CN113963195A - Method for identifying key nodes in complex network - Google Patents

Method for identifying key nodes in complex network Download PDF

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CN113963195A
CN113963195A CN202111124117.7A CN202111124117A CN113963195A CN 113963195 A CN113963195 A CN 113963195A CN 202111124117 A CN202111124117 A CN 202111124117A CN 113963195 A CN113963195 A CN 113963195A
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任晓龙
黄一鸣
吕琳媛
张符融
周玮
苏忠周
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Yangtze River Delta Research Institute of UESTC Huzhou
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Abstract

The invention discloses a method for identifying key nodes in a complex network, which comprises the following steps: (1) constructing a density operator rho (G) corresponding to the complex network according to the topology information of the network G; (2) calculating the spectral entropy S of the network Gτ(G) (ii) a (3) Constructing a disturbance network G generated by each nodex(ii) a (4) Spectral entropy of each perturbation network Sτ(Gx) Approximate calculation of (2); (5) and finally obtaining the entanglement degree E (x) of each node. The method not only considers the local structure and the topological information of the complex network, but also considers the global information of the network, and has simple calculation and wide application range.

Description

Method for identifying key nodes in complex network
Technical Field
The invention belongs to the technical field of complex networks, and relates to a method for identifying key nodes in a complex network.
Background
Many complex systems in the real world can be abstracted to networks with simple structures. In short, a network is composed of nodes abstracted from some elements actually existing in the system, and edges representing some kind of connection between such elements. Such as financial networks, biological systems, social systems, power and transportation systems, etc., can be abstracted into the structure of the network after limited processing, and the theory of complex networks is utilized to research and solve practical problems. In the last decade, the rapid development and application of complex network theory provides new methodology support for understanding and researching the world.
A critical node in a complex network refers to a small portion of special nodes that play a crucial role in the structure or function of the network. What vaccination strategies should be developed to prevent virus transmission in the face of a ravaged infectious disease? What species in the food chain system the change in survival status may have the greatest impact on the entire ecosystem? Which nodes of a complex network are disrupted will the functionality of the entire network be paralyzed? The method for identifying key nodes in the complex network provides a possible solution to the problems.
In the prior art, key nodes in a complex network are identified by using the measures of availability Centrality (Degree Centrality), Betweenness Centrality (Betweenness Centrality), proximity Centrality (Closeness Centrality) and the like.
The centrality method considers that more nodes can directly influence more neighbors, i.e. more nodes with more edges associated with the node. The betweenness centrality of a node, which describes the degree of distribution of a vertex on paths between other vertices, is defined as the sum of the weights of all geodetic paths (shortest paths) through the node. The betweenness centrality method considers that nodes with larger betweenness are more critical. The proximity centrality approach considers that a node is more critical the smaller the average distance it is from other nodes in the network.
However, these existing methods only focus on local structure or partial topology information of a complex network, and do not consider global information of the network.
Disclosure of Invention
The invention provides a method for identifying key nodes in a complex network, and aims to overcome the defects in the prior art.
In order to achieve the purpose, the invention adopts the technical scheme that:
a method for identifying key nodes in a complex network comprises the following steps:
(1) constructing a density operator rho (G) corresponding to the complex network according to the topology information of the network G;
(2) calculating the spectral entropy S of the network Gτ(G)；
(3) Constructing a perturbation network G generated by each node in the networkx；
(4) Spectral entropy of each perturbation network Sτ(Gx) Approximate calculation of (2);
(5) and finally obtaining the entanglement degree E (x) of each node.
Further, the step (1) of constructing a density operator ρ (G) corresponding to the complex network according to the topology information of the network G includes the following steps:
1) acquiring structural data of a topological network G (V, E), wherein the network G comprises an unauthorized network or a weighted network, the V is a set of all nodes in the network, and the E represents a set of all links in the network;
2) computing an adjacency matrix A of the network G, a of the element of the (i, j) th position in said adjacency matrix A if and only if (i, j) ∈ E is linkedijNot 0, if there are N nodes in the network, thenThe adjacent matrix A is an N-dimensional square matrix;
3) a graph laplacian matrix L of the network G is calculated,
L＝D-A；
where D is the diagonal matrix and the diagonal elements are the degrees of the nodes, i.e.
