High-order network vulnerability analysis method based on hypergraph

Abstract

The invention provides a high-order network vulnerability analysis method based on a hypergraph, relates to the technical field of systems and information, newly constructs a hypergraph network model with adjustable degree distribution as a bottom layer topology, and provides a novel vulnerability index suitable for a hypergraph network through theoretical analysis. And the relation between hypergraph degree distribution heterogeneity and vulnerability is disclosed. According to the method, high-order and randomness characteristics of interaction of an actual network system are fully considered, the constructed degree distribution adjustable hypergraph network model can be used for describing an ecological network, a social network and a traffic system, and the provided novel vulnerability analysis method can effectively evaluate the ability of the actual network system to deal with external interference. The method is wider in application range in practical application, and particularly has important technical value in the aspects of network security, survivability design, risk prediction and the like.

Description

Technical Field

[0001] The present invention relates to the field of system and information technology, and in particular to a high-order network vulnerability analysis method based on a hypergraph. Background Art

[0002] With the continuous deepening of complex network research, the traditional first-order network model can only describe the binary interaction relationship between nodes, but it is difficult to reflect the complex and diverse high-order interaction structure in the actual system. Therefore, the theory of high-order network has gradually become an important direction of complex network research, among which the hypergraph model has attracted much attention because it can naturally represent multivariate interaction relationships. By expanding the edges into "hyperedges" connecting multiple nodes, the hypergraph can more comprehensively reveal the high-order associations between nodes. However, the complexity of high-order networks also makes its fragility analysis face severe challenges. Fragility analysis is an important method to evaluate the performance and functional degradation of the network under structural failure, external interference or dynamic changes, and has important application value in communication networks, transportation systems, biological networks and other fields.

[0003] At present, most of the research on high-order networks focuses on the description of structural characteristics and dynamic behaviors, while there are still limited methods for systematic analysis of the vulnerability of high-order networks. Traditional complex network vulnerability analysis methods, such as indicators based on degree distribution, node importance or connectivity evaluation, are difficult to directly apply to high-order networks because the diversity and randomness of high-order interactions increase the complexity of the analysis. Therefore, it is urgent to combine hypergraph theory to develop a vulnerability analysis method suitable for high-order networks to reveal the impact of high-order interaction structures on the overall stability and robustness of the network, and provide theoretical basis and technical support for network design, optimization and security assurance. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a high-order network vulnerability analysis method based on a hypergraph; the present invention newly constructs a hypergraph network model with adjustable degree distribution as the underlying topology, proposes a new vulnerability index suitable for hypergraph networks through theoretical analysis, and reveals the relationship between the heterogeneity of hypergraph degree distribution and vulnerability. Compared with the prior art, the present invention fully considers the high-order and random characteristics of the interaction of actual network systems, and the constructed hypergraph network model with adjustable degree distribution can be used to describe ecological networks, social networks and transportation systems. The vulnerability analysis method provided can effectively evaluate the ability of actual network systems to cope with external interference, and has a wider range of applications in practical applications.

[0005] A high-order network vulnerability analysis method based on hypergraph, the specific steps are as follows:

[0006] Step 1: Construct a high-order network system consisting of N nodes;

[0007] Construct a high-order network system consisting of N nodes. The system dynamics equation is expressed by the following ordinary differential equation:

[0008]

[0009] Where X(t) = [X 1 (t),X 2 (t),…,X N (t)] T ∈R N represents the state vector of all nodes in the entire high-order network at time t, X N (t) represents the state of the Nth node, the superscript T represents transposition, and R N represents a set of N-dimensional real vectors; diag(X(t)) represents a diagonal matrix with diagonal elements X 1 (t),X 2 (t),…,X N (t), the rest of the elements are 0; the nonlinear function f(X(t)) represents the interaction relationship between nodes at time t; assuming that the equilibrium point of the high-order network system (1) is Satisfy the condition f(X * )=0, Indicates the equilibrium point X * The Nth component of , 0 represents a column vector whose elements are all 0; According to the linear system theory, the following linear system is obtained:

[0010]

[0011] Where x(t) = X(t) - X * , M is the high-order interaction matrix between nodes, defined as M = diag(X * )J, where diag(X * ) means the diagonal elements are The diagonal matrix of , J is the nonlinear function f(X(t)) at the equilibrium point X * The Jacobian matrix of the matrix M is M. ij , M ij Represents the element in the i-th row and j-th column of M, reflecting the effect of node j on node i;

[0012] Step 2: Propose a new vulnerability indicator;

