A hypergraph-based high-order network vulnerability analysis method

Abstract

The application provides a high-order network vulnerability analysis method based on a hypergraph, relates to the field of system and information technology, and newly constructs a hypergraph network model with adjustable degree distribution as a bottom topology, proposes a new vulnerability index suitable for the hypergraph network through theoretical analysis, and reveals the relationship between the heterogeneity of the hypergraph degree distribution and the vulnerability. The application fully considers the high-order and random characteristics of the interaction of actual network systems, the hypergraph network model with adjustable degree distribution can be used for describing ecological networks, social networks and traffic systems, the new vulnerability analysis method can effectively evaluate the ability of the actual network system to resist external interference, has a wider application range in practical application, and has important technical values in network security, invulnerability design and risk prediction and the like.

Description

Technical Field

[0001] This invention relates to the field of systems and information technology, and in particular to a high-order network vulnerability analysis method based on hypergraphs. Background Technology

[0002] As research into complex networks deepens, traditional first-order network models, which can only describe binary interactions between nodes, struggle to reflect the complex and diverse higher-order interaction structures in real-world systems. Therefore, high-order network theory has gradually become an important direction in complex network research, with hypergraph models attracting significant attention due to their ability to naturally represent multi-faceted interactions. Hypergraphs, by extending edges into "hyperedges" connecting multiple nodes, can more comprehensively reveal higher-order relationships between nodes. However, the complexity of high-order networks also presents significant challenges to their vulnerability analysis. Vulnerability analysis is a crucial method for assessing network performance and functional degradation under structural failure, external disturbances, or dynamic changes, and it has significant application value in fields such as communication networks, transportation systems, and biological networks.

[0003] Current research on high-order networks largely focuses on describing their structural characteristics and dynamic behavior, while methods for systematically analyzing their vulnerabilities remain limited. Traditional vulnerability analysis methods for complex networks, such as those based on degree distribution, node importance, or connectivity assessment, are difficult to apply directly to high-order networks because the diversity and randomness of high-order interactions increase the complexity of the analysis. Therefore, there is an urgent need to develop a vulnerability analysis method for high-order networks that incorporates hypergraph theory to reveal the impact of high-order interaction structures on the overall stability and robustness of the network, providing theoretical basis and technical support for network design, optimization, and security assurance. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a high-order network vulnerability analysis method based on hypergraphs. This invention novelly constructs a hypergraph network model with adjustable degree distribution as the underlying topology, and proposes a new vulnerability index applicable to hypergraph networks through theoretical analysis, revealing the relationship between the heterogeneity of hypergraph degree distribution and vulnerability. Compared with existing technologies, this invention fully considers the high-order and stochastic characteristics of interactions in real-world network systems. 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 assess the ability of real-world network systems to cope with external interference, and has a wider range of applications in practice.

[0005] A high-order network vulnerability analysis method based on hypergraphs, the specific steps of which 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 equations are expressed by the following ordinary differential equations:

[0008]

[0009] Where X(t)=[X1(t),X2(t),…,X N (t)] T ∈R N Let X represent the state vector of all nodes in the entire high-order network at time t. N (t) represents the state of the Nth node, the superscript T indicates transpose, and R N Let X(t) represent a set of N-dimensional real vectors; diag(X(t)) represents a diagonal matrix with diagonal elements X1(t), X2(t), ..., X N (t), with the remaining elements being 0; the nonlinear function f(X(t)) represents the interaction relationship between nodes at time t; assuming the equilibrium point of the high-order network system (1) is Satisfy condition f(X) * ) = 0, Represents the equilibrium point X * The Nth component, 0, represents a column vector with all elements equal to 0; according to linear system theory, the following linear system is obtained:

[0010]

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

[0012] Step 2: Propose a novel vulnerability index;

[0013] According to the theory of linear systems, the condition for the local asymptotic stability of linear system (2) at the equilibrium point is Re(λ). M,1 )<0, where λ M,1 Re(λ) represents the eigenvalue with the largest real part of matrix M. M,1 ) represents the characteristic root λ M,1The real part; the novel vulnerability index is defined as

[0014] F r =1 / |Re(λ) M,1 (3)

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

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

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

[0018] Step 3.2: Based on the number of nodes in each superedge, randomly select a specified number of nodes to place in the superedge;

[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. Where, k i and k j Let i and j represent the degrees of node i and node j, respectively. <k>Let N be the average degree, N be the total number of nodes, and r be the number of nodes contained in each hyperedge;

[0021] Step 4.2: Generate a higher-order interaction matrix M based on the interaction probabilities, where the elements M of the higher-order interaction matrix M are... ij With probability P ij From normal distribution The sample is drawn from the middle, with a probability of 1-P. ij Make M ij =0; symbol This means the distribution has a mean of 0 and a variance of σ. 2 ;

[0022] Step 5: Hypergraph Network with Adjustable Degree Distribution Combined with vulnerability index F r The impact of heterogeneity in the degree of analysis on vulnerability;

[0023] Specifically, this involves adjusting the expected average degree of nodes. <k>The degree of heterogeneity of the degree distribution is used to regulate the heterogeneity, which is denoted by the symbol ξ = <k 2 > / <k> 2 It means that, among them, <k 2 > represents the mean of the squares of the node degrees. <k> 2 Represents the square of the average degree; a hypergraph network model based on adjustable degree distribution. Combined with vulnerability index F r The impact of degree heterogeneity in the distribution of fragility on vulnerability is analyzed from a simulation perspective.

