A security control method for complex networks against proportional delay and DoS attacks

Abstract

The present invention discloses a security control method for complex networks against proportional delay and DoS attacks, comprising the following steps: Step 1, establishing a complex network model with proportional delay; Step 2, establishing an energy-constrained DoS attack model, using an average duration to describe and constrain the DoS attack, and introducing the average duration into the complex network model established in Step 1; Step 3, designing quantitative security controllers for different attack modes; Step 4, establishing a condition to ensure the stability of the error system under the first attack mode; Step 5, establishing a condition to ensure the stability of the error system under the second attack mode; Step 6, establishing a condition to ensure the stability of the error system under bounded delay, and if the system delay degenerates from an unbounded proportional delay to a general bounded delay, a less conservative stability condition can be obtained. The present invention can effectively achieve security consistency of complex networks with proportional delay under DoS attacks and limited bandwidth.

Description

Technical Field

[0001] The present invention relates to the technical field of complex network control, and in particular to a security control method for complex networks against proportional delay and DoS attacks. Background Art

[0002] With the rapid development of computer science and technology, complex networks have become one of the core tools in modern scientific research and practical applications. They are widely used in many important fields, such as communication networks, social networks, biological networks, economic networks, and transportation networks, demonstrating their enormous application potential. These networks exchange information through shared communication channels, allowing them to efficiently collaborate to complete tasks or solve complex problems. However, practical applications of complex networks often face multiple technical challenges, such as time delays, resource constraints, and insecure communication environments. These problems significantly affect the stability and reliability of the networks, posing enormous challenges and threats to industrial production and even human society. To address these challenges and threats, researchers are committed to designing robust and adaptable control strategies to ensure the stable operation of complex networks in harsh environments.

[0003] In complex networks, nodes exchange information through open, shared networks. While this facilitates collaboration and data transfer, it also makes the network vulnerable to various malicious attacks, severely impacting or even disrupting its performance. DoS attacks are a common and highly destructive attack method. Attackers disrupt normal information transmission by blocking communication channels or sending a large number of invalid requests, leading to system resource exhaustion or service failure. Therefore, constructing mathematical models for DoS attacks and enhancing the robustness of complex networks under attack has become an important direction in network security research. Existing research has made significant progress in analyzing and defending against DoS attacks. For example, DoS attack models based on Markov processes, periodic DoS attack models, and energy-constrained DoS attack models have been developed. Energy-constrained DoS attack models, in particular, are typically characterized by the average duration and frequency of attacks. These models have been recognized by the industry and widely used in subsequent research. A natural question is whether the constraints on average duration and frequency can be further relaxed. However, this remains a topic that requires further research, especially in systems with time delays, where stricter attack constraints are often imposed to ensure the performance of complex networks.

[0004] Time delay is a ubiquitous phenomenon in complex networks, significantly impacting coordination, communication, and decision-making processes between nodes. Therefore, analyzing complex networks with time delays is both interesting and challenging. Numerous studies have explored complex networks with both time-varying and time-invariant delays, achieving fruitful results. However, these studies typically assume an upper bound on delays, a stringent condition that limits their applicability. Unlike bounded delays, unbounded delays in complex networks arise from the system's long-term memory effects or history dependence, whereby a node's state changes depend not only on its current state but also on all past states. Such unbounded delays are widely present in real-world systems, such as synaptic plasticity in biological networks (the long-term effects of historical stimuli), price fluctuations in financial markets (the cumulative effects of historical trading behavior), and population dynamics in ecosystems (the long-term adaptation to environmental changes). The analysis of unbounded delays is crucial for studying the long-term stability, dynamic behavior prediction, and control design of complex networks. While some existing results on unbounded delays exist, none of them consider the impact of cyberattacks. A noteworthy fact is that the simultaneous analysis of resilience to delay and DoS attacks is currently limited to complex networks with bounded delays. This limitation raises a key question: Can complex networks with unbounded delays be studied under DoS attacks? In this context, we consider the security control of complex networks with proportional delays under DoS attacks. To our knowledge, this problem remains unexplored and unsolved, posing a significant challenge. Proportional delay, a type of unbounded delay, has important applications in fields such as the human brain and network science. When proportional delay is introduced into a complex network, the system's dynamic behavior depends on the current state x(t) and the past state x(pt), where x(pt) represents the ratio of x(t). Over time, this delay tends to infinity, causing the temporal evolution of nodes to abruptly change proportionally to pt with a proportional delay ratio p. Consequently, methods previously applied to bounded delays are no longer applicable. Furthermore, the presence of DoS attacks disrupts existing control strategies, leading to divergent system states. This further complicates the analysis of complex networks with proportional delays under DoS attacks.

