Vulnerability analysis method for time-delay high-order network

Abstract

The invention provides a vulnerability analysis method for a time-delay high-order network, and relates to the technical field of systems and information. The method comprises the following steps: firstly, constructing a time-delay high-order network model, describing a multi-body interaction relationship by a hypergraph structure, and introducing a time-delay kinetic equation to describe coupling characteristics among nodes; secondly, a novel vulnerability index is provided, and the quantitative relation between the vulnerability of the random interaction hypergraph network and node dynamics, a high-order interaction mode and time delay is established; and finally, a numerical simulation method is adopted to deeply research the coupling effect of the time delay influence and the high-order interaction, and the vulnerability evolution trend of the network under different parameter conditions is evaluated. Experimental results show that the method can improve the accuracy and reliability of vulnerability assessment, is suitable for the fields of communication networks, traffic systems, social networks, biological networks and the like, and provides theoretical basis and decision support for protection of key infrastructures and system risk management and control.

Description

Technical Field

[0001] The present invention relates to the field of systems and information technology, and particularly to a method for analyzing the vulnerability of a time-delay high-order network. Background Art

[0002] With the continuous in-depth research on complex systems, network science has been widely applied in many key fields such as communication, energy, power, transportation, finance, and social systems. Complex networks usually describe the interactions between individuals in a system through the connection relationships between nodes and edges. Among them, traditional network models mostly adopt network structures with low-order binary relationships, focusing on modeling static topological relationships and failing to fully consider the internal mechanisms in the dynamic evolution process of the system. However, in many actual systems, the node state is not only affected by the current neighbor nodes but also by the historical states, that is, there is a time-delay characteristic. At the same time, it may also involve more complex high-order relationships (such as triple interactions, etc.). These factors make the dynamic behavior of the network exhibit complex characteristics such as non-linearity and non-stationarity.

[0003] In the prior art, the methods for analyzing network vulnerability mainly perform vulnerability assessment based on static graph theory metrics (such as node degree, betweenness centrality, clustering coefficient, etc.). These methods can identify key nodes or vulnerable structures in the network to a certain extent, but most methods do not consider the time-delay effect existing in the evolution process of node states over time, nor fully introduce the influence of high-order network structures on system stability. Therefore, it is difficult to accurately evaluate the true vulnerability level of complex networks when facing perturbations, attacks, or failures. Therefore, there is an urgent need to propose a method for analyzing network vulnerability that can simultaneously consider time-delay factors and high-order interaction mechanisms to improve the accuracy and reliability of vulnerability assessment and provide a theoretical basis and decision-making support for the protection of critical infrastructure and system risk management and control. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, the present invention provides a method for analyzing the vulnerability of a time-delay high-order network, aiming to reveal the vulnerability characteristics of complex networks under time-delay effects and high-order interactions. Compared with the prior art, the present invention fully considers the high-order nature and time-delay characteristics of the interaction of the actual network system. The provided vulnerability analysis method can effectively evaluate the ability of the actual complex network system to cope with external disturbances and has a wider application range in practical applications.

[0005] A method for analyzing the vulnerability of a time-delay high-order network includes the following steps:

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

[0007] Construct a time-delay high-order network system composed of N nodes. The dynamic equation of the time-delay high-order network system is represented by the following time-delay differential equation:

[0008]

[0009] wherein, represents the state vector of all nodes in the entire high-order network system at time t, and x N (t) represents the state of the Nth node, and the superscript T represents the transpose. represents the set of N-dimensional real vectors; diag(x(t)) represents a diagonal matrix with diagonal elements x1(t), x2(t), …, x N (t), and the remaining elements are 0; the non-linear function f(x(t - τ)) represents the interaction relationship between nodes, and τ is the time delay; assume that the equilibrium point of the time-delay high-order network system (1) is satisfies the condition f(x * ) = 0. represents the Nth component of the equilibrium point x * , and 0 represents a column vector with all elements being 0; linearize the time-delay high-order network system (1) at the equilibrium point to obtain the following linear system:

[0010]

[0011] wherein, X(t) = x(t) - x * , A is the high-order interaction matrix between nodes, defined as A = diag(x * )J, wherein, diag(x * ) represents a diagonal matrix with diagonal elements being , and J is the Jacobian matrix of the non-linear function f(x(t - τ)) at the equilibrium point x * ; the elements of the matrix A are A ij , and A ij represents the element in the i-th row and j-th column of A, reflecting the influence of node j on node i;

[0012] Step 2: Propose a new type of vulnerability index;

