] Each time an edge is deleted, the edge connection relationship of the coupled double-layer network is modified, and the diffusion ability of the network changes. It is necessary to re-compare the diffusion ability of network G 1 and network G 2 , that is, re-compare the minimum non-zero eigenvalue λ 1 of network G 2 (L 1 ) and the minimum non-zero eigenvalue λ 2 of network G 2 (L 2 ). Set the threshold k1 for the proximity of the diffusion ability of the two networks. If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > k1, it means that although the minimum non-zero eigenvalue λ 1 representing the diffusion ability of network G 2 (L 1 ) has decreased after edge deletion, it has not decreased to a level close to the diffusion ability of network G 2 . Therefore, it is necessary to jump to step S3 for the edge addition operation of another network. Otherwise, it means that the minimum non-zero eigenvalue λ 1 representing the diffusion ability of network G 2 (L 1 ) has decreased to a level close to the diffusion ability of network G 2 . The expectation is achieved, and the current latest coupled double-layer network is output.

[0077] In one embodiment, the edge removal operation can be performed according to the following process:

[0078] Step S21: Randomly determine a node pair (i, j), where i ≠ j;

[0079] Step S22: Obtain the edge connection relationship l 1 between nodes i and j in the network G ij (G 1 ) and the edge connection relationship l 1 between nodes i and j in the network G ij (G 2 );

[0080] Step S23: Determine whether l ij (G 1 ) = l ij (G 2 ) = 1. If not, update the node pair (i, j) and return to Step S22. If so, execute Step S24;

[0081] Step S24: Determine whether If not, update the node pair (i, j) and return to Step S22. If so, set l ij (G 1 ) = 0 to complete the edge removal operation

[0082] Step S3: Perform an edge addition operation on the network G 2 and then re - compare λ 2 (L 1 ) and λ 2 (L 1 ). If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > k1, then jump to Step S2. Otherwise, the iteration ends and the updated coupled - type bilayer network is output.

[0083] Among them, the edge addition operation includes: calculating the shortest network distances between the nodes of the network G 1 and the network G 2 after they are merged into the network G to form a network distance set, determining the maximum value m max and the median m median ; finding the one that satisfies m ij ≥ m median , And for the node pair (i, j), where i ≠ j, add an edge between nodes i and j in network G 2 in the network G

[0084] In the present invention, the nodes for which edges are added satisfy three conditions:

[0085] The first condition is that m ij ≥ m median This condition means to select nodes i and j with longer network paths in the union network G 1 ∪ G 2 The longer the path, the more unfavorable it is for diffusion. Selecting nodes i and j with longer paths and adding connections can shorten the path, thereby reducing the average path length of networks G 2 and G 1 ∪ G 2 and improving the diffusion performance of the network;

[0086] The second condition is This condition means that in network G 2 the difference between nodes i and j is relatively large. Selecting two points with a relatively large difference and adding an edge can make the impact of adding the edge on enhancing the diffusion of network G 2 relatively large. Among them, k <2> is a proportional system for quantifying the difference. In this embodiment, it can be set as 0.5 < k <2> < 1 to obtain nodes i and j with a relatively large difference;

[0087] The third condition is This condition means that in network G 1 ∪ G 2 the difference between nodes i and j is also relatively large. Selecting two points with a relatively large difference and adding an edge can make the impact of adding the edge on enhancing the diffusion of network G 1 ∪ G 2 relatively large. Among them, k ′ is a proportional system for quantifying the difference. In this embodiment, it can be set as 0.5 < k ′ < 1 to obtain nodes i and j with a relatively large difference.

[0088] Each time a new edge is added, the edge connection relationship of the coupled double - layer network is modified, and the diffusion ability of the network changes. It is necessary to re - compare the diffusion abilities of network G 1 and network G 2 That is, re - compare the minimum non - zero eigenvalue λ 1 of network G 2 (L 1 ) and the minimum non - zero eigenvalue λ 2 of network G 2 (L 2 ). If λ2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > k1, indicating that for network G 2 after adding an edge, its minimum non - zero eigenvalue λ 2 (L 2 ) has increased to some extent, but still has not increased to a level close to the diffusion ability of network G 1 . Therefore, it is necessary to jump to step S2 to perform the edge - deletion operation on another network. Otherwise, it indicates that for network G 2 after adding an edge, its minimum non - zero eigenvalue λ 2 (L 2 ) has increased to a level close to the diffusion ability of network G 1 , the expectation is achieved, and the current latest coupled double - layer network is output.

