CN113037572A - Key node identification method and device based on graph signal analysis - Google Patents
Key node identification method and device based on graph signal analysis Download PDFInfo
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Abstract
The invention provides a method and a device for identifying key nodes based on graph signal analysis, wherein the method comprises the following steps: calculating the values of the graph signals of all nodes in the network; grouping all nodes in the network according to the values of the graph signals of the nodes to form two grouping sub-networks; obtaining a boundary network between grouping sub-networks; and solving the minimal cut set of the boundary network as a key node set. According to the scheme of the invention, the key node identification of network fracture caused by failure can be realized.
Description
Technical Field
The invention relates to the field of computer networks, in particular to a method and a device for identifying key nodes based on graph signal analysis.
Background
Many mechanisms and functions of the network are typically greatly affected by a small fraction of the nodes, which are typically defined as critical nodes. Relevant research shows that when the network is subjected to deliberate attack, 5% of the key nodes are destroyed and the network is almost paralyzed. The key nodes are identified, so that the monitoring burden is further reduced, and the network safety protection capability is improved.
The Problem of identifying a key Node, also referred to as the key Node Detection Problem (CNDP), is a Problem of finding a set of nodes that have a significant impact on the performance of the network. At present, the relevant research aiming at key node identification mainly relates to a plurality of fields such as social networks, biological networks, communication networks, power networks, traffic networks and the like, the relevant identification methods are mainly divided into two types, one type is that the importance of the nodes is equivalent to the criticality of the nodes, the importance measurement values of all the nodes in the network are obtained by setting the importance measurement indexes, the identification of the key nodes is further completed according to the sequencing result of the measurement values, and the relevant algorithm is divided into an identification method based on local characteristics and an identification method based on global characteristics according to whether the measurement indexes are regional indexes or global indexes; the other type is that the destructiveness of the deleted nodes to the network is equivalent to the criticality of the nodes, the variation value of the network robustness of the deleted nodes is measured by setting a network robustness evaluation index, the identification of the key nodes is further completed according to the sorting result of the variation value, and the larger the variation value is, the more critical the nodes are. The representing method mainly comprises a node deleting method and a node contraction method. The network robustness refers to the ability of the network to still work normally when part of the network is damaged (fails), and is mainly divided into static robustness and dynamic robustness according to whether the redistribution of the network load is considered in the research process, wherein the static robustness refers to the ability of the network to still keep connection after the nodes are deleted, and the problem of redistribution of the load is not considered.
The identification method based on the local features mainly identifies key nodes by evaluating the importance of neighbor nodes of the nodes or intra-hop neighbor nodes, and the representative algorithm mainly comprises Degree Centrality (Degree centricity), Centrality based on local information, a shell decomposition method and the like. The identification method based on the local features has the advantages of low computational complexity, capability of reflecting the importance of the nodes in the local network and reflecting the communication state of the nodes and the surrounding neighbor nodes, but does not consider the importance of the surrounding neighbor nodes.
The identification method based on the global characteristics mainly analyzes a network topological structure, evaluates the key of a node from the perspective of an information transmission path or comprehensively considers the importance of surrounding neighbors, and representative measurement indexes comprise Betweenness center, proximity center, centrifugal center, feature vector center and PageRank algorithm.
Based on the identification method with importance equivalent to criticality, the quality of the finally screened key nodes mainly depends on the quality of the measurement indexes, and if the selected measurement indexes have one-sidedness, the finally screened key nodes are only local key nodes.
In the node deletion method, in the process of measuring the nodes, the nodes are assumed to be invalid, and the importance of the nodes is evaluated by comparing the change of network robustness before and after the nodes are invalid. The method has the disadvantage that if a plurality of nodes in the network fail to cause network disconnection, the nodes have the same importance degree and cannot be distinguished. The node contraction method is characterized in that a node and a neighbor node connected with the node are contracted into a new node, the original edges connected with the neighbor node are connected with the new node, the change of the network condensation degree before and after the node contraction is solved, and the measurement of the node importance degree is completed by using the node importance degree. The more important the node shrinks, the larger the cohesion value of the network. The node shrinkage method has the disadvantage of large operation complexity, and is not suitable for a large-scale network.