[0005] In complex networks, network communication channels are inherently constrained by limited bandwidth, severely impacting information transfer between nodes. This limitation can lead to data distortion during transmission, as the received data deviates from the original, resulting in errors and reduced system performance. To mitigate this practical challenge, researchers have widely adopted signal quantization techniques. This paper employs a typical quantization mechanism, the logarithmic quantizer, to address limited bandwidth. Based on this, a control framework is designed to combat proportional delays and DoS attacks, providing a novel approach to secure control in complex networks. Summary of the Invention

[0006] Purpose of the invention: The present invention provides a security control method for complex networks against proportional delay and DoS attacks, which can effectively achieve security consistency of complex networks with proportional delay (a kind of unbounded delay) under more general DoS attacks and limited bandwidth.

[0007] Technical solution: The present invention provides a security control method for complex networks against proportional delay and DoS attacks, comprising the following steps:

[0008] Step 1: Establish a complex network model with proportional delay;

[0009] Step 2: Establish an energy-constrained DoS attack model, use the average duration to describe and constrain the DoS attack, and introduce it into the complex network model established in step 1;

[0010] Step 3: Design quantitative security controllers for different attack modes;

[0011] Step 4: Establish conditions to ensure the stability of the error system in the first attack mode;

[0012] Step 5: Establish conditions to ensure the stability of the error system in the second attack mode;

[0013] Step 6: Establish a condition to ensure the stability of the error system under bounded delay. If the system delay degenerates from unbounded proportional delay to general bounded delay, a less conservative stability condition can be obtained.

[0014] Furthermore, in step 1, a complex network model with proportional delay is established. Consider a system consisting of N nodes, and use a differential equation to express the evolution of the dynamic behavior of each node. The control signal of the i-th node is u i (k),i=1,2,…,N.

[0015] Furthermore, in step 2, the i-th DoS attack is expressed as: D i ={d i}∪[d i ,d i +h i ), d i Indicates the moment when the attack changes from 0 to 1, h i Indicates the duration of the attack. Depending on whether the system is attacked, the time axis is divided into the attack interval and the safety interval, that is,

[0016]

[0017] Here, i = 1, 2, ... and S0 = [t0, d1), the internal channels of the nodes and the external communication network connecting the nodes are potential targets of attack.

[0018] Furthermore, it is divided into two modes: mode 1, only the communication network between nodes is attacked; mode 2, both the internal channels of the nodes and the communication network between them are attacked.

[0019] Furthermore, in step 3, a quantitative safety controller is designed for each attack mode. Within the safety interval, node i designs a coupling controller based on the quantitative signals received from neighboring nodes and its own quantitative signals. Within the attack interval, in the first attack mode, node i cannot receive signals from its neighbors, but can still receive signals from itself. Therefore, the controller design is completely based on its own quantitative signal. The controller design is as follows:

[0020]

[0021] Among them, Q(x i (t)) represents the signal x i The quantized signal of (t), a ij >0(i≠j) indicates a directed edge from node i to node j, allowing node j to receive information from node i. Conversely, a ij = 0, it means that there is no communication edge from node i to node j. In addition, define c represents coupling gain and K represents feedback gain.

