Heterogeneous high-order network system containment control method based on simple complex

Abstract

The invention provides a heterogeneous high-order network system containment control method based on simple complex, and relates to the technical field of control and information. For a heterogeneous high-order network system constructed by a simple complex, a containment control law is designed to realize a network synchronization target, a network control efficiency evaluation index is constructed, and key factors influencing control effectiveness are revealed. The high-order correlation characteristics between the nodes are accurately described by means of a simple complex structure, and the dynamic characteristics of the heterogeneous nodes are described by means of a multi-dimensional simplex structure, so that the modeling precision of a complex network system is improved. Based on a containment control strategy, a network control efficiency index is taken as an optimization reference, and a small number of key nodes are selected to apply control signals, so that network synchronous control and stability adjustment are realized, and the control efficiency is improved while the control cost is reduced. The method can be widely applied to cooperative control and dynamic behavior adjustment of various complex systems such as a biological network, a social network and a traffic network, and has high practical value and wide popularization prospect.

Description

Technical Field

[0001] The present invention relates to the field of control and information technology, and in particular to a simplicial complex-based heterogeneous high-order network system containment control method. Background Art

[0002] With the development of complex network theory, network systems play an increasingly important role in the fields of society, economy, biology and engineering. Traditional network modeling is mostly based on graph theory, which describes the structure and dynamic behavior of the system through the binary relationship between nodes and edges. However, with the increase in network scale and structural complexity, multilateral relationships and high-order interactions between nodes gradually emerge. Simplical complex, as a mathematical tool that can describe multi-body interactions, has become an important means of modeling high-order network systems. Simplical complex can represent the multivariate association relationships between network nodes as simplices of different dimensions, such as nodes, edges, triangles and higher-dimensional structures, thereby effectively characterizing the high-order association characteristics in the network.

[0003] In complex network systems, pinning control is a widely used control method that guides the entire network system to a synchronous or stable state by selecting some nodes to apply control signals. Traditional pinning control methods are mainly aimed at graph-structured networks and are difficult to directly apply to high-order network systems that consider multi-body interactions. Therefore, studying a pinning control method for high-order network systems based on simplex complexes has important theoretical significance and practical value for improving the control performance of network systems and expanding the application scenarios of pinning control.

[0004] In addition, many real complex systems have nodes of multiple different dynamic types. For example, nodes in the power grid have different parameters, while the biochemical network that controls cell division in mammals contains a variety of substrates and enzymes. Therefore, the stability judgment conditions of the control of the control that ignores the heterogeneous characteristics of nodes and related research results are greatly limited in practical applications. The "heterogeneity" here refers to the heterogeneity of node dynamics, that is, each node has different self-dynamic behaviors. The combined effect of the self-dynamic behavior of a single node and the high-order structure of the network makes the difficulty and complexity of the control stability analysis of heterogeneous high-order network systems increase accordingly. Considering its practical significance, it is necessary to carry out technical research and development in this area. Summary of the invention

[0005] In view of the deficiencies of the prior art, the present invention provides a control method for heterogeneous high-order network systems based on simplicial complexes; the present invention designs a corresponding control strategy for heterogeneous high-order network systems constructed by simplicial complexes to achieve the synchronization goal of heterogeneous high-order network systems, and proposes indicators for evaluating network control efficiency, revealing the key factors affecting the effectiveness of network control. Compared with the prior art, the present invention fully considers the dynamic heterogeneity of nodes in actual network systems and the high-order interaction mode of structures. The control method provided can achieve effective regulation of heterogeneous high-order networks, reduce control costs while improving network control efficiency, and has a wider range of applications in practical applications.