Based on the identification method with the destructiveness equivalent to the criticality, the problems caused by unreasonable selection of the measurement indexes to the identification of the key nodes are avoided, however, higher requirements are provided for the selection of the network robustness measurement indexes, and the unreasonable selection of the measurement indexes directly influences the quality of the identification result.
In summary, most current key node identification methods do not identify key nodes for specific destructive influences (such as network rupture or paralysis).
Disclosure of Invention
In order to solve the technical problems, the invention provides a method and a device for identifying key nodes based on graph signal analysis, which are used for solving the technical problem of identifying the key nodes under the condition that a network is split (for example, the network is split into a plurality of sub-networks which cannot be communicated due to node failure) in the prior art.
According to a first aspect of the present invention, there is provided a method for identifying a key node based on graph signal analysis, the method comprising the steps of:
step S101: calculating the values of the graph signals of all nodes in the network;
step S102: grouping all nodes in the network according to the values of the graph signals of the nodes to form two grouping sub-networks;
step S103: obtaining a boundary network between grouping sub-networks;
step S104: and solving the minimal cut set of the boundary network as a key node set.
Further, the step S101: calculating the values of the graph signals of all nodes in the network, comprising:
calculating a node value matrix D in the network and an adjacent matrix A, wherein the node value matrix D comprises node values of all nodes in the network, namely the number of neighbor nodes of the nodes in the network; the adjacency matrix A represents the adjacency relation between nodes in the network, elements in the matrix represent vertexes in the network, if an edge exists between the nodes i and j, the ith row and the jth column elements in the adjacency matrix A are 1, otherwise, the jth column elements in the adjacency matrix A are 0, and the adjacency matrix gives the connection relation between the vertexes in the network;
the laplacian matrix corresponding to the network is L = D-a; performing feature decomposition on the Laplace matrix L:
L=U
UT （1）
wherein U is a matrix composed of L eigenvectors, and U = [ U =1,u2,…,uN], 
For diagonal matrices consisting of diagonal elements of eigenvalues in L, UTA transpose matrix representing U; for the ith eigenvector uiCorresponding characteristic value lambdaiIn the presence of λiL=uiL；
N is the number of the characteristic values;
all the obtained characteristic values are sorted according to the value, and the second smallest characteristic value is selected
λ2minCorresponding feature vector u2minAs a set of graph signals for all nodes in the network, the graph signal for each node k corresponds to a feature vector u2minThe kth component of (1).
Further, the step S102: according to the value of the graph signal of the node, all nodes in the network are grouped to form two grouping sub-networks, and the grouping sub-networks comprise:
presetting a grouping threshold value, wherein the grouping threshold value is lambda2minCorresponding feature vector u2minThe median among the components, the set criteria or criteria being the division of the sub-networks of packets by the grouping threshold;
when the value of the graph signal of the node is smaller than the grouping threshold, setting the corresponding label to be 1, otherwise, setting the label to be 0, and meeting the following conditions:
wherein g (i) is a label corresponding to the ith node, ucIs a grouping threshold, ucValue of u2minThe median of all graph signals;u l (i)is the value of the graph signal of the node;
after the labels of the nodes are obtained, the nodes with the labels of 1 are placed into the group A, and the rest nodes are placed into the group B.