Figure BDA0003278209480000031
Wherein k isiRepresenting the degree of the node i, namely the number of links related to the node i, wherein N is the dimension of the network;
4) density operator rho (G) of computing network G
The density operator ρ (G) corresponding to the network G can be expressed as:
Figure BDA0003278209480000032
wherein, the distribution function Z ═ tr (e)-τL) And tr (·) is a trace-taking function.
Further, the step (2) of calculating the spectral entropy S of the network Gτ(G) The method comprises the following steps:
1) the graph Laplace matrix L is spectrally decomposed, i.e.
Figure BDA0003278209480000033
Wherein λ isiAnd viRespectively are the eigenvalue of L and the corresponding eigenvector, and N represents the dimension of the network;
2) the calculation process of the spectral entropy can be simplified by utilizing the spectral decomposition theorem, and the spectral entropy S can be obtained by simplificationτ(G) The expression of (a) is:
Figure BDA0003278209480000034
wherein the distribution function
Figure BDA0003278209480000035
Further, the step (3) is to construct a disturbance network G generated by each nodexThe construction mode comprises the following steps:
a subgraph formed by the node x and the neighbor nodes becomes a complete probability graph, and other link relations in the network are kept unchanged; link probabilities of the probabilistic full graph
Figure BDA0003278209480000036
Obtaining a simplified form of link probabilities for an unauthorized network
Figure BDA0003278209480000037
CxIs the clustering coefficient of node x;
and executing the same operation on each node in the network to obtain N disturbance networks.
Further, the spectral entropy S of each perturbation network in the step (4)τ(Gx) Comprises the following steps:
1) calculating each perturbation network GxCharacteristic value of
For large scale networks, note that each perturbation network GxThe graph laplacian matrix and the original network G have only a few items of slight disturbance, which is recorded as Δ L (Δ L-L'), in order to reduce the complexity of the algorithm, an approximation method is used to calculate the eigenvalues of the disturbed network and the corresponding spectral entropy, and the approximation calculation process is as follows:
eigenvalue λ 'of graph Laplacian matrix of disturbance network'iThe expression of (L) is:
λ′i(L)＝v′iTL′v′i；
wherein, v'iIs lambda'iCorresponding feature vectors, since L' and L have only a few terms changed, use viApproximately instead of v'iIt is possible to obtain:
Figure BDA0003278209480000041
in particular, it is noted that Δ L has only a few non-degenerate terms, so only these terms need to be calculated;
disturbance network GxThe eigenvalues of the corresponding density operators ρ can be approximately expressed as:
Figure BDA0003278209480000045
wherein the distribution function of the disturbance network
Figure BDA0003278209480000042
2) Computing disturbance network GxSpectral entropy ofSτ(Gx)
Substituting the characteristic value of the disturbance network into the definition formula of the spectral entropy to obtain
Figure BDA0003278209480000043
Wherein the distribution function of the disturbance network
Figure BDA0003278209480000044
Further, the finally obtaining the entanglement degree e (x) of each node in the step (4) includes:
defining the entanglement degree of the node x as follows:
ET(x)＝Sτ(Gx)-Sτ(G)；
wherein, the propagation time tau is an optional parameter. When the connectivity of the disturbed network is not influenced by the parameter τ as much as possible, the parameter τ should be selected in such a way that the entanglement of the node is minimal, in which case the entanglement e (x) of the node x can be expressed as:
Figure BDA0003278209480000051
compared with the prior art, the invention has the following advantages and beneficial effects:
the invention is inspired by quantum information theory, creatively provides the concept of node entanglement in the complex network, and utilizes the node entanglement to identify key nodes in the complex network. The method not only considers the local structure and topology information of the complex network, but also considers the global information of the network, and has simple calculation and wide application range.