[0013] According to the linear system theory, the condition for the linear system (2) to be locally asymptotically stable at the equilibrium point is Re(λ M,1 )<0, where λ M,1Re(λ M,1 ) represents the characteristic root λ M,1 The real part of the new vulnerability index is defined as

[0014] F r =1 / |Re(λ M,1 )| (3)

[0015] Among them, F r represents the vulnerability index, λ M,1 Re(λ M,1 ) represents the characteristic root λ M,1 The real part of , the symbol |·| represents the absolute value;

[0016] Step 3: Construct a hypergraph network with adjustable degree distribution As the underlying topology map;

[0017] Step 3.1: Based on the given total number of nodes N, the number of nodes contained in each hyperedge r, and the average degree of the expected nodes <k>, calculate the target hyperedge number E, where E = N <k> / r;

[0018] Step 3.2: According to the number of nodes on each hyperedge, randomly select a specified number of nodes and put them into the hyperedge;

[0019] Step 4: Determine the high-order interaction matrix M based on the generated hypergraph network structure;

[0020] Step 4.1: Calculate the interaction probability between any two nodes i and j Among them, k i and k j denote the degrees of node i and node j respectively, <k>is the average degree, N is the total number of nodes, and r is the number of nodes contained in each hyperedge;

[0021] Step 4.2: Generate a high-order interaction matrix M based on the interaction probability, where the element M of the high-order interaction matrix M is ij With probability P ij From the normal distribution is drawn from the ij Make M ij =0; symbol Indicates that the mean of the distribution is 0 and the variance is σ 2 ;

[0022] Step 5: Hypergraph network based on adjustable degree distribution , combined with the vulnerability index F r , analyze the impact of degree distribution heterogeneity on vulnerability;

[0023] Specifically: By adjusting the expected average node degree <k>To adjust the degree of heterogeneity of the degree distribution, the heterogeneity is represented by the symbol ξ = <k 2 > / <k> 2 It indicates that, <k 2 > represents the mean of the square of the node degree, <k> 2 Represents the square of the average degree; hypergraph network model based on adjustable degree distribution , combined with the vulnerability index F r ,The impact of degree distribution heterogeneity on vulnerability is analyzed from a simulation perspective.

[0024] The beneficial effects of adopting the above technical solution are:

[0025] The present invention provides a high-order network vulnerability analysis method based on a hypergraph, which specifically includes the following beneficial effects:

[0026] (1) Traditional network models mostly analyze network structures based on graphs of first-order relationships (nodes and edges), while this method incorporates higher-order relationships (such as multilateral associations between node groups) into the analysis scope through a hypergraph model, which can more comprehensively characterize the structural characteristics of actual complex network systems, thereby more accurately evaluating the vulnerability of the network; (2) The vulnerability analysis method based on the hypergraph can not only evaluate the impact of a single node or a single connection, but also consider the impact of the failure of a node group or substructure on the entire network, realizing multi-dimensional risk assessment from local to global, and providing more comprehensive data support for the optimal design of complex networks. This method is applicable to various types of complex networks, including communication networks, transportation networks, social networks, and ecological networks. In practical applications, it can provide a unified theoretical framework for network security and robustness analysis in different industries. In summary, the present invention provides advanced tools for high-order structural analysis of complex networks, helps promote theoretical research and practical applications in related fields, and has important technical value in network security, anti-destruction design, and risk prediction. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 A flow chart of a high-order network vulnerability analysis method based on a hypergraph provided for the implementation of the present invention;

[0028] Figure 2 A relationship diagram between the degree distribution heterogeneity and the average degree of the hypergraph network in the implementation of the present invention;

[0029] Figure 3 A diagram showing the relationship between the vulnerability of a hypergraph network and the degree of heterogeneity of its degree distribution in the implementation of the present invention. DETAILED DESCRIPTION

[0030] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0031] A high-order network vulnerability analysis method based on hypergraph, such as Figure 1 As shown, the specific steps are as follows:

[0032] Step 1: Construct a high-order network system consisting of N nodes;

[0033] Construct a high-order network system consisting of N nodes. The system dynamics equation is expressed by the following ordinary differential equation:

[0034]

[0035] Where X(t) = [X 1 (t),X 2 (t),…,X N (t)] T ∈R N represents the state vector of all nodes in the entire high-order network at time t, X N (t) represents the state of the Nth node, the superscript T represents transposition, and R N represents a set of N-dimensional real vectors; diag(X(t)) represents a diagonal matrix with diagonal elements X 1 (t),X 2 (t),…,X N (t), the rest of the elements are 0; the nonlinear function f(X(t)) represents the interaction relationship between nodes at time t; assuming that the equilibrium point of the high-order network system (1) is Satisfy the condition f(X * )=0, Indicates the equilibrium point X * The Nth component of , 0 represents a column vector whose elements are all 0; According to the linear system theory, the following linear system is obtained:

[0036]

[0037] Where x(t) = X(t) - X * , M is the high-order interaction matrix between nodes, defined as M = diag(X * )J, where diag(X * ) means the diagonal elements are The diagonal matrix of , J is the nonlinear function f(X(t)) at the equilibrium point X * The Jacobian matrix of the matrix M is M. ij , M ij Represents the element in the i-th row and j-th column in M, reflecting the effect of node j on node i.