[0022] Furthermore, in the second attack mode, both the internal channels of the nodes and the external communication network between them are attacked. Assume that the DoS attacks suffered by the internal and external channels of the nodes are synchronized, that is, they occur and end at the same time. Under this attack, the controller does not receive any signal, so the control input is set to zero. The controller in this part is as follows:

[0023]

[0024] Furthermore, in step 4, the error system of the complex network is constructed using the state and average state of each node. Then, according to different attack modes, a simple Lyapunov function for the error system is established. For the first attack mode, the Lyapunov function established in the error system stability analysis process is:

[0025]

[0026] in, is the error state between the state of the ith node and the average state, W=(ω ij ) N×N =E-ηη I ,η=[η1,η2,…,η N ] Tis the eigenvector corresponding to the zero eigenvalue of the Laplace matrix A, E=diag{η1,η2,…,η N}, and η satisfies η j >0,j=1,2,…,N and Based on this, the sufficient condition for ensuring the stability of the error system is:

[0027]

[0028] in, a=min{a1,a2}, δ∈(0,1),l 01 , l 02 , α b1 , α b2 , α c2 and β1 are positive constants, B1, B2, C1 and C2 are system parameters.

[0029] Furthermore, in step 5, for the second attack mode, the Lyapunov function established in the error system stability analysis process is:

[0030]

[0031] Furthermore, the sufficient condition for ensuring the stability of the error system is obtained as follows:

[0032]

[0033] Among them, β2 is a positive constant, p is the coefficient of proportional delay, T0>0 is a regularization term, is the ratio of attack duration to total time.

[0034] Furthermore, in step 6, the unbounded proportional delay pt,p∈(0,1) is replaced by a time-varying bounded delay t-τ(t), where τ(t)∈(0,τ) and τ is a positive constant. Then the complex network becomes:

[0035]

[0036] By constructing the Lyapunov function, the sufficient condition for ensuring the stability of the error system is obtained:

[0037] a1<0,a3>0

[0038]

[0039] Among them, β * is a positive constant.

[0040] Beneficial effects: Compared with the existing technology, the present invention has the following significant advantages: The present invention provides a simple and easy-to-use security control method for the security problem of complex networks with proportional delays under DoS attacks: First, a class of nonlinear complex network models with proportional delays is considered. The model may be subject to network attacks and has limited communication bandwidth. The introduced DoS attack is energy-limited, which only constrains the average duration of the attack and relaxes the previous average frequency condition; at the same time, for different attack scenarios, the interference characteristics of two typical attack modes on system performance are studied in detail. The delay considered is an unbounded delay, which is different from the The results of delay and DoS attacks are analyzed simultaneously with the existing ones. In addition, by constructing the error system and its Lyapunov function, and utilizing Lyapunov stability theory, mathematical induction, integral inequality, and matrix theory, the sufficient conditions for designing corresponding quantitative controllers and maintaining the stability of the error system under different attack modes are provided. At the same time, by replacing the proportional delay with a bounded delay, a less conservative condition for maintaining the stability of the error system can be obtained. Finally, simulation experiments show that the present invention can achieve security control of complex systems against proportional delay and DoS attacks, thus providing a new idea for the security control of a class of complex networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 Schematic diagram of the communication process of the present invention.

[0042] Figure 2 This is a state trajectory diagram of the system in the first attack mode of the present invention.

[0043] Figure 3 This is the error trajectory diagram of the error system in the first attack mode of the present invention.

[0044] Figure 4 This is the state trajectory diagram of the system when the proportional parameter is p=0.48.

[0045] Figure 5 This is the error trajectory diagram of the error system of the present invention when the proportional parameter is p=0.48.

[0046] Figure 6 This is the state trajectory diagram of the system in the second attack mode of the present invention.

[0047] Figure 7 This is the error trajectory diagram of the error system in the second attack mode of the present invention.

[0048] Figure 8 This is a state trajectory diagram of the system under secure communication of the present invention.

[0049] Figure 9 This is the error trajectory diagram of the error system under secure communication of the present invention. DETAILED DESCRIPTION

[0050] like Figure 1 As shown, a security control method for complex networks against proportional delay and DoS attacks includes the following steps:

[0051] Step 1: Establish a complex network model with proportional delay.