The invention can be applied to network attack, and the network can be quickly decomposed into a plurality of communication networks with smaller number of nodes when a small number of nodes with stronger entanglement in the network are attacked, which can cause the breakdown of the whole network, namely when a small number of nodes with stronger entanglement are deleted from the network. The invention can also be used in the aspects of network marketing, key crowd vaccination, large-scale power grid performance analysis and the like.
In particular, the invention may be applied to the identification of key brain regions of autistic patients, i.e. the identification of key nodes in a functional brain network of autistic patients. In addition, the identification method of the key nodes in the complex network provided by the invention can be applied to the behavior prediction of the autism patient, and the prediction method is as follows. Firstly, respectively calculating entanglement degrees of all nodes in functional brain networks of autistic patients and control groups (namely normal testees), and recording the entanglement degrees as yi(i ═ 1, 2, …, N); then, calculating the mean value of the entanglement degree of each node of the normal testee, and recording as xi(i ═ 1, 2, …, N); then, a set of points (x)i，yi-xi) Linear fitting is performed to obtain the slope k. The index k can reflect the topological abnormal condition of the functional brain network of the testee, the experiment also finds that the k values of the autism patient and the normal testee are obviously different (the k value of the autism patient is obviously lower than that of the normal testee), and the index k has strong positive correlation with the intelligence quotient of the autism patient.
The invention has wide application range, including and not limited to the application cases listed above.
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FIG. 1 is a flow chart of a method of identifying key nodes in a complex network in accordance with the present invention;
FIG. 2 is a schematic diagram of the construction of the perturbation network of the present invention.
Detailed Description
In order to make the objects, technical problems to be solved, and technical solutions of the present invention clearer, the present invention is further described below with reference to the accompanying drawings and specific embodiments.
First, method step
Aiming at the defect that the prior art only utilizes local or partial information of a complex network, the invention provides a key node identification method considering the global information of a network structure, as shown in figure 1, and the detailed steps are as follows:
1. constructing a density operator rho (G) corresponding to the complex network according to the topology information of the network G (corresponding to the steps 1-2 of the flow chart)
(1) Structural data of a topological network G (V, E) is obtained, wherein the network G can be an unauthorized network or a privileged network, the V is a set of all nodes in the network, and the E represents a set of all links in the network.
(2) An adjacency matrix a for the network G is computed. A of the element of the (i, j) th position in the adjacency matrix A if and only if (i, j) ∈ E is linkedijIs not 0. If there are N nodes in the network, the adjacency matrix A is an N-dimensional square matrix.
(3) And calculating a graph Laplace matrix L of the network G in the following way:
L＝D-A；
where D is the diagonal matrix and the diagonal elements are the degrees of the nodes, i.e.
Figure BDA0003278209480000061
Wherein k isiRepresenting the degree of node i, i.e., the number of links associated with node i, and N is the dimension of the network.
(4) Density operator rho (G) of computing network G
The density operator ρ (G) corresponding to the network G can be expressed as:
Figure BDA0003278209480000071
wherein, the distribution function Z ═ tr (e)-τL) And tr (·) is a trace-taking function.
2. Calculating the spectral entropy S of the network Gτ(g) (corresponding to Steps 3-4 of the flow chart)
With the density operator ρ (G) of the network, the network spectral entropy can be defined as:
Sτ(G)＝Sτ(ρ)＝-tr(ρlog2p)；
where the parameter τ physically represents the diffusion time of the network. The entropy of the traditional complex network is mainly defined based on partial information of the network, and other information is ignored; spectral entropy treats a complex system from a global perspective of the network.
Direct calculation of spectral entropy S with definitionτ(G) The invention simplifies the G spectrum entropy S of the networkτ(G) The detailed steps of the specific calculation process are as follows.
(1) The graph Laplace matrix L is spectrally decomposed, i.e.
Figure BDA0003278209480000072
Wherein λ isiAnd viThe eigenvalues and corresponding eigenvectors, respectively, of L, with N representing the dimensionality of the network.