[0006] A method for controlling a heterogeneous high-order network system based on a simplicial complex comprises the following steps:

[0007] Step 1: Construct a heterogeneous simplicial complex network system with N nodes;

[0008] Construct a heterogeneous simplicial complex network system with N nodes. The dynamic equation of the i-th node is expressed as follows:

[0009]

[0010] in, represents the state vector of the i-th node in the network, represents a set of m-dimensional real vectors, x im represents the mth component of the state vector of the ith node, Represents node j 1 The state vector of represents the first-order derivative of the state of the i-th node with respect to time, m represents the dimension of a single node, superscript T represents transposition, and N represents the total number of network nodes. is the self-dynamic function of the ith node; H (d) , d=1,2,…,D: represents the internal coupling function between node state variables in the d-simplex, satisfying the condition Where d represents the order of the simplex, D represents the dimension of the simplex-complex network, x represents the node state, represents a set of m-dimensional real vectors; the constant σ d >0, d=1,2,…,D represents the coupling strength; The definition of is as follows: when (i,j 1 ,j 2 ,…,j d ) belongs to a d-simplex, otherwise

[0011] The node dynamics of the heterogeneous simplicial complex network system meets the following assumptions:

[0012] Assumption 1: There exists a continuously differentiable Lyapunov function V(x) satisfying So that for each node function f i (x i ), there exists a scalar θ i The following conditions are established:

[0013]

[0014] in, x i represents the state vector of the i-th node, x s Indicates the synchronization state, f i (·) represents the self-dynamic function of the i-th node, θ i represents a scalar related to the dynamics of node i itself, Indicates that the condition ‖x is satisfied i -x s ‖<δx i The set of , ‖·‖ represents the 2-norm, Represents the function V(x i ) for x i The first-order partial derivative of , the symbol ∪ represents the union;

[0015] Step 2: Design the pinning control law to achieve the synchronization goal of the heterogeneous simplicial complex network system;

[0016] The control law is designed as follows:

[0017] u i =b i σ 1 H (1) (x i ,x s ), i=1,2,…,N, (3)

[0018] Among them, the constant σ 1 >0 indicates coupling strength, H (1) : represents the internal coupling function between node state variables; b i represents the control gain. When the control action is applied to node i, b i >0, otherwise b i =0; Indicates synchronization status, meeting the conditions and f(x s )=0,x sm The mth component representing the synchronization state; represents a set of m-dimensional real vectors;

[0019] Step 3: Analyze the synchronization stability conditions of the heterogeneous simplicial complex network system under pinning control based on Lyapunov function;

[0020] The condition for the heterogeneous simplex complex network system (1) to be locally asymptotically stable in the synchronization state under the control law (3) is:

[0021]

[0022] Θ+σ 1 C+σ 2 L (2) +…+σ D L (D) ≥0, (4)

[0023] Among them, H (d) , d=1,2,…,D represents the internal coupling function between the node state variables in the d-simplex, d represents the order of the simplex, D represents the dimension of the simplicial complex network, x i represents the state vector of node i, x j1 Represents node j 1 The state vector of 1 ,θ 2 ,…,θ N}, θ i represents a scalar related to the dynamics of node i itself, and the constant σ d >0, d=1,2,…,D represents coupling strength; C=L (1) + B, B = diag{b 1 ,b 2 ,…b N} is a diagonal matrix, b i , i=1,2,…,N is the control gain corresponding to node i, and the symbol diag represents a diagonal matrix; L (i) , i=1,2,…,D is the generalized Laplacian matrix of the i-simplex;

[0024] Step 4: Propose indicators for evaluating network control efficiency, reveal the key factors affecting the effectiveness of network control, and improve network control efficiency.

[0025] Define a feature matrix Recorded as

[0026] Let λ 1 is the feature matrix The smallest characteristic root of 1 It can be used to evaluate the effectiveness of network control and is called the network control efficiency index; the network control efficiency index λ1 The larger it is, the easier the network is to control; the network control efficiency can be improved by adjusting the layout of the controlled nodes and the size of the control gain.