[0089] In one embodiment, the operation of adding an edge can be performed according to the following process:

[0090] Step S31: Randomly determine a node pair (i, j), where i ≠ j;

[0091] Step S32: Calculate the shortest network distances between each pair of nodes in the network G 1 and the network G 2 after the two networks are merged into network G, form a network distance set, and determine the maximum value m max and the median value m median ;

[0092] Step S33: Determine whether m ij ≥ m median . If not, update the node pair (i, j) and return to step S32. If so, execute step S34;

[0093] Step S34: Determine whether and . If not, update the node pair (i, j) and return to step S32. If so, set l ij (G 2 ) = 1 to complete the edge - adding operation.

[0094] Among them, in step S32, it includes: generating the distance matrix M of network G, where the element in the i - th row and j - th column of the distance matrix is the network distance m ij between the node pair (i, j), and recording the maximum value m max, extract the non - diagonal elements of the upper triangular or lower triangular part of the matrix M and sort them monotonically by network distance to determine the median m median .

[0095] The literature R.F. Andrade, J.G.V. Miranda, S.T.R. Pinho, T.P. Measuring distances between complex networks[J]. Physics Letters A, 2008, 372(32):5265 - 5269 defines the distance matrix M, which contains all the information about the shortest path between any two nodes i and j. The distance between nodes i and j in the network G is d ij , define a set of adjacency matrices M(l) to describe the node pairs with d ij = l in the network G. The elements in the matrix are described as:

[0096]

[0097] where l = 1, 2, … D, D is the network diameter, the identity matrix I = M(0), the usual adjacency matrix A(G) = M(1), and all M(l) can be calculated using the following recurrence equation:

[0098]

[0099] where the symbols and ⊙ are the Boolean operators sum and multiplication respectively.

[0100] Then the distance matrix M can be described by the following formula:

[0101]

[0102] The j - th element m ij in the i - th row of the matrix represents the value of the shortest path between node i and node j.

[0103] In one embodiment, it is also possible to first perform an edge - addition operation on the network G 2 , and then perform an edge - removal operation on the network G 1 to alternately achieve the regulation of super - diffusion performance.

[0104] It should be noted that the value ranges of the proportionality coefficients k <1> , k <2> and k ′ can be adjusted according to different network structure characteristics and requirements.

[0105] Next, the following specific examples are used to verify the technical solution.

[0106] Step S1: Generate networks G with the number of nodes N = 200 respectively 1 and network G 2 , as Figure 3 shown in the network structure diagram of network G in an embodiment 1 , as Figure 4 shown in the network structure diagram of network G in an embodiment 2 Calculate the minimum non - zero eigenvalue λ 1 = 2.76 of the initial network G 2 , and the minimum non - zero eigenvalue of network G 2 is λ 2 = 1.39. Let the set proximity threshold k1 = 0.137, and the proportionality coefficients k <1> = 0.08, k <2> = 0.55, k ′ = 0.55.

[0107] Step S2: Select and delete the edge between nodes i and j in network G 1 . Nodes i and j satisfy: l ij (G 1 ) = l ij (G 2 ) = 1 and Let l ij (G 1 ) = 0, calculate and compare λ 2 (L 1 ) and λ 2 (L 2 ). If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > 0.137, then continue the following steps.

[0108] Step S3: Generate the distance matrix M of the union network G 1 ∪G 2 . Select to add an edge between nodes i and j in network G 2 . Nodes i and j satisfy m ij ≥m median , And Let l ij (G 2 ) = 1, calculate and compare λ 2 (L 1 ) and λ 2 (L 2), if λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > 0.137, then return to step S2 for iteration; otherwise, the iteration ends.