Further, the step S103: determining a boundary network between packet subnetworks, comprising:
for any node v in the group Ai AIs traversed, i =1,2, …, NAIn which N isAIs the number of nodes in the A group of nodes, if vi AIf the node B belongs to the B group, the node v is judgedi AAnd node b is a boundary node, node vi AThe link between the node b and the node b is a boundary link;
the boundary nodes and the boundary links jointly form a boundary network, and the boundary network is a bipartite network;
the step S104: the step of obtaining the minimal cut set of the boundary network as a key node set comprises the following steps:
step S1041: adding two virtual nodes v in the border networkSAnd vTWherein v isSWith each node in group A there is a link, vTA link exists between the node and each node in the B group; initializing a key node set to be empty;
step S1042: judging whether a link exists in the boundary network; if yes, go to step S1043; if not, the step S1045 is carried out;
step S1043: calculating the node value of each node in the boundary network, sequencing all the nodes in the boundary network from large to small according to the node values, and deleting the node with the maximum node value in the boundary network and the edge connected with the node; recording the deleted nodes into a key node set;
step S1044: taking the current boundary network as the boundary network, and entering step S1042;
step S1045: and taking the key node set as a minimum cut set, wherein all nodes in the minimum cut set are key nodes.
According to a second aspect of the present invention, there is provided a key node identification apparatus based on graph signal analysis, the apparatus comprising:
a calculation module: configured to compute values of graph signals for all nodes in the network;
a grouping module: the network node grouping method comprises the steps that all nodes in a network are grouped according to values of graph signals of the nodes to form two grouping sub-networks;
a boundary network computing module: obtaining a boundary network between grouping sub-networks;
a key node acquisition module: and solving the minimal cut set of the boundary network as a key node set.
Further, the calculation module includes:
a first calculation submodule: the method comprises the steps that a node value matrix D in the network and an adjacent matrix A are configured to be calculated, wherein the node value matrix D comprises node values of all nodes in the network, namely the number of neighbor nodes of the nodes in the network; the adjacency matrix A represents the adjacency relation between nodes in the network, elements in the matrix represent vertexes in the network, if an edge exists between the nodes i and j, the ith row and the jth column elements in the adjacency matrix A are 1, otherwise, the jth column elements in the adjacency matrix A are 0, and the adjacency matrix gives the connection relation between the vertexes in the network;
the laplacian matrix corresponding to the network is L = D-a; performing feature decomposition on the Laplace matrix L:
L=U
UT （1）
wherein U is a matrix composed of L eigenvectors, and U = [ U =1,u2,…,uN], 
For diagonal matrices consisting of diagonal elements of eigenvalues in L, UTA transpose matrix representing U; for the ith eigenvector uiCorresponding characteristic value lambdaiIn the presence of λiL=uiL；
N is the number of the characteristic values;
a graph signal calculation submodule: all the obtained characteristic values are sorted according to the value, and the second small characteristic value lambda is selected2minCorresponding feature vector u2minAs a set of graph signals for all nodes in the network, the graph signal for each node k corresponds to a feature vector u2minThe kth component of (1).
Further, the grouping module includes:
set grouping threshold submodule: is configured as a preset grouping threshold value which is lambda2minCorresponding feature vector u2minThe median among the components, the set criteria or criteria being the division of the sub-networks of packets by the grouping threshold;
when the value of the graph signal of the node is smaller than the grouping threshold, setting the corresponding label to be 1, otherwise, setting the label to be 0, and meeting the following conditions:
wherein g (i) is a label corresponding to the ith node, ucIs a grouping threshold, ucValue of u2minThe median of all graph signals;u l (i)is the value of the graph signal of the node;
a grouping submodule: after the label of the node is obtained, the node with the label of 1 is placed into the group A, and the rest nodes are placed into the group B.
Further, the border network computing module includes:
traversing the sub-modules: configured to any node v in the A groupi AIs traversed, i =1,2, …, NAIn which N isAIs the number of nodes in the A group of nodes, if vi AIf the node B belongs to the B group, the node v is judgedi AAnd node b is a boundary node, node vi AAnd a link between node bThe way is a boundary link;
a boundary network acquisition submodule: configuring boundary nodes and boundary links to jointly form a boundary network, wherein the boundary network is a bipartite network;
the key node acquisition module includes:
initializing a submodule: configured to add two virtual nodes v in the border networkSAnd vTWherein v isSWith each node in group A there is a link, vTA link exists between the node and each node in the B group; initializing a key node set to be empty;
a judgment submodule: configured to determine whether a link exists in the border network;
a second calculation submodule: the method comprises the steps that a node value of each node in a boundary network is calculated, all nodes in the boundary network are sequenced from large to small according to the node values, and the node with the maximum node value in the boundary network and an edge connected with the node are deleted; recording the deleted nodes into a key node set;
and the boundary network updating submodule: configuring to take the current boundary network as the boundary network;
a key node set acquisition submodule: and configuring a key node set as a minimum cut set, wherein all nodes in the minimum cut set are key nodes.