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

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

[0026] (1) Traditional network models mostly analyze network structures based on first-order graphs (nodes and edges), while this method incorporates higher-order relationships (such as multi-edge 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 and thus more accurately assess network vulnerability; (2) The vulnerability analysis method based on hypergraphs can not only assess the impact of a single node or single connection, but also consider the impact of the failure of node groups or substructures on the overall network, realizing multi-dimensional risk assessment from local to global, and providing more comprehensive data support for the optimization design of complex networks. This method is applicable to various types of complex networks, including communication networks, transportation networks, social networks, and ecological networks, and can provide a unified theoretical framework for the network security and robustness analysis of different industries in practical applications. In summary, this invention provides an advanced tool for the high-order structure analysis of complex networks, which helps to promote theoretical research and practical applications in related fields, especially in network security, survivability design, and risk prediction, and has important technical value. Attached Figure Description

[0027] Figure 1 Flowchart of the high-order network vulnerability analysis method based on hypergraph provided for the implementation of this invention;

[0028] Figure 2 This is a graph showing the relationship between the degree distribution heterogeneity and the average degree of the hypergraph network in this invention.

[0029] Figure 3 This is a graph showing the relationship between the fragility and degree distribution heterogeneity of the hypergraph network in the implementation of this invention. Detailed Implementation

[0030] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0031] A high-order network vulnerability analysis method based on hypergraphs, 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 equations are expressed by the following ordinary differential equations:

[0034]

[0035] Where X(t)=[X1(t),X2(t),…,X N (t)] T ∈R N Let X represent the state vector of all nodes in the entire high-order network at time t. N (t) represents the state of the Nth node, the superscript T indicates transpose, and R N Let X(t) represent a set of N-dimensional real vectors; diag(X(t)) represents a diagonal matrix with diagonal elements X1(t), X2(t), ..., X N (t), with the remaining elements being 0; the nonlinear function f(X(t)) represents the interaction relationship between nodes at time t; assuming the equilibrium point of the high-order network system (1) is Satisfy condition f(X) * ) = 0, Represents the equilibrium point X * The Nth component, 0, represents a column vector with all elements equal to 0; according to linear system theory, the following linear system is obtained:

[0036]

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

[0038] In this embodiment, we 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 elements M of matrix M are... ij This represents the influence of species j on species i near the equilibrium point;

[0039] Step 2: Propose a novel vulnerability index;

[0040] According to the theory of linear systems, the condition for the local asymptotic stability of linear system (2) at the equilibrium point is Re(λ). M,1 )<0, where λ M,1 Re(λ) represents the eigenvalue with the largest real part of matrix M. M,1 ) represents the characteristic root λ M,1 The real part; the novel vulnerability index is defined as

[0041] F r =1 / |Re(λ) M,1 (3)

[0042] Among them, F r This represents a vulnerability index; the higher the value of this index, the more vulnerable the linear system is. M,1 Re(λ) represents the eigenvalue with the largest real part of matrix M. M,1 ) represents the characteristic root λ M,1 The real part of , the symbol |·| denotes absolute value;

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

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

[0045] Step 3.2: Based on the number of nodes in each superedge, randomly select a specified number of nodes to place in the superedge;

[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. Where, k i and k j Let i and j represent the degrees of node i and node j, respectively. <k>Let N be the average degree, N be the total number of nodes, and r be the number of nodes contained in each hyperedge;

[0048] Step 4.2: Generate a higher-order interaction matrix M based on the interaction probabilities, where the elements M of the higher-order interaction matrix M are... ij With probability P ij From normal distribution The sample is drawn from the middle, with a probability of 1-P. ij Make M ij =0; symbol This means the distribution has a mean of 0 and a variance of σ. 2 ;

[0049] Step 5: Hypergraph Network with Adjustable Degree Distribution Combined with vulnerability index F r The impact of heterogeneity in the degree of analysis on vulnerability;

[0050] Specifically, this involves adjusting the expected average degree of nodes. <k>The degree of heterogeneity of the degree distribution is used to regulate the heterogeneity, which is denoted by the symbol ξ = <k 2 > / <k> 2 It means that, among them, <k 2 > represents the mean of the squares of the node degrees. <k> 2 Represents the square of the average degree; a hypergraph network model based on adjustable degree distribution. Combined with vulnerability index F r This study analyzes the impact of degree distribution heterogeneity on vulnerability from a simulation perspective, revealing the mechanism by which degree distribution heterogeneity in high-order networks affects vulnerability.