(2) The calculation process of the spectral entropy can be simplified by utilizing the spectral decomposition theorem, and the spectral entropy S can be obtained by simplificationτ(the expression of G is:
Figure BDA0003278209480000073
wherein the distribution function
Figure BDA0003278209480000074
3. Constructing a perturbation network G generated by each node in the networkx(corresponding to step 5 of the flow chart)
GxDenotes the network formed after removing node x in network G, thus GxReferred to as a "perturbation network". One of the simplest configurations is to delete node x directly from the network without any further adjustment, which would disturb the network by the number of nodes N, links M, and the degree of averaging<k>Etc. the basic properties of the network change. More reasonable way requires GxThe structural features of the original network G are maintained as much as possible.
Based on the thought, the invention does not directly remove the node x, but selects to change the subgraph formed by the node x and the neighbor nodes thereof into a probabilistic complete graph, and the link relation among other nodes is kept unchanged.
The probability complete graph represents the complete graph with probability links between any two nodes, namely the weight w of the links between any two nodes satisfies 0 ≦ w ≦ 1 (unweighted network), or wmin≤w≤wmax(network of ownership, wminAnd wmaxRepresenting the minimum and maximum link weights in the weighted network, respectively).
Disturbance network GxIs schematically shown in fig. 2.
The invention requires that a subgraph formed by the node x and the neighbor nodes becomes a complete probability graph, and other link relations in the network are kept unchanged; link probabilities of the probabilistic full graph
Figure BDA0003278209480000081
Simplified forms of link probabilities are available for unlicensed networks
Figure BDA0003278209480000082
CxIs the clustering coefficient of node x.
The same operation is performed for each node in the network, and N perturbation networks can be obtained.
4. Spectral entropy of each perturbation network Sτ(Gx) Approximate calculation of (corresponding to the steps 6-7 of the flow chart)
(1) Calculating each perturbation network GxIs characterized byValue of
For large scale networks, note that each perturbation network GxThe graph laplacian matrix of (a) and the original network G have only a few entries with slight disturbance, which is recorded as Δ L (Δ L ═ L-L'). In order to reduce the algorithm complexity, the invention uses an approximate method to calculate the characteristic value of the disturbance network and the corresponding spectral entropy, and the approximate calculation process is as follows.
Eigenvalue λ 'of graph Laplacian matrix of disturbance network'iThe expression of (L) is:
λ′i(L)＝v′i TL′v′i；
wherein, v'iIs lambda'iThe corresponding feature vector. Since L' and L have only a few changes, use viApproximately instead of v'iIt is possible to obtain:
Figure BDA0003278209480000083
in particular, it is noted that Δ L has only a few non-degenerate terms (non-0 terms), so only these terms need to be calculated.
Disturbance network GxThe eigenvalues of the corresponding density operators ρ can be approximately expressed as:
Figure BDA0003278209480000095
wherein the distribution function of the disturbance network
Figure BDA0003278209480000091
(2) Computing disturbance network GxSpectral entropy S ofτ(Gx)
Substituting the characteristic value of the disturbance network into the definitional expression of the spectrum entropy can obtain:
Figure BDA0003278209480000092
wherein the distribution function of the disturbance network
Figure BDA0003278209480000093
5. Finally, the entanglement degree E (x) of each node is obtained (corresponding to the step 8 of the flow chart)
Defining the entanglement degree of the node x as follows:
Eτ(x)＝Sτ(Gx)-Sτ(G)；
wherein, the propagation time tau is an optional parameter. Because the invention identifies the key nodes in the network from the perspective of the network structure, and the disturbance of the network connectivity is hoped to be not influenced by the parameter tau as much as possible, the selection mode of the parameter tau recommended by the invention should minimize the entanglement degree of the nodes. In this case, the entanglement level e (x) of the node x may be expressed as:
Figure BDA0003278209480000094
when the entanglement degree of the first node is calculated, the parameter tau can be obtained through an optimization algorithm*(ii) a When calculating the entanglement degrees of other nodes, the tau of one other node can be randomly selected*As an initial value and optimized.