According to a third aspect of the present invention, there is provided a key node identification system based on graph signal analysis, comprising:
a processor for executing a plurality of instructions;
a memory to store a plurality of instructions;
wherein the instructions are used for being stored by the memory and loaded and executed by the processor to perform the key node identification method based on graph signal analysis.
According to a fourth aspect of the present invention, there is provided a computer readable storage medium having a plurality of instructions stored therein; the instructions are used for loading and executing the key node identification method based on graph signal analysis.
According to the scheme of the invention, the key node identification of network fracture caused by failure can be realized.
The foregoing description is only an overview of the technical solutions of the present invention, and in order to make the technical solutions of the present invention more clearly understood and to implement them in accordance with the contents of the description, the following detailed description is given with reference to the preferred embodiments of the present invention and the accompanying drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention. In the drawings:
FIG. 1 is a flow chart of a method for identifying key nodes based on graph signal analysis according to an embodiment of the present invention;
FIG. 2 is a network topology diagram of one embodiment of the present invention;
FIG. 3 is a diagram illustrating a node grouping scenario according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a border network according to an embodiment of the present invention;
fig. 5 is a block diagram of a key node identification apparatus based on graph signal analysis according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be clearly and completely described below with reference to the specific embodiments of the present invention and the accompanying drawings. It is to be understood that the described embodiments are merely exemplary of the invention, and not restrictive of the full scope of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Defining:
CNDP: critical Node Detection Problem, key Node Detection Problem
DC: degree center, Degree Centrality
BC: betweenness centricity, mesomeric Centrality
CC: closeness center, near Centrality
EC: eigenvector center, feature vector Centrality
PR: PageRank algorithm
Network splitting: refers to a state in which a network is split into several sub-networks that cannot be connected to each other due to a failure of a single or partial node in the network.
Boundary nodes: referring to nodes that are critical to connectivity between two subnets, failure of a single or multiple border nodes often causes network fragmentation.
Boundary link: refers to a link connecting different subnets, and nodes connected at two ends of the link belong to different subnets.
Boundary network: and a network consisting of border nodes and border links.
Node degree value: the number of neighbor nodes of the nodes in the network is the size of the value.
And (3) neighbor nodes: if a connecting edge exists between two nodes, the two nodes are mutually adjacent nodes.
First, a flowchart of a key node identification method based on graph signal analysis according to an embodiment of the present invention is described with reference to fig. 1. As shown in fig. 1, the method comprises the steps of:
step S101: calculating the values of the graph signals of all nodes in the network;
step S102: grouping all nodes in the network according to the values of the graph signals of the nodes to form two grouping sub-networks;
step S103: obtaining a boundary network between grouping sub-networks;
step S104: and solving the minimal cut set of the boundary network as a key node set.