[0013] The condition for the stability of the linear system (2) at the equilibrium point is that all roots of the characteristic equation H(z) = z - λe -zτ = 0 have negative real parts, that is, Re(z1) < 0, wherein, z is the root of the characteristic equation, λ is the eigenvalue of the matrix A, z1 represents the root with the largest real part in the characteristic equation, and Re(z1) represents the real part of the root z1;

[0014] The new type of vulnerability index is defined as

[0015] F r = 1 / |Re(z1)| (3)

[0016] Among them, F r represents the vulnerability index, z1 represents the root with the largest real part in the characteristic equation, Re(z1) represents the real part of the root z1, and the symbol |·| represents the absolute value;

[0017] Step 3: Construct a hypergraph network of random interaction type;

[0018] Step 3.1: Given the total number of nodes N and the total number of hyperedges m, let each node belong to K hyperedges;

[0019] Step 3.2: Let any two nodes in each hyperedge interact with each other with probability C, and the interaction strength has a mean of 0 and a variance of σ 2 Normal distribution of

[0020] Step 3.3: Generate a random interaction hypergraph network and determine the high-order interaction matrix A;

[0021] Step 4: Analyze the vulnerability of high-order network systems with time delays;

[0022] When the interaction between nodes in a high-order network system is of random interaction type, that is, the high-order interaction matrix A is a random matrix, when λ takes the following boundary value and satisfies Re(z1)<0, then this random interaction time-delay high-order network system is stable:

[0023] Left border:

[0024] Right border:

[0025] Upper bound:

[0026] Where N is the total number of nodes, m is the total number of hyperedges, K is the number of hyperedges to which each node belongs, C is the probability of interaction between any two nodes in each hyperedge, and σ 2 Represents the variance of the normal distribution.

[0027] Step 5: Based on the random interaction hypergraph network, combined with the vulnerability index F r , analyze the relationship between the vulnerability of time-delayed high-order network systems and time delay;

[0028] Specifically, by adjusting the time lag τ, we observe the inverse of the corresponding vulnerability index 1 / F when the time lag τ changes from small to large. r The network vulnerability first decreases and then increases with the increase of time delay. There is a threshold for time delay that makes the vulnerability reach the minimum value. At this time, the high-order network with time delay is the most robust against external interference.

[0029] The beneficial effects of adopting the above technical solution are:

[0030] The present invention provides a vulnerability analysis method for time-delay high-order networks, which can effectively make up for the limitations of existing network vulnerability assessment methods and has the following beneficial effects:

[0031] (1) Introduce high-order interaction relationships to characterize the characteristics of real systems

[0032] Traditional network analysis is mainly based on binary interaction relationships, that is, only considering the interaction between nodes. However, complex systems in the real world often involve high-order correlations (such as group interactions, etc.). The present invention combines high-order network modeling methods, which can more comprehensively characterize the complex interaction relationships between nodes, making the vulnerability analysis more in line with the evolution mechanism of real systems.

[0033] (2) Consider the time-delay effect to improve the analysis accuracy

[0034] Existing network vulnerability assessment methods are mostly based on static topological structures or simple dynamic models, ignoring the time-delay effect caused by factors such as information propagation and state evolution during the actual operation of the system. By introducing a time-delay dynamic model, the present invention can more accurately describe the change characteristics of the states of complex network nodes over time, thereby improving the accuracy of vulnerability analysis. By constructing a vulnerability assessment framework for time-delay high-order networks, the quantitative relationships between network vulnerability, node dynamics, high-order interaction patterns, and time-delay are revealed, which can provide more targeted risk protection strategies for critical infrastructure, communication networks, power networks, ecosystems, etc., thereby improving the overall robustness and anti-destruction ability of the network.

[0035] In summary, by simultaneously considering the time-delay effect and high-order interaction mechanism, the present invention breaks through the limitations of traditional vulnerability analysis methods, can more accurately evaluate the response characteristics of the network under external interference or faults, and provides an efficient and accurate analysis tool for the security analysis, risk prediction, and optimization of complex networks, having important theoretical value and application prospects. Brief Description of the Drawings

[0036] Figure 1 It is a flowchart of the vulnerability analysis method for time-delay high-order networks provided for the implementation of the present invention;

[0037] Figure 2 It is a relationship diagram between the vulnerability of a time-delay high-order network with a random interaction type and time-delay in the implementation of the present invention. Detailed Embodiment

[0038] The following combines the drawings and embodiments to further describe in detail the specific implementation manners of the present invention. The following embodiments are used to illustrate the present invention but are not used to limit the scope of the present invention.