[0027] The beneficial effects of adopting the above technical solution are:

[0028] The present invention provides a method for controlling heterogeneous high-order network systems based on simplicial complexes, which can effectively solve the problem that traditional graph theory network models are difficult to describe multi-body interaction relationships. By introducing a simplicial complex structure, this method can not only accurately characterize the high-order correlation characteristics between nodes, but also use multidimensional simplices to describe the dynamic characteristics of heterogeneous nodes in the network, thereby improving the modeling accuracy of complex network systems. By adopting a control strategy and taking the network control efficiency index as a reference, a small number of key nodes are selected to apply control signals, which can achieve synchronous control and stability regulation of the entire network system, reduce control costs, and improve control efficiency. This method can be widely used in the coordinated control and dynamic behavior regulation of various complex systems such as biological networks, social networks, and transportation networks, and has good practical value and promotion prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 A flow chart of a simplicial complex-based heterogeneous high-order network system containment control method provided for the implementation of the present invention;

[0030] Figure 2 It is a schematic diagram of a simple complex network structure in the implementation of the present invention;

[0031] Figure 3 is a state evolution diagram of a heterogeneous controlled network node composed of a Lorenz system and a Chen system in the implementation of the present invention;

[0032] Where (a) is σ 1 =94,σ 2 =95, (b) is the node state diagram, 1 =5,σ 2 Node state diagram when =10;

[0033] Figure 4 It is a network control efficiency index diagram in the implementation of the present invention;

[0034] Among them, (a) is the relationship between the network control efficiency index and the control gain, and (b) is the relationship between the network control efficiency index and the number of controlled nodes. DETAILED DESCRIPTION

[0035] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0036] A simplicial complex-based pinning control method for heterogeneous high-order network systems, such as Figure 1 As shown, the following steps are included:

[0037] Step 1: Construct a heterogeneous simplicial complex network system with N nodes;

[0038] Construct a heterogeneous simplicial complex network system with N nodes. The dynamic equation of the i-th node is expressed as follows:

[0039]

[0040] in, represents the state vector of the i-th node in the network, represents a set of m-dimensional real vectors, x im represents the mth component of the state vector of the ith node, Represents node j 1 The state vector of represents the first-order derivative of the state of the i-th node with respect to time, m represents the dimension of a single node, superscript T represents transposition, and N represents the total number of network nodes. is the self-dynamic function of the ith node; H (d) , d=1,2,…,D: represents the internal coupling function between node state variables in the d-simplex, satisfying the condition Where d represents the order of the simplex, D represents the dimension of the simplex-complex network, x represents the node state, represents a set of m-dimensional real vectors; the constant σ d >0, d=1,2,…,D represents the coupling strength; The definition of is as follows: when (i,j 1 ,j 2 ,…,j d ) belongs to a d-simplex, otherwise

[0041] The node dynamics of the heterogeneous simplicial complex network system meets the following assumptions:

[0042] Assumption 1: There exists a continuously differentiable Lyapunov function V(x) that satisfies V(x s )=0, So that for each node function f i (x i ), there exists a scalar θ i The following conditions are established:

[0043]

[0044] in, x i represents the state vector of the i-th node, x s Indicates the synchronization state, f i (·) represents the self-dynamic function of the i-th node, θ i represents a scalar related to the dynamics of node i itself, Indicates that the condition ‖x is satisfied i -x s ‖<δx i The set of , ‖·‖ represents the 2-norm, Represents the function V(x i ) for x i The first-order partial derivative of , the symbol ∪ represents the union;

[0045] In this embodiment, the constructed simple complex network structure is as follows: Figure 2 As shown, there is a 2-simplex, D = 2. For the node's own dynamics equation f i (x i ), consider it as Lorenz system and Chen system. The dynamic equation of the Lorenz system itself is:

[0046]

[0047] Among them, when the parameter value is a 1 =10,a 2 =8 / 3,a 3 =28, we get a balance point (i.e. synchronous state) x s =[0,0,0] T ; The self-dynamic equation of the Chen system is:

[0048]

[0049] Among them, when the parameter value is β 1 =35,β 2 =3,β 3 =28, we get a balance point (i.e. synchronous state) x s =

[0050] [0,0,0] T ;

[0051] The goal of this embodiment is to control the state of all nodes in the network to the synchronization state [0,0,0] T ; According to assumption 1, we can calculate the θ corresponding to the Lorenz system and the Chen system. i They are: L =-29,θ C =-2;

[0052] Step 2: Design the pinning control law to achieve the synchronization goal of the heterogeneous simplicial complex network system;