[0052] Consider a system consisting of N nodes, the dynamic equation of the i-th node is as follows:

[0053]

[0054] Among them, x i (t)∈R n is the state vector of node i (i=1,2,…,N) at time t≥t0; B j and C j (j=1,2) is the coefficient matrix of the corresponding dimension; g1(·) and g2(·) represent nonlinear terms; pt is the proportional delay, which satisfies 0 <p<1.u i (t) is the control input of node i, which will be designed in step 3. Assume that the function g m (·) satisfies the Lipschitz condition, that is, there exists a constant l 0m , so that (g m (z1)-g m (z2)) T (g m (z1)-g m (z2))≤l 0m (z1-z2) T (z1-z2). Here, z1,z2∈R n and m=1,2.

[0055] The complex system under consideration has a directed and connected network topology, and its Laplace matrix is ​​A=(a ij ) N×N , where a ij >0(i≠j) indicates a directed edge from node i to node j, allowing node j to receive information from node i. Conversely, if a ij = 0, it means that there is no communication edge from node i to node j. In addition, define Therefore, the matrix A is irreducible. The communication process of complex networks is as follows Figure 1 shown.

[0056] Step 2: Establish an energy-limited DoS attack model.

[0057] The i-th DoS attack is represented as: D i ={di}∪[d i ,d i +h i ).d i Indicates the moment when the attack changes from 0 to 1, h i Represents the duration of the attack. Depending on whether the system is attacked, we divide the time axis into the attack interval and the safety interval, that is,

[0058]

[0059] Here, i = 1, 2, ... and S0 = [t0, d1). ∪D i represents the set of intervals where communication fails due to DoS attacks, ∪S i A set of intervals representing successful communication. Importantly, both the internal channels of a node and the external communication network connecting them are potential targets for attack. This paper explores two modes: Mode 1, where only the communication network between nodes is attacked; Mode 2, where both the internal channels of a node and the external communication network between them are attacked.

[0060] Step 3: Design quantitative security controllers for different attack modes.

[0061] Step 31: Give the quantization function. The quantization level of the logarithmic quantizer Q(·) is: q={±q k ,q k =q0ρ k ,k=0,±1,±2,…}∪{0}, where ρ∈(0,1) and q0>0. The quantization mechanism of Q(·) is as follows:

[0062]

[0063] here, Can be verified

[0064] Step 32: Design a controller. Within the safe range, node i designs a coupled controller based on the quantized signals received from neighboring nodes and its own quantized signal. Within the attack range, under the first attack mode, node i cannot receive signals from its neighbors, but can still receive its own signal. Therefore, the controller design is entirely based on its own quantized signal. Based on the above discussion, the quantized controller design is as follows:

[0065]

[0066] Among them, Q(x i (t)) represents the signal x i (t), c represents the coupling gain and K represents the feedback gain.

[0067] In the second attack mode, both the internal channels of the nodes and the external communication network between them are attacked. It is assumed that the DoS attacks on the internal and external channels of the nodes are synchronous, that is, they occur and end at the same time. Under this attack, the controller does not receive any signal, so the control input is set to zero. Similarly, we adopt the coupled control in formula (4). In summary, the controller in this part is as follows:

[0068]

[0069] Step 4: Establish conditions to ensure the stability of the error system in the first attack mode.

[0070] The error system of the complex network is constructed using the state and average state of each node. Then, according to different attack modes, a simple Lyapunov function for the error system is established. For the first attack mode, the Lyapunov function established in the error system stability analysis process is:

[0071]

[0072] in, is the error state between the state of the ith node and the average state, W=(ω ij ) N×N =E-ηη I . Here, η = [η1, η2,…, η N ] T is the eigenvector corresponding to the zero eigenvalue of the Laplace matrix A, E=diag{η1,η2,…,η N}, and η satisfies η j >0,j=1,2,…,N and Under controller (4), based on Lyapunov stability theory, mathematical induction, integral inequality and matrix theory, its dynamic behavior in the safe range and attack range is analyzed respectively. On this basis, V(t) is further analyzed as a whole, and finally the sufficient condition for ensuring the stability of the error system is obtained:

[0073] Criterion I. For system (1), if the following conditions hold:

[0074]

[0075]

[0076]

[0077] in, a=min{a1,a2}, l 01 , l02 , α b1 , α b2 , α c3 and β1 are positive constants. Then, under controller (4), system (1) can achieve secure consistency against proportional delay and arbitrary DoS attacks.