[0038] In this embodiment, consider an ecological network system, where nodes represent species, X i (t) represents the abundance of species i at time t; the nonlinear function f(X(t)) represents the interaction relationship between species at time t, and the element M of the matrix M ij represents the impact of species j on species i near the equilibrium point;

[0039] Step 2: Propose a new vulnerability indicator;

[0040] According to the linear system theory, the condition for the linear system (2) to be locally asymptotically stable at the equilibrium point is Re(λ M,1 )<0, where λ M,1 Re(λ M,1 ) represents the characteristic root λ M,1 The real part of the new vulnerability index is defined as

[0041] F r =1 / |Re(λ M,1 )| (3)

[0042] Among them, F r Represents the vulnerability index. The larger the index value, the more fragile the linear system is. M,1 Re(λ M,1 ) represents the characteristic root λ M,1 The real part of , the symbol |·| represents the absolute value;

[0043] Step 3: Construct a hypergraph network with adjustable degree distribution As the underlying topology map;

[0044] Step 3.1: Based on the given total number of nodes N, the number of nodes contained in each hyperedge r, and the average degree of the expected nodes <k>, calculate the target hyperedge number E, where E = N <k> / r;

[0045] Step 3.2: According to the number of nodes on each hyperedge, randomly select a specified number of nodes and put them into the hyperedge;

[0046] Step 4: Determine the high-order interaction matrix M based on the generated hypergraph network structure;

[0047] Step 4.1: Calculate the interaction probability between any two nodes i and j Among them, k i and k j denote the degrees of node i and node j respectively, <k>is the average degree, N is the total number of nodes, and r is the number of nodes contained in each hyperedge;

[0048] Step 4.2: Generate a high-order interaction matrix M based on the interaction probability, where the element M of the high-order interaction matrix M is ij With probability P ij From the normal distribution is drawn from the ij Make M ij =0; symbol Indicates that the mean of the distribution is 0 and the variance is σ 2 ;

[0049] Step 5: Hypergraph network based on adjustable degree distribution , combined with the vulnerability index F r , analyze the impact of degree distribution heterogeneity on vulnerability;

[0050] Specifically: By adjusting the expected average node degree <k>To adjust the degree of heterogeneity of the degree distribution, the heterogeneity is represented by the symbol ξ = <k 2 > / <k> 2 It indicates that, <k 2 > represents the mean of the square of the node degree, <k> 2 Represents the square of the average degree; hypergraph network model based on adjustable degree distribution , combined with the vulnerability index F r ,From the simulation perspective, the impact of degree distribution heterogeneity on vulnerability is analyzed, and the ,influence mechanism of degree distribution heterogeneity on vulnerability of high-order ,networks is revealed.

[0051] In this embodiment, when the total number of nodes (i.e., species in the ecological network) of the constructed hypergraph network model is N=1200, the number of nodes contained in each hyperedge is r=6, and the average node degree is <k>When the degree distribution heterogeneity index ξ changes from 10 to 300, the relationship between the average degree and the degree distribution is shown in the figure below. Figure 2 As shown in Figure 2, it is easy to see that the heterogeneity of the degree distribution of the hypergraph network is negatively correlated with the average degree. The smaller the average degree, the more heterogeneous the degree distribution. <k>The degree of heterogeneity of the hypergraph network can be adjusted, which provides theoretical guidance for designing the desired network structure in practice.

[0052] When the total number of nodes in the constructed hypergraph network model is N = 500, and the number of nodes contained in each hyperedge is r = 6, the average node degree <k>When the vulnerability index F changes from 10 to 100, r The relationship diagram with the change of degree distribution heterogeneity index ξ is as follows Figure 3 As shown. It is easy to know that the vulnerability of the hypergraph network is negatively correlated with the degree distribution heterogeneity. The more heterogeneous the degree distribution is, the more vulnerable the network system is. It can be seen that the simulation results are consistent with the theoretical results, proving that the vulnerability index proposed in the present invention is reasonable and effective.