It should be noted that the entanglement degrees of all the nodes are negative by definition, so that the smaller the entanglement degree is in value, indicating that the entanglement is stronger. Correspondingly, the nodes with stronger entanglement are more critical in the network; in short, E (numerically smaller x nodes are more critical in the network).
Second, discussion analysis
1. Physical interpretation of the partition function Z
On the one hand, τ → ∞ time,
Figure BDA0003278209480000101
c represents the number of communication branches of the network, indicating that Z is related to the connectivity of the network. On the other hand, the allocation function Z and any node can return to the initial position at time τThe average probabilities are directly proportional, with higher average return probabilities indicating that the network structure tends to trap the flow in the initial location and that it is difficult for a random walker to effectively explore the rest of the network. In addition, Z may also be increased by the deletion of links or nodes in the network. The above analysis shows that a larger value of Z reflects a poorer connectivity of the network.
2. Entanglement degree of nodeτ(x) Multi-scale analysis of
Entanglement E of node xτ(x) Influenced by the parameter tau, for tau with different scales, the node entanglement degree is different in form:
on the microscopic scale, i.e., when τ → 0,
Eτ(x)＝Sτ(Gx)-Sτ(c)＝0。
on a macroscopic scale, i.e. when τ → ∞,
Figure BDA0003278209480000102
wherein C represents the number of connected branches of the original network, CxRepresenting the number of communication branches of the perturbed network of node x. If the initial network is a connected network, Eτ(x)＝log2Cx(τ→∞)。
On an intermediate scale, i.e. τ ═ τm，0＜τmWhen the temperature is less than infinity, the temperature is controlled,
Figure BDA0003278209480000103
where Δ Z represents the distribution function Z of the disturbance networkxDifference from the distribution function Z of the original network, i.e. Δ Z ═ Zx-Z。
The micro, macro and intermediate scale analysis of the propagation time tau shows that the entanglement degree of the nodes in the network is closely related to the structure of the network, and the importance of the nodes can be measured from the network structure.
3. Optimization parameter tau*Is proved by the existence of
According to the stepsStep 5, parameter τ selected by the invention*It should satisfy:
Figure BDA0003278209480000111
an important problem is that of*Whether or not, tau will be demonstrated below*Presence of (a).
Order to
Figure BDA0003278209480000112
Since the function f is continuous and f (0+), 1, let
Figure BDA0003278209480000114
Then τ > 0 is present such that f (τ) < γ. At this time, the process of the present invention,
Eτ(x)＝Sτ(Gx)-Sτ(G)＜0，
the inequality is true because the arbitrary eigenvalues λ of the graph laplacian matrix L are satisfiedi(L) all satisfy the inequality
Figure BDA0003278209480000113
And because when τ → ∞ is reached, Eτ(x)＝log2CxIs more than or equal to 0. So for any node x, Eτ(x) There must be a minimum in the interval τ e (0, γ, i.e. τ*Are present. After the syndrome is confirmed.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art, having the benefit of this disclosure, may effect numerous modifications thereto and changes may be made without departing from the scope of the invention in its broader aspects.

Claims (6)

1. A method for identifying key nodes in a complex network is characterized by comprising the following steps:
(1) constructing a density operator rho (G) corresponding to the complex network according to the topology information of the network G;
(2) calculating the spectral entropy S of the network Gτ(G)；
(3) Constructing a perturbation network G generated by each node in the networkx；
(4) Spectral entropy of each perturbation network Sτ(Gx) Approximate calculation of (2);
(5) and finally obtaining the entanglement degree E (x) of each node.
2. The method for identifying key nodes in a complex network according to claim 1, wherein the step (1) of constructing a density operator p (G) corresponding to the complex network according to the topology information of the network G comprises the following steps:
1) acquiring structural data of a topological network G (V, E), wherein the network G comprises an unauthorized network and a weighted network, the V is a set of all nodes in the network, and the E represents a set of all links in the network;
2) computing an adjacency matrix A of the network G, a of the element of the (i, j) th position in said adjacency matrix A if and only if (i, j) ∈ E is linkedijIf the number of the nodes in the network is not 0, the adjacency matrix A is an N-dimensional square matrix;
3) and calculating a graph Laplace matrix L of the network G in the following way:
L＝D-A；
where D is the diagonal matrix and its diagonal elements are the degrees of the nodes, i.e.
