Security control method for resisting proportional delay and DoS attack of complex network

Abstract

The invention discloses an anti-proportional delay and DoS attack security control method for a complex network. The method comprises the following steps: step 1, establishing a complex network model with proportional delay; 2, establishing an energy-limited DoS attack model, describing and constraining the DoS attack by adopting average duration, and introducing the DoS attack into the complex network model established in the step 1; 3, aiming at different attack modes, respectively designing quantitative security controllers; 4, establishing a condition for ensuring the stability of an error system in the first attack mode; 5, establishing a condition for ensuring the stability of the error system in a second attack mode; and step 6, establishing a condition for ensuring the stability of the error system under bounded delay, and if the delay of the system is degraded from unbounded proportional delay to general bounded delay, obtaining a stability condition with lower conservative property. According to the method, the security consistency of the complex network with proportional delay can be effectively realized under DoS attack and limited bandwidth.

Description

Technical Field

[0001] The present invention relates to the technical field of complex network control, and in particular to a complex network security control method for resisting 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 and technological 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, showing their huge application potential. These networks exchange information through shared communication channels, so as to efficiently collaborate to complete tasks or solve complex problems. However, the actual application scenarios of complex networks often face multiple technical difficulties, such as time delays, resource constraints, and insecure communication environments. These problems significantly affect the stability and reliability of the network, bringing huge challenges and threats to industrial production and even human society. In order to cope with these challenges and threats, researchers are committed to designing a control strategy with strong robustness and high adaptability to ensure that complex networks can still operate stably in harsh environments.

[0003] In complex networks, nodes interact with each other through open and shared networks. Although this feature promotes collaboration and data transmission, it also makes the network vulnerable to various malicious attacks, which seriously affects or even destroys the normal performance of the network. DoS attack is a common and extremely destructive attack method. Attackers interfere with normal information transmission by blocking communication channels or sending a large number of invalid requests, resulting in system resource exhaustion or inability to provide services. Therefore, building a mathematical model of DoS attacks and enhancing the robustness of complex networks under attacks 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. In particular, energy-constrained DoS attack models are usually characterized by the average duration and frequency of attacks. This model has been recognized by the industry and has been widely used in subsequent research. A natural question is, can we further relax the restrictions on average duration and frequency? However, this is still a topic that needs further research, especially in systems with time delays, which often have stricter restrictions on attacks in order to ensure the performance of complex networks.

[0004] Time delay is a common phenomenon in complex networks, which significantly affects the coordination, communication and decision-making processes between nodes. Therefore, analyzing complex networks with time delay becomes both interesting and challenging. At present, a large number of studies have explored complex networks with time-varying or time-invariant delays and achieved rich results. However, these studies usually assume that the delay has an upper bound, which is a strict condition that limits its applicability. Unlike bounded delays, the phenomenon of unbounded delay in complex networks originates from the long-term memory effect or historical dependence of the system, that is, the state change of a node depends not only on the current state, but also on all past states. This unbounded delay is widely present in practical systems, such as the synaptic plasticity of neurons in biological networks (the long-term impact of historical stimuli), price fluctuations in financial markets (the cumulative effect of historical trading behaviors), and population dynamics in ecosystems (long-term adaptation to environmental changes). The analysis of unbounded delay is of great significance in studying the long-term stability, dynamic behavior prediction and control design of complex networks. Although there are some results on unbounded delay, none of them take into account the impact of network attacks. A noteworthy fact is that the research on simultaneous analysis of resistance 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? Based on this, the problem of secure control of complex networks with proportional delays under DoS attacks is considered. To the best of our knowledge, this problem remains unexplored and unsolved, which is an obvious challenge. Proportional delay, as a type of unbounded delay, has important applications in fields such as the human brain and network science. When proportional delay is introduced in a complex network, the dynamic behavior of the system depends on the current state x(t) and the historical state x(pt), where x(pt) represents the ratio of x(t). Over time, this delay tends to infinity, causing the time evolution of the nodes to mutate proportionally to pt with the proportional delay ratio p. Therefore, the methods previously applied to bounded delays are no longer applicable. In addition, the presence of DoS attacks destroys the original control strategy and causes the system state to diverge. This further increases the difficulty of analyzing complex networks with proportional delays under DoS attacks.

[0005] In complex networks, network communication channels are inherently constrained by limited bandwidth, which seriously affects the information transmission between nodes. This limitation will cause data distortion during transmission, because the received data will deviate from the original data, resulting in errors and reduced system performance. In order to alleviate this practical challenge, researchers have widely adopted signal quantization technology. The present invention adopts a typical quantization mechanism: logarithmic quantizer to deal with limited bandwidth, and on this basis designs a control framework that is resistant to proportional delay and DoS attacks, providing a new idea for the security control of 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 (an unbounded delay) under more general DoS attacks and limited bandwidth.