[0053] The above description is only a preferred embodiment of the present disclosure and an explanation of the technical principles used. Those skilled in the art should understand that the scope of the invention involved in the embodiments of the present disclosure is not limited to the technical solutions formed by a specific combination of the above-mentioned technical features, but should also cover other technical solutions formed by any combination of the above-mentioned technical features or their equivalent features without departing from the above-mentioned inventive concept. For example, the above-mentioned features are replaced with the technical features with similar functions disclosed in the embodiments of the present disclosure (but not limited to) to form a technical solution.< / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k> < / k>

Claims

1. A high-order network vulnerability analysis method based on hypergraph, characterized in that: The following steps are involved: Step 1: Construct a high-order network system consisting of N nodes; Step 2: Propose a new vulnerability indicator; According to the linear system theory, the condition for the linear system (2) to be locally asymptotically stable at the equilibrium point is Re(λ M,1 )<0, where λ M,1 Re(λ M,1 ) represents the characteristic root λ M,1 The real part of the new vulnerability index is defined as F r =1 / |Re(λ M,1 )| (3) Among them, F r represents the vulnerability index, λ M,1 Re(λ M,1 ) represents the characteristic root λ M,1 The real part of , the symbol |·| represents the absolute value; Step 3: Construct a hypergraph network with adjustable degree distribution As the underlying topology map; Step 4: Determine the high-order interaction matrix M based on the generated hypergraph network structure; Step 5: Hypergraph network based on adjustable degree distribution Combined with vulnerability index F r , analyze the impact of degree distribution heterogeneity on vulnerability.

2. According to claim 1, a high-order network vulnerability analysis method based on hypergraph is characterized in that: The step 1 is specifically as follows: construct a high-order network system consisting of N nodes, and the system dynamics equation is represented by the following ordinary differential equation: Where X(t)=[X1(t),X2(t),…,X N (t)] T ∈R N represents the state vector of all nodes in the entire high-order network at time t, X N (t) represents the state of the Nth node, the superscript T represents transposition, and R N represents a set of N-dimensional real vectors; diag(X(t)) represents a diagonal matrix with diagonal elements X1(t),X2(t),…,X N (t), the rest of the elements are 0; the nonlinear function f(X(t)) represents the interaction relationship between nodes at time t; assuming that the equilibrium point of the high-order network system (1) is Satisfy the condition f(X * )=0, Indicates the equilibrium point X * The Nth component of , 0 represents a column vector whose elements are all 0; According to the linear system theory, the following linear system is obtained: Where x(t) = X(t) - X * , M is the high-order interaction matrix between nodes, defined as M = diag(X * )J, where diag(X * ) means the diagonal elements are The diagonal matrix of , J is the nonlinear function f(X(t)) at the equilibrium point X * The Jacobian matrix of the matrix M is M. ij , M ij Represents the element in the i-th row and j-th column in M, reflecting the effect of node j on node i.

3. According to the hypergraph-based high-order network vulnerability analysis method of claim 1, it is characterized in that: The step 3 comprises the following steps: Step 3.1: Based on the given total number of nodes N, the number of nodes contained in each hyperedge r, and the average degree of the expected nodes <k>, calculate the target hyperedge number E, where E = N <k> / r；< / k> < / k> Step 3.2: According to the number of nodes in each hyperedge, a specified number of nodes are placed into the hyperedge through random selection.

4. According to claim 1, a high-order network vulnerability analysis method based on hypergraph is characterized in that: The step 4 comprises the following steps: Step 4.1: Calculate the interaction probability between any two nodes i and j Among them, k i and k j denote the degrees of node i and node j respectively, <k> is the average degree, N is the total number of nodes, and r is the number of nodes contained in each hyperedge;< / k> Step 4.2: Generate a high-order interaction matrix M based on the interaction probability, where the element M of the high-order interaction matrix M is ij With probability P ij From the normal distribution is drawn from the ij Make M ij =0; symbol Indicates that the mean of the distribution is 0 and the variance is σ 2 .

5. The hypergraph-based high-order network vulnerability analysis method according to claim 1, characterized in that: The step 5 is specifically as follows: by adjusting the expected node average degree <k>To adjust the degree of heterogeneity of the degree distribution, the heterogeneity is represented by the symbol ξ = <k 2 > / <k> 2 express,< / k> < / k> in, <k 2 > represents the mean of the square of the node degree, <k> 2 Represents the square of the average degree; hypergraph network model based on adjustable degree distribution Combined with vulnerability index F r ,The impact of degree distribution heterogeneity on vulnerability is analyzed from a simulation perspective.< / k>

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