[0039] A vulnerability analysis method for time-delay high-order networks, such asFigure 1 As shown in the figure, it includes the following steps:

[0040] Step 1: Construct a time-delay high-order network system composed of N nodes;

[0041] Construct a time-delay high-order network system composed of N nodes. The dynamic equation of the time-delay high-order network system is represented by the following time-delay differential equation:

[0042]

[0043] where represents the state vector of all nodes in the entire high-order network system at time t, x N (t) represents the state of the Nth node, and the superscript T represents the transpose. represents the set of N-dimensional real vectors; diag(x(t)) represents a diagonal matrix with diagonal elements x1(t), x2(t), …, x N (t), and the remaining elements are 0; the nonlinear function f(x(t - τ)) represents the interaction relationship between nodes, and τ is the time delay; assume that the equilibrium point of the time-delay high-order network system (1) is satisfies the condition represents the Nth component of the equilibrium point x * , and 0 represents a column vector with all elements being 0; perform linearization on the time-delay high-order network system (1) at the equilibrium point to obtain the following linear system:

[0044]

[0045] where X(t) = x(t) - x * , A is the high-order interaction matrix between nodes, defined as A = diag(x * )J, where diag(x * ) represents a diagonal matrix with diagonal elements , and J is the Jacobian matrix of the nonlinear function f(x(t - τ)) at the equilibrium point x * ; the elements of the matrix A are A ij , and A ij represents the element in the i-th row and j-th column of A, reflecting the influence of node j on node i;

[0046] In this embodiment, consider a time-delay ecological network system, where the nodes represent species, and x i (t) represents the abundance of species i at time t; the nonlinear function f(x(t - τ)) represents the interaction relationship between species, and the element A ij of the matrix A represents the influence of species j on species i;

[0047] Step 2: Propose a new type of vulnerability index;

[0048] Through theoretical analysis, it can be obtained that the condition for the linear system (2) to be stable at the equilibrium point is that all roots of the characteristic equation H(z) = z - λe -zτ = 0 have negative real parts, that is, Re(z1) < 0, where z is the root of the characteristic equation, λ is the eigenvalue of matrix A, z1 represents the root with the largest real part in the characteristic equation, and Re(z1) represents the real part of the root z1;

[0049] The new vulnerability index is defined as

[0050] F r = 1 / |Re(z1)| (3)

[0051] where F r represents the vulnerability index, z1 represents the root with the largest real part in the characteristic equation, Re(z1) represents the real part of the root z1, and the symbol |·| represents the absolute value;

[0052] Step 3: Construct a hypergraph network of random interaction type;

[0053] Step 3.1: Given the total number of nodes N and the total number of hyperedges m, let each node belong to K hyperedges;

[0054] Step 3.2: Let any two nodes in each hyperedge interact with each other with probability C, and the interaction strength follows a normal distribution with a mean of 0 and a variance of σ 2 ;

[0055] Step 3.3: According to the given conditions, generate a random interaction hypergraph network and determine the high-order interaction matrix A;

[0056] Step 4: Analyze the vulnerability of the time-delay high-order network system;

[0057] When the interaction between nodes in the high-order network system is of random interaction type, that is, the high-order interaction matrix A is a random matrix, through theoretical analysis, it can be obtained that when λ takes the following boundary values and satisfies Re(z1) < 0, this random interaction time-delay high-order network system is stable:

[0058] Left boundary:

[0059] Right boundary:

[0060] Upper bound:

[0061] where N represents the total number of nodes, m represents the total number of hyperedges, K represents the number of hyperedges to which each node belongs, C represents the probability of interaction between any two nodes in each hyperedge, σ2 Represents the variance of the normal distribution.

[0062] Step 5: Based on the random interaction hypergraph network, combined with the vulnerability index F r , analyze the relationship between the vulnerability of time-delayed high-order network systems and time delay;

[0063] Specifically, by adjusting the time lag τ, we observe the inverse of the corresponding vulnerability index 1 / F when the time lag τ changes from small to large. r Theoretically, the vulnerability of time-delayed hypergraph networks is closely related to time delay. Short and long time delays have different effects on network vulnerability. With the increase of time delay, network vulnerability first decreases and then increases. There is a threshold for time delay that makes the vulnerability reach the minimum value. At this time, the time-delayed high-order network is the most robust against external interference. The research results can provide guidance and basis for the design of robust high-order network systems in practice.