[0078] Step 5: Establish conditions to ensure the stability of the error system in the second attack mode.

[0079] During the error system stability analysis process, the V(t) function is established as in step 4. The second attack mode means that both the internal channels of the nodes and the communication network between them are attacked. In this case, the V(t) function converges in the safe interval but diverges in the attack interval. Through variable transformation and comparison principles, V(t) is converted into the form of a comparison system μ(t) that is easy to analyze. Combined with the results of step 4, the upper bound functions of μ(t) in the two intervals are analyzed respectively. Furthermore, in order to perform stability analysis on the system μ(t) with proportional delay, the time axis is divided into several intervals J k =[t0p -k ,t0p -k-1 ), k=0,1,2..., respectively analyze the system μ(t) in each interval J k According to the convergence and divergence rates of the system and the proportion of attack time to total time, the sufficient condition to ensure the stability of the error system is derived:

[0080] Criterion II. For system (1), if the following conditions hold:

[0081]

[0082]

[0083]

[0084]

[0085]

[0086] Among them, T0>0 is the regularization term, is the proportion of attack duration to total time, and β2 is a positive constant. Then, under controller (4), system (1) can achieve security consistency against DoS attacks with proportional delay and average duration (13).

[0087] Step 6: Establish the conditions for ensuring the stability of the error system under bounded delay.

[0088] If the system's delay degenerates from an unbounded proportional delay to a bounded delay, a less conservative stability condition can be obtained. Replace the proportional delay pt,p∈(0,1) with a time-varying bounded delay t-τ(t), where τ(t)∈(0,τ) and τ is a positive constant. The complex network then becomes:

[0089]

[0090] Furthermore, constructing the same V(t) function as in step 5, the sufficient condition for ensuring the stability of the error system is:

[0091] Criterion III. For system (15), if the following conditions hold:

[0092] a1<0,a3>0 (16)

[0093]

[0094]

[0095]

[0096] Among them, β * is a positive constant. Then, under controller (4), system (1) can achieve security consistency against DoS attacks with proportional delay and average duration (18).

[0097] In order to verify the effectiveness of the method of the present invention, the following simulation experiments are conducted: Consider a three-dimensional system (1) with N = 5 nodes, and select the following system parameters: p = 0.9, g1(·) = g2(·) = sin(·), C1 = -0.2I n , C2=0.1I n , B2=1.5B1,

[0098]

[0099] In addition, select c = 3, K = -3I n , T0=5, T=2, q0=1, ρ=0.8, δ=0.1112α b1 =5,α b2 =0.5,α c2 =0.8, β1=1 and β2=6.2.

[0100] Figure 2 and Figure 3The evolution of the system state and its error state are described respectively. It can be seen that under the designed controller, the system can reach consistency. It is worth noting that the above simulation is performed with a delay parameter p = 0.9. In order to observe the effect of delay on the system, we set p = 0.48. In this case, the evolution of the system state and its error state is shown as follows Figure 4 and Figure 5 As shown. Figure 2 ( Figure 3 )and Figure 4 ( Figure 5 ), we found that the smaller the p-value, the slower the convergence speed, indicating that delay has a negative impact on consistency.

[0101] like Figure 6 and Figure 7 As shown in Figure 2, the designed quantized controller can achieve consensus performance when subjected to both inter-node and intra-node DoS attacks with parameters T0 = 5 and T = 2. Furthermore, if the systems communicate in a secure network, consensus can be reached faster. Figure 6 ( Figure 7 )and Figure 8 ( Figure 9 ) supports this observation, highlighting the adverse impact of DoS attacks on system performance.

[0102] In summary, this paper successfully designs a controller based on quantized signals that can resist DoS attacks and proportional delays to achieve security consistency in nonlinear complex networks. This provides a new approach to solving the security control problem of complex networks with proportional delays.