[0007] Technical solution: The security control method for complex network against proportional delay and DoS attack described in the present invention comprises 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 conditions to ensure the stability of the error system under bounded delay. If the system delay degenerates from an unbounded proportional delay to a 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 an attack interval and a safety interval, that is,

[0016]

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

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

[0019] Furthermore, in step 3, quantized safety controllers are designed for different attack modes. In the safety interval, node i designs a coupling controller based on the quantized signals received from neighboring nodes and its own quantized signals. In 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 design of the controller is completely based on its own quantized signal. The controller design is as follows:

[0020]

[0021] Among them, Q(x i (t)) represents the signal x i (t) is the quantized signal, 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. It is assumed 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 concise 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 i-th node state 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 to ensure 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 the time-varying bounded delay t-τ(t), where τ(t)∈(0,τ), τ is a positive constant, and the complex network becomes:

[0035]

[0036] By constructing the Lyapunov function, the sufficient condition to ensure the stability of the error system is:

[0037] a1<0,a3>0

[0038]

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

[0040] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages: Aiming at the security problem of complex networks with proportional delays under DoS attacks, the present invention provides a simple and easy-to-use security control method: First, a class of nonlinear complex network models with proportional delays is considered, which may be subject to network attacks and have 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 results of delay and DoS attacks are analyzed while existing ones are being analyzed; in addition, by constructing the error system and its Lyapunov function, and using knowledge such as Lyapunov stability theory, mathematical induction, integral inequality and matrix theory, the corresponding quantitative controller design and sufficient conditions for 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 delays 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 It is a communication flow diagram 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 an error trajectory diagram of the error system of the present invention in the first attack mode.

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

[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 a state trajectory diagram of the system in the second attack mode of the present invention.

[0047] Figure 7 This is an 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] Fig. 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 network against proportional delay and DoS attack 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. On the contrary, 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-constrained DoS attack model.

[0057] The i-th DoS attack is represented by: D i ={di}∪[d i ,d i +h i ). 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 timeline into an attack interval and a 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 The set of intervals representing successful communication. Importantly, both the internal channels of the nodes and the external communication network connecting the nodes are potential targets of attack. The present invention explores two modes: mode 1, only the communication network between the nodes is attacked; mode 2, both the internal channels of the nodes and the external communication network between them are attacked.

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

[0061] Step 31: Give a 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 the controller. In the safe interval, node i designs a coupling controller based on the quantized signal received from the neighbor node and its own quantized signal. In 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 design of the controller is completely 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 suffered by 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 coupling 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 concise Lyapunov function for the error system is established. For the first attack mode, the Lyapunov function is established in the error system stability analysis process as follows:

[0071]

[0072] in, is the error state between the i-th node state 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 interval and attack interval is analyzed respectively. On this basis, V(t) is analyzed as a whole, and finally the sufficient condition to ensure 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 safe 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] In the process of error system stability analysis, the V(t) function as in step 4 is established. 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 rate 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 the attack duration to the 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 conditions to ensure 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), τ is a positive constant. The complex network then becomes:

[0089]

[0090] Further, construct the same V(t) function as in step 5, and the sufficient condition to ensure 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 the controller (4), the system (1) can achieve safety 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 experiment is carried out: 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 in Figure 4 and Figure 5 As shown. Figure 2 ( Figure 3 )and Figure 4 ( Figure 5 ), we find that the smaller the p value, the slower the convergence speed, indicating that the 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 consistent performance when both inter-node and intra-node DoS attacks are simultaneously performed with parameters T0 = 5 and T = 2. In addition, if the system communicates in a secure network, it can reach consensus faster. Figure 6 ( Figure 7 )and Figure 8 ( Fig. 9 ) supports this observation, highlighting the adverse impact of DoS attacks on system performance.

[0102] In summary, the present invention successfully designs a controller based on quantized signals, which can resist DoS attacks and proportional delays to achieve security consistency of nonlinear complex networks, and provides a new idea for solving the security control problem of complex networks with proportional delays.

Claims

1. A security control method for complex network against proportional delay and DoS attack, characterized in that: The steps include: Step 1: Establish a complex network model with proportional delay; 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; Step 3: Design quantitative security controllers for different attack modes; Step 4: Establish conditions to ensure the stability of the error system in the first attack mode; Step 5: Establish conditions to ensure the stability of the error system in the second attack mode; Step 6: Establish conditions to ensure the stability of the error system under bounded delay. If the system delay degenerates from an unbounded proportional delay to a 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 as claimed in 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 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.

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 an attack interval and a safety interval, that is, Here, i = 1, 2, … and S0 = [t0, d1), the internal channels of the nodes and the external communication networks 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 3, characterized in that: There are two modes: in mode one, only the communication network between nodes is attacked; in mode two, both the internal channels of the nodes and the communication network between them are attacked.

5. The security control method for complex network against proportional delay and DoS attack as claimed in claim 1, characterized in that: In step 3, quantized safety controllers are designed for different attack modes. In the safety interval, node i designs a coupling controller based on the quantized signals received from neighboring nodes and its own quantized signals. In 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 design of the controller is completely based on its own quantized signal. The controller design is as follows: Among them, Q(x i (t)) represents the signal x i (t) is the quantized signal, 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.

6. The security control method for complex network against proportional delay and DoS attack as claimed in claim 5, characterized in that: 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 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:

7. The security control method for complex network against proportional delay and DoS attack as claimed in claim 1, characterized in that: 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 concise 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: in, is the error state between the i-th node state 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 to ensure 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.

8. The security control method for complex network against proportional delay and DoS attack as claimed in claim 1, characterized in that: In step 5, for the second attack mode, the Lyapunov function established in the error system stability analysis process is: Furthermore, the sufficient condition for ensuring the stability of the error system is obtained as follows: 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.

9. 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 the time-varying bounded delay t-τ(t), where τ(t)∈(0,τ), τ is a positive constant, and the complex network becomes: By constructing the Lyapunov function, the sufficient condition to ensure the stability of the error system is: a1<0,a3>0 a1+β * +be β*τ =0 Among them, β * is a positive constant.

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