[0109] Finally, the minimum non - zero eigenvalue λ 1 of network G 2 = 2.612, and the minimum non - zero eigenvalue of network G 2 is λ 2 = 2.611. It can be found that under the condition of keeping the total number of edges in the network unchanged, through the in - layer edge reconnection of the present invention, the optimization of the in - layer edge distribution is realized. The removed edges have a very weak impact on the diffusion performance of network G 1 , and the removed edges are effectively reorganized into network G 2 , significantly improving the diffusion performance of this layer. After the iteration ends, each single - layer network can maintain a relatively high diffusion level.

[0110] As Figure 5 shown is the dynamic change process diagram of parameter ζ during the algorithm iteration. Parameter ζ is a quantization index of the super - diffusion degree, which represents the relative ratio of the super - diffusion performance of the bilayer network to the diffusion performance of the single - layer with the fastest diffusion speed:

[0111]

[0112] When ζ > 0, λ s > max{λ 2 (L 1 ), λ 2 (L 2 )}, the network is super - diffusible.

[0113] From Figure 5 it can be seen that as the iteration progresses, the quantization index ζ of the super - diffusion degree gradually increases, indicating that as the iteration progresses, the super - diffusion performance of the bilayer network gradually enhances compared to the diffusion performance of the single - layer with the fastest diffusion speed.

[0114] As Figure 6 shown is the dynamic change process diagram of parameter λ s during the algorithm iteration. From Figure 6 it can be seen that as the iteration progresses, the convergence value λ s gradually increases, indicating that as the iteration progresses, the super - diffusion performance of the bilayer network gradually enhances.

[0115] As Figure 7The figure shows the variation curves of the minimum non-zero eigenvalues of each single-layer network and double-layer network with the inter-layer coupling strength d after the algorithm iteration ends. From Figure 7 it can be seen that for the updated coupled double-layer network after the iteration ends, its minimum non-zero eigenvalue increases with the increase of the inter-layer coupling strength d and finally converges, indicating that the double-layer network adjusted by the algorithm is super-diffusive.

[0116] Embodiment 2

[0117] This application also relates to a super-diffusion performance regulation system for a coupled double-layer network, which includes an acquisition unit, an edge removal unit, an edge addition unit, and a control unit:

[0118] The acquisition unit is used to acquire the original coupled double-layer network, including networks G 1 and G 2 with different diffusion performances and exactly the same nodes. The nodes of network G 1 and the nodes of network G 2 are connected in one-to-one correspondence to achieve coupling. The minimum non-zero eigenvalue λ 1 (L 2 ) of network G 1 is greater than the minimum non-zero eigenvalue λ 2 (L 2 ) of network G 2 ;

[0119] The edge removal unit is used to perform the operation of edge removal, including: finding node pairs (i, j) that satisfy l ij (G 1 ) = l ij (G 2 ) = 1 and , i ≠ j, and deleting the edge between nodes i and j in network G 1 ; where l ij (X) represents whether there is an edge connection between nodes i and j in network X, and the value of 1 means there is an edge connection, respectively represent the Fiedler components of node i and node j in network X, respectively represent the maximum component and the minimum component in the Fiedler vector of network X, and k <1> is a set proportionality coefficient less than 1;

[0120] The edge addition unit is used to perform the operation of edge addition, including: calculating the shortest network distance between the nodes of the current network G 1 and network G 2 after merging into network G, forming a network distance set, and determining the maximum value m max and the median m median in the network distance set; finding those that satisfy m ij≥m median 、 and For node pairs (i, j) where i ≠ j, add an edge between nodes i and j in network G 2 where m ij is the network distance between nodes i and j, and k <2> and k ′ are both set proportionality coefficients less than 1;

[0121] The control unit is used to control the edge removal unit and the edge addition unit to alternately and iteratively perform edge removal and edge addition. During the alternating iteration, every time at least one edge modification is performed, λ 2 (L 1 ) and λ 2 (L 1 ) are re - compared. If λ 2 (L 1 ) > λ 2 (L 2 ) and |λ 2 (L 1 ) - λ 2 (L 2 )| > k1, then continue to perform the alternating iteration. Otherwise, the iteration ends and the updated coupled double - layer network is output. k1 is a set proximity threshold.