[0053] The control law is designed as follows:

[0054] u i =b i σ 1 H (1) (x i ,x s ), i=1,2,…,N, (3)

[0055] Among them, the constant σ 1 >0 indicates coupling strength, H (1) : represents the internal coupling function between node state variables; b i represents the control gain. When the control action is applied to node i, b i >0, otherwise b i =0; Indicates synchronization status, meeting the conditions and f(x s )=0,x sm The mth component representing the synchronization state; represents a set of m-dimensional real vectors;

[0056] Step 3: Analyze the synchronization stability conditions of the heterogeneous simplicial complex network system under pinning control based on Lyapunov function;

[0057] The condition for the heterogeneous simplicial complex network system (1) to be locally asymptotically stable in the synchronized state under the control law (3) is:

[0058]

[0059]

[0060] Among them, H (d) , d=1,2,…,D represents the internal coupling function between the node state variables in the d-simplex, d represents the order of the simplex, D represents the dimension of the simplicial complex network, x i represents the state vector of node i, Represents node j 1 The state vector of 1 ,θ 2 ,…,θ N}, θ i represents a scalar related to the dynamics of node i itself, and the constant σ d >0, d=1,2,…,D represents coupling strength; C=L (1)+ B, B = diag{b 1 ,b 2 ,…b N} is a diagonal matrix, b i , i=1,2,…,N is the control gain corresponding to node i, and the symbol diag represents a diagonal matrix; L (i) , i=1,2,…,D is the generalized Laplacian matrix of the i-simplex;

[0061] In this embodiment, nodes 1, 2, and 3 are Lorenz systems, and nodes 4, 5, and 6 are Chen systems; nodes 1, 2, and 3 are selected for containment control. 1 =94,σ 2 =95, when the control gain is 50, the stability condition (4) is met after verification, and the node state diagram of the controlled network is as follows Figure 3 As shown in (a), all nodes are controlled to the synchronous state. 1 =5,σ 2 =10, when the control gain is 15, the stability condition (4) is not satisfied after verification, and the node state diagram of the controlled network is as follows Figure 3 As shown in (b), the network does not achieve the synchronization goal. This proves that the simulation results are consistent with the theoretical results;

[0062] Step 4: Propose indicators for evaluating network control efficiency, reveal the key factors affecting the effectiveness of network control, and improve network control efficiency.

[0063] Define a feature matrix Recorded as From the stability condition (4), it can be seen that the synchronization control problem of a complex heterogeneous simplicial complex network system is transformed into a semi-positive definite problem of determining a characteristic matrix, which is related to the node dynamics, the high-order interaction mode of the network structure, the coupling strength, and the containment control strategy;

[0064] Let λ 1 is the feature matrix The smallest characteristic root of 1 It can be used to evaluate the effectiveness of network control, which is called "network control efficiency index". Through analysis, it can be obtained that the network control efficiency index λ 1 The larger it is, the easier the network is to control. In practical applications, the network control efficiency can be improved by adjusting the layout of the controlled nodes and the size of the control gain.

[0065] In this embodiment, in order to reveal the key factors affecting the effectiveness of network control, we first analyze Figure 2 The relationship between control efficiency and control gain in the network shown in Figure 1 is as follows. 1 =σ 2 =σ, the control efficiency index λ can be obtained1 The relationship between the control gain b and σ is shown in the figure below: Figure 4 (a) As shown; it is easy to see that the larger the control gain, the larger λ 1 The larger the network, the easier it is to control. Figure 2 The relationship between control efficiency and the number of controlled nodes in the network shown. When the control gain b = 20, the control efficiency index λ can be obtained 1 The relationship between the number of controlled nodes l and σ is shown in the figure below: Figure 4 (b) As shown; it is easy to see that the larger the number of controlled nodes, the larger the 1 The larger it is, the easier the network is to control.