The step S101: calculating the values of the graph signals of all nodes in the network, comprising:
calculating a node value matrix D in the network and an adjacent matrix A, wherein the node value matrix D comprises node values of all nodes in the network, namely the number of neighbor nodes of the nodes in the network; the adjacency matrix A represents the adjacency relation between nodes in the network, elements in the matrix represent vertexes in the network, if an edge exists between the nodes i and j, the ith row and the jth column elements in the adjacency matrix A are 1, otherwise, the jth column elements in the adjacency matrix A are 0, and the adjacency matrix gives the connection relation between the vertexes in the network;
the laplacian matrix corresponding to the network is L = D-a; performing feature decomposition on the Laplace matrix L:
L=U
UT （1）
wherein U is a matrix composed of L eigenvectors, and U = [ U =1,u2,…,uN], 
For diagonal matrices consisting of diagonal elements of eigenvalues in L, UTA transpose matrix representing U; for the ith eigenvector uiCorresponding characteristic value lambdaiIn the presence of λiL=uiL；
N is the number of the characteristic values; in this embodiment, all eigenvalues in the matrix L are calculated, and generally the N-th order matrix has N eigenvalues, the ith eigenvector uiCorresponding characteristic value lambdai。
For all the obtained characteristic values lambdaiSorting according to the value size, and selecting the second smallest characteristic value
λ2minCorresponding feature vector u2minAs a set of graph signals for all nodes in the network, the graph signal for each node k corresponds to a feature vector u2minThe kth component of (1). Generally, the values of the components of the feature vector are different from each other.
In this embodiment, a laplacian matrix corresponding to the network is obtained according to the network topology, the laplacian matrix is subjected to feature decomposition, and feature vectors corresponding to the second small eigenvalue are selected as graph signals of all nodes in the network.
The step S102: according to the value of the graph signal of the node, all nodes in the network are grouped to form two grouping sub-networks, and the grouping sub-networks comprise:
presetting a grouping threshold value, wherein the grouping threshold value is lambda2minCorresponding feature vector u2minThe median number in the component, the criterion or criterion set, i.e. the division of the sub-network of packets by the packet threshold.
When the value of the graph signal of the node is smaller than the grouping threshold, setting the corresponding label to be 1, otherwise, setting the label to be 0, and meeting the following conditions:
wherein g (i) is a label corresponding to the ith node, ucFor grouping threshold, u in this embodimentcIs u2minThe median of all graph signals;u l (i)is the value of the graph signal of the node;
in this embodiment, the magnitude of each component value in the feature vector is compared in consideration of the value of each component in the feature vector.
After the labels of the nodes are obtained, the nodes with the labels of 1 are placed into the group A, and the rest nodes are placed into the group B.
In the embodiment, the graph signals of all the nodes are sequenced, and the median of the signal size is selected as a grouping threshold; and traversing the size of the graph signals corresponding to all the nodes, comparing the graph signals with a grouping threshold, putting the nodes corresponding to the signals larger than the grouping threshold into the group A, and putting the rest nodes into the group B.
The step S103: determining a boundary network between packet subnetworks, comprising:
for any node v in the group Ai AIs traversed, i =1,2, …, NAIn which N isAIs the number of nodes in the A group of nodes, if vi AIf the node B belongs to the B group, the node v is judgedi AAnd node b is a boundary node, node vi AThe link between the node b and the node b is a boundary link;
the boundary nodes and the boundary links jointly form a boundary network, and the boundary network is a bipartite network.
In this embodiment, the boundary nodes in the nodes are identified according to the grouping result, and all the boundary nodes and the boundary links are extracted as the boundary network.
The step S104: the step of obtaining the minimal cut set of the boundary network as a key node set comprises the following steps:
step S1041: adding two virtual nodes v in the border networkSAnd vTWherein v isSWith each node in group A there is a link, vTA link exists between the node and each node in the B group; initializing a key node set to be empty;
step S1042: judging whether a link exists in the boundary network; if yes, go to step S1043; if not, the step S1045 is carried out;
step S1043: calculating the node value of each node in the boundary network, sequencing all the nodes in the boundary network from large to small according to the node values, and deleting the node with the maximum node value in the boundary network and the edge connected with the node; recording the deleted nodes into a key node set;
in this embodiment, when the node values are the same, the node with the smaller sequence number is deleted.
Step S1044: taking the current boundary network as the boundary network, and entering step S1042;
step S1045: and taking the key node set as a minimum cut set, wherein all nodes in the minimum cut set are key nodes.
In this embodiment, the backbone nodes in the acquired physical topology are identified according to the rule sequence.