[0064] In this embodiment, when the total number of nodes (i.e., species in the ecological network) of the constructed random interactive time-delay hypergraph network model is N = 500, and each node is randomly assigned to K = 4 hyperedges, when the time lag τ changes from 0 to 0.5, the inverse of the vulnerability index 1 / F r The relationship diagram of the time lag τ change is as follows Figure 2 As shown in the figure, it is easy to see that the vulnerability of the time-delay hypergraph network is closely related to the time delay. As the time delay increases, the vulnerability first decreases and then increases, revealing the influence mechanism of short and long time delays on network vulnerability. The simulation results are consistent with the theoretical results, proving that the vulnerability index proposed in this invention is reasonable and effective. This provides theoretical guidance and basis for enhancing the risk resistance of the network through design in practice.

[0065] 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.

Claims

1. A vulnerability analysis method for a time-delay high-order network, characterized in that: The following steps are involved: Step 1: Construct a time-delay high-order network system consisting of N nodes; Step 2: Propose a new vulnerability indicator; Step 3: Construct a hypergraph network of random interaction type; Step 4: Analyze the vulnerability of high-order network systems with time delays; Step 5: Based on the random interaction hypergraph network, combined with the vulnerability index F r , analyze the relationship between the vulnerability of time-delay high-order network systems and time delay.

2. The vulnerability analysis method of a time-delay high-order network according to claim 1 is characterized in that: The steps are specifically as follows: construct a time-delay high-order network system consisting of N nodes, and the dynamic equation of the time-delay high-order network system is represented by the following time-delay differential equation: in, represents the state vector of all nodes in the entire high-order network system at time t, x N (t) represents the state of the Nth node, and the superscript T represents transposition. 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-τ)) represents the interaction relationship between nodes, τ is the time delay; assuming that the equilibrium point of the time-delay high-order network system (1) is Satisfy the conditions represents the equilibrium point x * The Nth component of , 0 represents a column vector whose elements are all 0; linearizing the time-delay high-order network system (1) at the equilibrium point, we obtain the following linear system: Where X(t) = x(t) - x * , A is the high-order interaction matrix between nodes, defined as A = diag(x * )J, where diag(x * ) means the diagonal elements are The diagonal matrix of , J is the nonlinear function f(x(t-τ)) at the equilibrium point x * The Jacobian matrix of the matrix A is A. ij , A ij Represents the element in the i-th row and j-th column in A, reflecting the effect of node j on node i.

3. The vulnerability analysis method of a time-delay high-order network according to claim 2 is characterized in that: The condition for the linear system (2) in step 2 to be stable at the equilibrium point is: characteristic equation H(z) = z-λe -zτ =0 have negative real parts, that is, Re(z1)<0, where z is the root of the characteristic equation, λ is the characteristic root of the matrix A, z1 represents the root with the largest real part in the characteristic equation, and Re(z1) represents the real part of the root z1.

4. The vulnerability analysis method of a time-delay high-order network according to claim 3 is characterized in that: The new vulnerability indicator is defined as F r =1 / |Re(z1)| (3) Among them, F r represents the vulnerability index, z1 represents the root with the largest real part in the characteristic equation, Re(z1) represents the real part of the root z1, and the symbol |·| represents the absolute value.

5. The vulnerability analysis method of a time-delay high-order network according to claim 1, characterized in that: The step 3 specifically comprises the following steps: Step 3.1: Given the total number of nodes N and the total number of hyperedges m, let each node belong to K hyperedges; Step 3.2: Let any two nodes in each hyperedge interact with each other with probability C, and the interaction strength has a mean of 0 and a variance of σ 2 Normal distribution of Step 3.3: Generate a random interaction hypergraph network and determine the high-order interaction matrix A.

6. The vulnerability analysis method of a time-delay high-order network according to claim 3 is characterized in that: The step 4 is specifically as follows: when the interaction between nodes in the high-order network system is of a random interaction type, that is, the high-order interaction matrix A is a random matrix, when λ takes the following boundary value and satisfies Re(z1)<0, then the random interaction time-delay high-order network system is stable: Left border: Right border: Upper bound: Where N is the total number of nodes, m is the total number of hyperedges, K is the number of hyperedges to which each node belongs, C is the probability of interaction between any two nodes in each hyperedge, and σ 2 Represents the variance of the normal distribution.

7. The vulnerability analysis method of a time-delay high-order network according to claim 4 is characterized in that: The step 5 is specifically as follows: by adjusting the time lag τ, observing the inverse of the corresponding vulnerability index 1 / F when the time lag τ changes from small to large r The network vulnerability first decreases and then increases with the increase of time delay. There is a threshold for time delay that makes the vulnerability reach the minimum value. At this time, the high-order network with time delay is the most robust against external interference.

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