[0066] 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 control method for heterogeneous high-order network systems based on simplicial complexes, characterized in that: The following steps are involved: Step 1: Construct a heterogeneous simplicial complex network system with N nodes; Construct a heterogeneous simplicial complex network system with N nodes. The dynamic equation of the i-th node is expressed as follows: in, represents the state vector of the i-th node in the network, represents a set of m-dimensional real vectors, x im represents the mth component of the state vector of the ith node, represents the state vector of node j1, represents the first-order derivative of the state of the i-th node with respect to time, m represents the dimension of a single node, superscript T represents transposition, and N represents the total number of network nodes. is the self-dynamic function of the ith node; H (d) , d=1,2,…,D: represents the internal coupling function between node state variables in the d-simplex, satisfying the condition Where d represents the order of the simplex, D represents the dimension of the simplex-complex network, x represents the node state, represents a set of m-dimensional real vectors; the constant σ d >0, d=1,2,…,D represents the coupling strength; The definition of is as follows: when (i, j1, j2, …, j d ) belongs to a d-simplex, otherwise Step 2: Design the pinning control law to achieve the synchronization goal of the heterogeneous simplicial complex network system; Step 3: Analyze the synchronization stability conditions of the heterogeneous simplicial complex network system under pinning control based on Lyapunov function; Step 4: Propose indicators for evaluating network control efficiency, reveal the key factors affecting the effectiveness of network control, and improve network control efficiency.

2. According to the method for controlling heterogeneous high-order network systems based on simplicial complexes in claim 1, it is characterized in that: The node dynamics of the heterogeneous simplicial complex network system described in step 1 satisfies the following assumptions: Assumption 1: There exists a continuously differentiable Lyapunov function V(x) that satisfies V(x s )=0, So that for each node function f i (x i ), there exists a scalar θ i The following conditions are established: in, δ>0, x i represents the state vector of the i-th node, x s Indicates the synchronization state, f i (·) represents the self-dynamic function of the i-th node, θ i represents a scalar related to the dynamics of node i itself, Indicates that the condition ‖x is satisfied i -x s ‖<δx i The set of , ‖·‖ represents the 2-norm, Represents the function V(x i ) for x i The first-order partial derivative of , where the symbol ∪ represents the union.

3. The method for controlling a heterogeneous high-order network system based on a simplicial complex according to claim 1, characterized in that: The control law described in step 2 is designed as follows: u i =b i σ1H (1) (x i ,x s ), i=1,2,…,N, (3) where the constant σ1>0 represents the coupling strength, represents the internal coupling function between node state variables; b i represents the control gain. When the control action is applied to node i, b i >0, otherwise b i =0; Indicates synchronization status, meeting the conditions and f(x s )=0,x sm The mth component representing the synchronization state; Represents a set of m-dimensional real vectors.

4. The method for controlling a heterogeneous high-order network system based on a simplicial complex according to claim 1, characterized in that: The step 3 is specifically as follows: the condition for the heterogeneous simplicial complex network system (1) to be locally asymptotically stable in the synchronization state under the action of the pinning control law (3) is: Θ+σ1C+σ2L (2) +…+s D L (D) ≥0, (4) Among them, H (d) , d=1,2,…,D represents the internal coupling function between the node state variables in the d-simplex, d represents the order of the simplex, D represents the dimension of the simplicial complex network, x i represents the state vector of node i, represents the state vector of node j1; θ=diag{θ1,θ2,…,θ N }, θ i represents a scalar related to the dynamics of node i itself, and the constant v d >0, d=1,2,…,D represents coupling strength; C=L (1) + B, B = diag{b1, b2, ... b N } is a diagonal matrix, b i , i=1,2,…,N is the control gain corresponding to node i, and the symbol diag represents a diagonal matrix; L (i) ,i=1,2,…,D is the generalized Laplacian matrix of the i-simplex.

5. The method for controlling heterogeneous high-order network systems based on simplicial complexes according to claim 1, characterized in that: The step 4 is specifically as follows: define a feature matrix Recorded as Let λ1 be the feature matrix The smallest characteristic root of , then λ1 can be used to evaluate the effectiveness of network control, which is called the network control efficiency index; the larger the network control efficiency index λ1, the easier the network is to be controlled; the network control efficiency can be improved by adjusting the layout of the controlled nodes and the size of the control gain.

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