In this embodiment, the node with the largest node value in the boundary network is continuously deleted until all links in the boundary network are deleted, and the deleted node set is the key node identification result.
The technical effect of the key node identification method based on graph signal analysis is verified by using the EXata software to build a simulation environment:
step 1, setting up a scene: establishing an experimental scene by using EXata software;
step 2, basic parameters are configured: 44 nodes are randomly deployed in the scene by using a random deployment model, and relevant parameters are shown in table 1.
TABLE 1
The topology of the Ad Hoc network obtained according to the relevant parameter settings is shown in fig. 2.
According to the network topology in fig. 2, the graph signal size of the node is obtained, and 2-dimensional and 3-dimensional views of the network are obtained.
A schematic diagram of the node grouping situation obtained according to the graph signal size is shown in fig. 3.
And sorting according to the node values in the boundary network, continuously deleting the nodes with the large values in the network until the edges between the two groups of nodes do not exist, wherein the deleted node set [48,43,47,35,36,53,57 and 69] is the key node identification result. A schematic diagram of the border network is shown in fig. 4.
An embodiment of the present invention further provides a key node identification apparatus based on graph signal analysis, as shown in fig. 5, the apparatus includes:
a calculation module: configured to compute values of graph signals for all nodes in the network;
a grouping module: the network node grouping method comprises the steps that all nodes in a network are grouped according to values of graph signals of the nodes to form two grouping sub-networks;
a boundary network computing module: obtaining a boundary network between grouping sub-networks;
a key node acquisition module: and solving the minimal cut set of the boundary network as a key node set.
The embodiment of the invention further provides a key node identification system based on graph signal analysis, which comprises the following steps:
a processor for executing a plurality of instructions;
a memory to store a plurality of instructions;
wherein the instructions are used for being stored by the memory and loaded and executed by the processor to perform the key node identification method based on graph signal analysis.
The embodiment of the invention further provides a computer readable storage medium, wherein a plurality of instructions are stored in the storage medium; the instructions are used for loading and executing the key node identification method based on graph signal analysis.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
In the embodiments provided in the present invention, it should be understood that the disclosed system, apparatus and method may be implemented in other ways. For example, the above-described apparatus embodiments are merely illustrative, and for example, the division of the units is only one logical division, and there may be other divisions in actual implementation, for example, a plurality of units or components may be combined or integrated into another system, or some features may be omitted, or not executed. In addition, the shown or discussed mutual coupling or direct coupling or communication connection may be an indirect coupling or communication connection through some interfaces, devices or units, and may be in an electrical, mechanical or other form.
The units described as separate parts may or may not be physically separate, and parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of the embodiment.
In addition, functional units in the embodiments of the present invention may be integrated into one processing unit, or each unit may exist alone physically, or two or more units are integrated into one unit. The integrated unit can be realized in a form of hardware, or in a form of hardware plus a software functional unit.
The integrated unit implemented in the form of a software functional unit may be stored in a computer readable storage medium. The software functional unit is stored in a storage medium and includes several instructions to enable a computer device (which may be a personal computer, a physical machine Server, or a network cloud Server, etc., and needs to install a Windows or Windows Server operating system) to perform some steps of the method according to various embodiments of the present invention. And the aforementioned storage medium includes: various media capable of storing program codes, such as a usb disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk, or an optical disk.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention in any way, and any simple modification, equivalent change and modification made to the above embodiment according to the technical spirit of the present invention are still within the scope of the technical solution of the present invention.
Claims (10)
1. A key node identification method based on graph signal analysis is characterized by comprising the following steps:
step S101: calculating the values of the graph signals of all nodes in the network;
step S102: grouping all nodes in the network according to the values of the graph signals of the nodes to form two grouping sub-networks;
step S103: obtaining a boundary network between grouping sub-networks;
step S104: and solving the minimal cut set of the boundary network as a key node set.