Claims

1. A security control method for complex networks against proportional delay and DoS attacks, characterized in that: The steps include: Step 1: Establish a complex network model with proportional delay; Step 2: Build an energy-constrained DoS attack model. Use the average duration to describe and constrain DoS attacks, and introduce it into the complex network model built in Step 1. There are two modes: Mode 1, where only the communication network between nodes is attacked; Mode 2, where both the internal channels of the nodes and the communication network between them are attacked. Step 3: Design quantitative security controllers for different attack modes; Step 4: Establish the conditions for ensuring the stability of the error system under the first attack mode; construct the error system of the complex network using the state and average state of each node. Then, according to different attack modes, establish a simple Lyapunov function for the error system. For the first attack mode, the Lyapunov function established in the error system stability analysis process is: in, is the error state between the state of the ith node and the average state, W=(ω ij ) N×N =E-ηη T ,η=[η1,η2,…,η N ] T is the eigenvector corresponding to the zero eigenvalue of the Laplace matrix A, E=diag{η1,η2,…,η N }, and η satisfies η j >0,j=1,2,…,N and Based on this, the sufficient condition for ensuring the stability of the error system is: in, a=min{a1,a2}, δ∈(0,1),l 01 , l 02 , α b1 , α b2 , α c2 and β1 are positive constants, B1, B2, C1 and C2 are system parameters; c represents coupling gain, K represents feedback gain; Step 5: Establish the conditions for ensuring the stability of the error system under the second attack mode. For the second attack mode, the sufficient conditions for ensuring the stability of the error system are: Among them, β2 is a positive constant, p is the coefficient of proportional delay, T0>0 is a regularization term, is the proportion of attack duration to total time; Step 6: Establish a condition to ensure the stability of the error system under bounded delay. If the system delay degenerates from unbounded proportional delay to general bounded delay, a less conservative stability condition can be obtained.

2. The security control method for complex network against proportional delay and DoS attack according to claim 1, characterized in that: In step 1, a complex network model with proportional delay is established. Consider a system consisting of N nodes and use differential equations to express the evolution of the dynamic behavior of each node. The control signal of the i-th node is u i (k),i=1,2,…,N.

3. The security control method for complex network against proportional delay and DoS attack as claimed in claim 1, characterized in that: In step 2, the i-th DoS attack is represented as: D i ={d i }∪[d i ,d i +h i ), d i Indicates the moment when the attack changes from 0 to 1, h i Indicates the duration of the attack. Depending on whether the system is attacked, the time axis is divided into the attack interval and the safety interval, that is, Here, i = 1, 2, ... and S0 = [t0, d1), the internal channels of the nodes and the external communication network connecting the nodes are potential targets of attack.

4. The security control method for complex network against proportional delay and DoS attack as claimed in claim 1, characterized in that: In step 3, a quantitative safety controller is designed for each attack mode. In the safety interval, node i designs a coupling controller based on the quantitative signals received from neighboring nodes and its own quantitative signals. In the attack interval, in the first attack mode, node i cannot receive signals from its neighbors but receives signals from itself. Therefore, the controller design is completely based on its own quantitative signals. The controller design is as follows: Among them, Q(x i (t)) represents the signal x i The quantized signal of (t), a ij >0(i≠j) indicates a directed edge from node i to node j, allowing node j to receive information from node i; conversely, a ij = 0, it means that there is no communication edge from node i to node j; In addition, define c represents coupling gain and K represents feedback gain.

5. The security control method for complex network against proportional delay and DoS attack as claimed in claim 4, characterized in that: In the second attack mode, both the internal channels of the nodes and the external communication network between them are attacked. Assume that the DoS attacks suffered by the internal and external channels of the nodes are synchronized, that is, they occur and end at the same time. Under this attack, the controller does not receive any signal, so the control input is set to zero. The controller in this part is as follows:

6. The security control method for complex network against proportional delay and DoS attack as claimed in claim 1, characterized in that: In step 6, the unbounded proportional delay pt,p∈(0,1) is replaced by a time-varying bounded delay t-τ(t), where τ(t)∈(0,τ) and τ is a positive constant. Then the complex network becomes: By constructing the Lyapunov function, the sufficient condition for ensuring the stability of the error system is obtained: a1<0,a3>0 a1+β * +be β*τ =0 Among them, β * is a positive constant.

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