[0051] In this embodiment, when the total number of nodes (i.e., species in the ecological network) in the constructed hypergraph network model is N = 1200, and each hyperedge contains r = 6 nodes, the average node degree is... <k>The graph showing the relationship between the degree distribution heterogeneity index ξ and the mean degree as the degree varies from 10 to 300 is shown below. Figure 2 As shown, it is easy to see that the heterogeneity of the degree distribution in this hypergraph network is negatively correlated with the average degree; the smaller the average degree, the more heterogeneous the degree distribution. Therefore, it can be seen that by adjusting the parameters... <k>The degree distribution heterogeneity of hypergraph networks can be adjusted. This provides theoretical guidance for designing desired network structures in practice.

[0052] When the constructed hypergraph network model has a total of N = 500 nodes and each hyperedge contains r = 6 nodes, the average node degree... <k>When the vulnerability index F changes from 10 to 100, r The relationship between the heterogeneity index ξ and the degree distribution is shown in the graph. Figure 3 As shown, it is easy to see that the vulnerability of this hypergraph network is negatively correlated with the degree distribution heterogeneity; the more heterogeneous the degree distribution, the more vulnerable the network system. Therefore, the simulation results are consistent with the theoretical results, proving that the vulnerability index proposed in this invention is reasonable and effective.

[0053] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.< / 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 hypergraphs, characterized in that, Includes the following steps: Step 1: Based on the real-world network system, construct a high-order network system consisting of N nodes; The higher-order network system is an ecological network system, where nodes represent species, and the elements M of matrix M are... ij This represents the influence of species j on species i near the equilibrium point; Step 2: Propose a novel vulnerability index; According to the theory of linear systems, the condition for a linear system (2) to be locally asymptotically stable at its equilibrium point is: ,in, This represents the eigenvalue with the largest real part in matrix M. Representing characteristic roots The real part; the novel vulnerability index is defined as: (3)； in, Indicators of vulnerability This represents the eigenvalue with the largest real part in matrix M. Representing characteristic roots The real part, symbol Represents absolute value; Step 3: Construct a hypergraph network with adjustable degree distribution As the underlying topology graph; Step 4: Determine the high-order interaction matrix M based on the generated hypergraph network structure; Step 4.1: Calculate the interaction probability between any two nodes i and j. ,in, and Let i and j represent the degrees of node i and node j, respectively. Let N be the average degree, N be the total number of nodes, and r be the number of nodes contained in each hyperedge; Step 4.2: Generate a higher-order interaction matrix M based on the interaction probabilities, where the elements of the higher-order interaction matrix M are... With probability From normal distribution Extracted from the middle, and based on probability. Make ;symbol This indicates that the mean of the distribution is 0 and the variance is . ; Step 5: Hypergraph Network with Adjustable Degree Distribution Combined with vulnerability indicators The impact of heterogeneity in the degree of analysis on vulnerability.

2. The method for high-order network vulnerability analysis based on hypergraphs according to claim 1, characterized in that, Step 1 specifically involves constructing a high-order network system consisting of N nodes, whose dynamic equations are represented by the following ordinary differential equations: (1)； in, This represents the state vector of all nodes in the entire high-order network at time t. This represents the state of the Nth node, with the superscript T indicating transpose, and R... N Represents the set of N-dimensional real vectors; This represents a diagonal matrix with diagonal elements as follows: All other elements are 0; nonlinear function This represents the interaction relationship between nodes at time t; assuming the equilibrium point of the high-order network system (1) is... The conditions are met. , Indicates the equilibrium point The Nth component, Let represent a column vector whose elements are all 0; according to the theory of linear systems, the following linear system is obtained: (2)； in, , The higher-order interaction matrix between nodes is defined as follows: ,in, Indicates that the diagonal element is diagonal matrix, nonlinear function At the equilibrium point Jacobian matrix; matrix The elements are , express The element in the i-th row and j-th column reflects the relationship between node j and node i.

3. The method for high-order network vulnerability analysis based on hypergraphs according to claim 1, characterized in that, Step 3 includes the following steps: Step 3.1: Based on the given total number of nodes N, the number of nodes r contained in each superedge, and the expected average degree of the nodes. Calculate the target hyperedge number E, where, ; Step 3.2: Based on the number of nodes in each superedge, randomly select a specified number of nodes to place in the superedge.

4. The method for high-order network vulnerability analysis based on hypergraphs according to claim 1, characterized in that, Step 5 specifically involves: adjusting the expected average node degree To moderate the degree of heterogeneity in the degree distribution, the degree of heterogeneity is represented by the symbol. It means that, among them, This represents the mean of the squared degree of a node. Represents the square of the average degree; a hypergraph network model based on adjustable degree distribution. Combined with vulnerability indicators The impact of degree heterogeneity in the distribution of fragility on vulnerability is analyzed from a simulation perspective.

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