2. The graph signal analysis-based key node identification method according to claim 1, wherein the step S101: calculating the values of the graph signals of all nodes in the network, comprising:
calculating a node value matrix D in the network and an adjacent matrix A, wherein the node value matrix D comprises node values of all nodes in the network, namely the number of neighbor nodes of the nodes in the network; the adjacency matrix A represents the adjacency relation between nodes in the network, elements in the matrix represent vertexes in the network, if an edge exists between the nodes i and j, the ith row and the jth column elements in the adjacency matrix A are 1, otherwise, the jth column elements in the adjacency matrix A are 0, and the adjacency matrix gives the connection relation between the vertexes in the network;
the laplacian matrix corresponding to the network is L = D-a; performing feature decomposition on the Laplace matrix L:
L=U
UT （1）
wherein U is a matrix composed of L eigenvectors, and U = [ U =1,u2,…,uN], 
For diagonal matrices consisting of diagonal elements of eigenvalues in L, UTA transpose matrix representing U; for the ith eigenvector uiCorresponding characteristic value lambdaiIn the presence of λiL=uiL；
N is the number of the characteristic values;
all the obtained characteristic values are sorted according to the value, and the second smallest characteristic value is selected
λ2minCorresponding feature vector u2minAs a set of graph signals for all nodes in the network, the graph signal for each node k corresponds to a feature vector u2minThe kth component of (1).
3. The graph signal analysis-based key node identification method according to claim 2, wherein the step S102: according to the value of the graph signal of the node, all nodes in the network are grouped to form two grouping sub-networks, and the grouping sub-networks comprise:
presetting a grouping threshold value, wherein the grouping threshold value is lambda2minCorresponding feature vector u2minThe median among the components, the set criteria or criteria being the division of the sub-networks of packets by the grouping threshold;
when the value of the graph signal of the node is smaller than the grouping threshold, setting the corresponding label to be 1, otherwise, setting the label to be 0, and meeting the following conditions:
wherein g (i) is a label corresponding to the ith node, ucIs a grouping threshold, ucValue of u2minThe median of all graph signals;u l (i)is the value of the graph signal of the node;
after the labels of the nodes are obtained, the nodes with the labels of 1 are placed into the group A, and the rest nodes are placed into the group B.
4. The graph signal analysis-based key node identification method according to claim 3, wherein the step S103: determining a boundary network between packet subnetworks, comprising:
for any node v in the group Ai AIs traversed, i =1,2, …, NAIn which N isAIs the number of nodes in the A group of nodes, if vi AIf the node B belongs to the B group, the node v is judgedi AAnd node b is a boundary node, node vi AThe link between the node b and the node b is a boundary link;
the boundary nodes and the boundary links jointly form a boundary network, and the boundary network is a bipartite network;
the step S104: the step of obtaining the minimal cut set of the boundary network as a key node set comprises the following steps:
step S1041: adding two virtual nodes v in the border networkSAnd vTWherein v isSWith each node in group A there is a link, vTA link exists between the node and each node in the B group; initializing a key node set to be empty;
step S1042: judging whether a link exists in the boundary network; if yes, go to step S1043; if not, the step S1045 is carried out;
step S1043: calculating the node value of each node in the boundary network, sequencing all the nodes in the boundary network from large to small according to the node values, and deleting the node with the maximum node value in the boundary network and the edge connected with the node; recording the deleted nodes into a key node set;
step S1044: taking the current boundary network as the boundary network, and entering step S1042;
step S1045: and taking the key node set as a minimum cut set, wherein all nodes in the minimum cut set are key nodes.
5. An apparatus for identifying key nodes based on graph signal analysis, the apparatus comprising:
a calculation module: configured to compute values of graph signals for all nodes in the network;
a grouping module: the network node grouping method comprises the steps that all nodes in a network are grouped according to values of graph signals of the nodes to form two grouping sub-networks;
a boundary network computing module: obtaining a boundary network between grouping sub-networks;
a key node acquisition module: and solving the minimal cut set of the boundary network as a key node set.