Figure FDA0003278209470000011
Wherein k isiRepresenting the degree of the node i, namely the number of links related to the node i, wherein N is the dimension of the network;
4) density operator rho (G) of computing network G
The density operator ρ (G) corresponding to the network G can be expressed as:
Figure FDA0003278209470000012
wherein the partition function z ═ tr (e)-τL) And tr (·) is a trace-taking function.
3. The method for identifying key nodes in complex network according to claim 1, wherein the spectral entropy S of the network G is calculated in step (2)τ(G) The method comprises the following steps:
1) the graph Laplace matrix L is spectrally decomposed, i.e.
Figure FDA0003278209470000021
Wherein λ isiAnd viRespectively are the eigenvalue of L and the corresponding eigenvector, and N represents the dimension of the network;
2) the calculation process of the spectral entropy can be simplified by utilizing the spectral decomposition theorem, and the spectral entropy S can be obtained by simplificationτ(G) The expression of (a) is:
Figure FDA0003278209470000022
wherein the distribution function
Figure FDA0003278209470000023
4. The method according to claim 1, wherein the step (3) is implemented by constructing a perturbation network G generated by each nodexThe construction mode comprises the following steps:
a subgraph formed by the node x and the neighbor nodes becomes a complete probability graph, and other link relations in the network are kept unchanged; linking of the probabilistic full graphProbability of
Figure FDA0003278209470000024
Obtaining a simplified form of link probabilities for an unauthorized network
Figure FDA0003278209470000025
CxIs the clustering coefficient of node x;
and executing the same operation on each node in the network to obtain N disturbance networks.
5. Method for identifying key nodes in a complex network according to claim 1, wherein the spectral entropy S of each perturbation network in step (4)τ(Gx) Comprises the following steps:
1) calculating each perturbation network GxCharacteristic value of
For large scale networks, note that each perturbation network GxThe graph laplacian matrix and the original network G have only a few items of slight disturbance, which is recorded as Δ L (Δ L-L'), in order to reduce the complexity of the algorithm, an approximation method is used to calculate the eigenvalues of the disturbed network and the corresponding spectral entropy, and the approximation calculation process is as follows:
eigenvalue λ 'of graph Laplacian matrix of disturbance network'iThe expression of (L) is:
λ′i(L)＝v′i TL′v′i；
wherein, v'iIs lambda'iCorresponding feature vectors, since L' and L have only a few terms changed, use viApproximately instead of v'iIt is possible to obtain:
Figure FDA0003278209470000031
notably, Δ L is noted to have only a few non-degenerate terms (non-0 terms), so only these terms need to be calculated;
disturbance network GxThe eigenvalues of the corresponding density operators ρ can be approximately expressed as:
Figure FDA0003278209470000032
wherein the distribution function of the disturbance network
Figure FDA0003278209470000033
2) Computing disturbance network GxSpectral entropy S ofτ(Gx)
Substituting the characteristic value of the disturbance network into the definitional expression of the spectrum entropy can obtain:
Figure FDA0003278209470000034
wherein the distribution function of the disturbance network
Figure FDA0003278209470000035
6. The method for identifying key nodes in a complex network according to claim 1, wherein the finally obtaining the entanglement degree E (x) of each node in the step (4) comprises:
defining the entanglement degree of the node x as follows:
Eτ(x)＝Sτ(Gx)-Sτ(G)；
wherein, the propagation time tau is an optional parameter; when the connectivity of the disturbed network is not influenced by the parameter τ as much as possible, the parameter τ should be selected in such a way that the entanglement of the node is minimal, in which case the entanglement e (x) of the node x can be expressed as:
Figure FDA0003278209470000041
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