[0122] Among them, the above - mentioned super - diffusion performance regulation system of the coupled double - layer network can be used to execute the super - diffusion performance regulation method of the coupled double - layer network in Embodiment 1. Each unit module therein is used to implement the relevant steps in the corresponding method. The specific implementation process can refer to the introduction of Embodiment 1 and will not be elaborated here.

[0123] Embodiment 3

[0124] This application also relates to an electronic device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the above - mentioned method are implemented.

[0125] The electronic device can be a computing device such as a desktop computer, a notebook, a palm computer, and a cloud server. The so-called processor can be a Central Processing Unit (CPU), or can also be other general-purpose processors, Digital Signal Processors (DSPs), Application-Specific Integrated Circuits (ASICs), Field-Programmable Gate Arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The memory can be used to store computer programs and / or modules. The processor, by running or executing the computer programs and / or modules stored in the memory, and by calling the data stored in the memory, realizes various functions of the electronic device.

[0126] Embodiment 4

[0127] This application also relates to a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the above method are realized.

[0128] Specifically, the memory can include high-speed random access memory, and can also include non-volatile memory, such as a hard disk, a memory, a plug-in hard disk, a Smart Media Card (SMC), a Secure Digital (SD) card, a Flash Card, at least one magnetic disk storage device, a flash memory device, or other volatile solid-state storage devices.

[0129] Embodiment 5

[0130] This application embodiment provides a computer program product or a computer program. The computer program product or the computer program includes computer instructions, and the computer instructions are stored in a computer-readable storage medium. The processor of the computer device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the computer device executes the steps of the method in the above embodiments of this application.

[0131] The technical features of the above-described embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered to be within the scope described in this specification. It should be noted that "in an embodiment of this application", "for example", "similarly", etc. are intended to illustrate this application, rather than to limit this application.

[0132] The above-described embodiments merely represent several implementation manners of the present application. The description thereof is relatively specific and detailed, but it should not be construed as a limitation to the scope of the patent application. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all fall within the protection scope of the present application.

Claims

1. A method for regulating the superdiffusion performance of a coupled double-layer network, characterized in that: include: Obtain the original coupled two-layer network, including network G1 and network G2 with different diffusion performance and completely identical nodes. Both network G1 and network G2 are unweighted and undirected networks with N nodes. The Laplacian matrix of network G1 is L1, and the Laplacian matrix of network G2 is L2. Each node of network G1 and each node of network G2 are connected one-to-one to achieve coupling. The minimum non-zero eigenvalue λ2(L1) of network G1 is greater than the minimum non-zero eigenvalue λ2(L2) of network G2. The edge removal and edge addition are performed in alternating iterations. During the alternating iterations, λ2(L1) and λ2(L1) are re-compared after at least one edge modification. If λ2(L1)>λ2(L2) and |λ2(L1)-λ2(L2)|>k1, the alternating iterations are continued. Otherwise, the iteration ends and the updated coupled two-layer network is output. k1 is the set proximity threshold. The operations to perform edge removal are: Finding satisfaction ij (G1) = l ij (G2) = 1 and Node pair (i, j), i≠j, delete the edge between nodes i and j in network G1; where l ij (X) indicates whether there is an edge connection between nodes i and j in the network X. A value of 1 indicates that there is an edge connection. They represent the Fiedler components of the Fiedler vector of network X corresponding to nodes i and j, respectively. Respectively represent the maximum and minimum components in the Fiedler vector of network X, k <1> is a proportional coefficient set to be less than 1; The operations to perform edge addition include: Calculate the shortest network distance between the nodes of the network G after the current network G1 and the network G2 are connected to form a network distance set, and determine the maximum value m in the network distance set max and the median m median ; Find satisfying m ij ≥m median , and The node pair (i, j), i≠j, increases the edge between nodes i and j in network G2, where m ij is the shortest network distance between nodes i and j, k <2> and k ′ All of them are set to a proportional coefficient less than 1.

2. The method for controlling the super diffusion performance of a coupled double-layer network according to claim 1, characterized in that: The edge removal operation includes: Step S21: randomly determine a node pair (i, j), i≠j; Step S22: Obtain the edge connection relationship l between nodes i and j in the network G1 ij (G1) and the edge connection relationship l between nodes i and j in network G1 ij (G2); Step S23: Determine whether l is satisfied ij (G1) = l ij (G2) = 1, if not, update the node pair (i, j) and return to step S22, if yes, execute step S24; Step S24: Determine whether If not, update the node pair (i, j) and return to step S22. If yes, set l ij (G1)=0, edge removal operation is completed.