6. The graph signal analysis-based key node identification apparatus of claim 5, wherein the computation module comprises:
a first calculation submodule: the method comprises the steps that a node value matrix D in the network and an adjacent matrix A are configured to be calculated, wherein the node value matrix D comprises node values of all nodes in the network, namely the number of neighbor nodes of the nodes in the network; the adjacency matrix A represents the adjacency relation between nodes in the network, elements in the matrix represent vertexes in the network, if an edge exists between the nodes i and j, the ith row and the jth column elements in the adjacency matrix A are 1, otherwise, the jth column elements in the adjacency matrix A are 0, and the adjacency matrix gives the connection relation between the vertexes in the network;
the laplacian matrix corresponding to the network is L = D-a; performing feature decomposition on the Laplace matrix L:
L=U
UT （1）
wherein U is a matrix composed of L eigenvectors, and U = [ U =1,u2,…,uN], 
For diagonal matrices consisting of diagonal elements of eigenvalues in L, UTA transpose matrix representing U; for the ith eigenvector uiCorresponding characteristic value lambdaiIn the presence of λiL=uiL；
N is the number of the characteristic values;
a graph signal calculation submodule: all the obtained characteristic values are sorted according to the value, and the second small characteristic value lambda is selected2minCorresponding feature vector u2minAs a set of graph signals for all nodes in the network, the graph signal for each node k corresponds to a feature vector u2minThe kth component of (1).
7. The graph signal analysis-based key node identification apparatus of claim 6, wherein the grouping module comprises:
set grouping threshold submodule: is configured as a preset grouping threshold value which is lambda2minCorresponding feature vector u2minThe median among the components, the set criteria or criteria being the division of the sub-networks of packets by the grouping threshold;
when the value of the graph signal of the node is smaller than the grouping threshold, setting the corresponding label to be 1, otherwise, setting the label to be 0, and meeting the following conditions:
wherein g (i) is a label corresponding to the ith node, ucIs a grouping threshold, ucValue of u2minThe median of all graph signals;u l (i)is the value of the graph signal of the node;
a grouping submodule: after the label of the node is obtained, the node with the label of 1 is placed into the group A, and the rest nodes are placed into the group B.
8. The graph signal analysis-based key node identifying apparatus of claim 7,
the border network computing module comprises:
traversing the sub-modules: configured to any node v in the A groupi AIs traversed, i =1,2, …, NAIn which N isAIs the number of nodes in the A group of nodes, if vi AIf the node B belongs to the B group, the node v is judgedi AAnd node b is a boundary node, node vi AThe link between the node b and the node b is a boundary link;
a boundary network acquisition submodule: configuring boundary nodes and boundary links to jointly form a boundary network, wherein the boundary network is a bipartite network;
the key node acquisition module includes:
initializing a submodule: configured to add two virtual nodes v in the border networkSAnd vTWherein v isSWith each node in group A there is a link, vTA link exists between the node and each node in the B group; initializing a key node set to be empty;
a judgment submodule: configured to determine whether a link exists in the border network;
a second calculation submodule: the method comprises the steps that a node value of each node in a boundary network is calculated, all nodes in the boundary network are sequenced from large to small according to the node values, and the node with the maximum node value in the boundary network and an edge connected with the node are deleted; recording the deleted nodes into a key node set;
and the boundary network updating submodule: configuring to take the current boundary network as the boundary network;
a key node set acquisition submodule: and configuring a key node set as a minimum cut set, wherein all nodes in the minimum cut set are key nodes.
9. A graph signal analysis-based key node identification system, comprising:
a processor for executing a plurality of instructions;
a memory to store a plurality of instructions;
wherein the instructions are for being stored by the memory and loaded and executed by the processor to perform the graph signal analysis-based key node identification method of any one of claims 1-4.
10. A computer-readable storage medium having stored therein a plurality of instructions; the plurality of instructions for being loaded by a processor and for performing the method for graph signal analysis based key node identification according to any of claims 1-4.
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