3. The method for controlling the super diffusion performance of a coupled double-layer network according to claim 1 or 2, characterized in that: Proportional coefficient k <1> Satisfy 0 <k <1> <0.

1.

4. The method for controlling the super diffusion performance of a coupled double-layer network according to claim 1, characterized in that: The operation of performing edge addition includes: Step S31: randomly determine a node pair (i, j), i≠j; Step S32: Calculate the shortest network distance between nodes of the network G after the current network G1 and the network G2 are connected to form a network distance set, and determine the maximum value m in the network distance set. max and the intermediate value m median ; Step S33: Determine whether m is satisfied ij ≥m median If not, update the node pair (i, j) and return to step S32. If yes, execute step S34; Step S34: Determine whether and If not, update the node pair (i, j) and return to step S32. If yes, set l ij (G2)=1, the edge addition operation is completed.

5. The method for controlling the superdiffusion performance of a coupled double-layer network according to claim 1 or 4, characterized in that: The shortest network distance between each node of the network G after the current network G1 and the network G2 are connected to the network is calculated to form a network distance set, and the maximum value m in the network distance set is determined. max and the intermediate value m median ,include: Generate the distance matrix M of the network G. The element in the i-th row and j-th column of the distance matrix is ​​the network distance m between the node pair (i, j) ij , record the maximum value m in the distance matrix M max , extract the off-diagonal elements of the upper or lower triangular part of the matrix M and sort them monotonically by network distance to determine the median m median .

6. The method for controlling the superdiffusion performance of a coupled double-layer network according to claim 1 or 4, characterized in that: Proportional coefficient k <2> and k ′ Satisfied: 0.5 <k <2> <1,0.5 <k ′ <1.

7. A coupled double-layer network superdiffusion performance control system, characterized in that: include: An acquisition unit is used to acquire an original coupled two-layer network, including a network G1 and a network G2 with different diffusion performances and completely identical nodes, wherein both the network G1 and the network G2 are unweighted and undirected networks with N nodes, the Laplacian matrix of the network G1 is L1, the Laplacian matrix of the network G2 is L2, each node of the network G1 is connected to each node of the network G2 in a one-to-one correspondence to achieve coupling, and the minimum non-zero eigenvalue λ2(L1) of the network G1 is greater than the minimum non-zero eigenvalue λ2(L2) of the network G2; The edge removal unit is used to perform edge removal operations, including: finding an edge that satisfies l ij (G1) = l ij (G2) = 1 and Node pair (i, j), i≠j, delete the edge between nodes i and j in network G1; where l ij (X) indicates whether there is an edge connection between nodes i and j in the network X. A value of 1 indicates that there is an edge connection. They represent the Fiedler components of the Fiedler vector of network X corresponding to nodes i and j, respectively. Respectively represent the maximum and minimum components in the Fiedler vector of network X, k <1> is a proportional coefficient set to be less than 1; The edge adding unit is used to perform edge adding operations, including: calculating the shortest network distance between each node of the network G after the current network G1 and the network G2 are connected to form a network distance set, and determining the maximum value m in the network distance set. max and the median m median ; Find satisfying m ij ≥m median , and The node pair (i, j), i≠j, increases the edge between nodes i and j in network G2, where m ij is the network distance between nodes i and j, k <2> and k ′ All of them are set to a proportional coefficient less than 1; The control unit is used to control the edge removal unit and the edge addition unit to perform edge removal and edge addition alternately and iteratively. During the alternating iteration, each time at least one edge modification is performed, λ2(L1) and λ2(L1) are re-compared. If λ2(L1)>λ2(L2) and |λ2(L1)-λ2(L2)|>k1, the alternating iteration continues. Otherwise, the iteration ends and an updated coupled two-layer network is output. k1 is a set proximity threshold.

8. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, wherein: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 6 are implemented.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

10. A computer program product comprising a computer program or instructions, characterized in that When the computer program or instruction is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

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