A simplicial complex-based random high-order network pinning control method

Abstract

The application provides a simplicial complex-based random high-order network pinning control method and relates to the technical fields of control and information. The application is directed to a simplicial complex network affected by noise described by a nonlinear stochastic differential equation, and a reasonable and effective pinning control law is designed to achieve the synchronization target. By introducing a simplicial complex framework, the model can accurately describe the high-order correlation and complex interaction relationship between nodes in the network, and is more close to the characteristics of the actual system. The description method of the nonlinear Ito type stochastic differential equation is adopted, and the network model can truly reproduce the nonlinear characteristics of the actual system and the influence of the modeling noise. On this basis, the pinning control strategy designed has a wider application range, a lower control cost, can enhance the stability of the network, and can avoid the system from falling into instability or functional degradation, thereby providing strong theoretical support and technical tools for the regulation and control of the dynamic behavior of complex systems such as biological networks, social networks and communication networks.

Description

Technical Field

[0001] This invention relates to the field of control and information technology, and in particular to a method for constrained control based on a simple complex stochastic high-order network. Background Technology

[0002] With the continuous development of science and technology, complex networks are increasingly widely used in fields such as communication, transportation, biological systems, sociology, and artificial intelligence. How to effectively control the dynamic behavior of nodes in these networks has become a key research focus. In recent years, high-order network models have become an important direction in complex network research due to their ability to describe higher-order interactions between nodes. Traditional complex networks are usually modeled based on first-order interactions, while high-order networks, by introducing the mathematical framework of simplical complexes, can more realistically reflect the higher-order relationships between nodes. In reality, higher-order interactions in networks often exhibit nonlinear and stochastic characteristics, making the modeling and control of high-order networks more challenging. In theoretical research, network models can be described using nonlinear stochastic differential equations to capture the complex characteristics of the dynamic behavior of stochastic high-order networks.

[0003] The dynamic behavior of high-order networks is influenced by both their structural characteristics and external random disturbances, potentially leading to performance degradation or instability. Existing control strategies primarily focus on low-order networks, lacking effective control methods for high-order networks, especially those with nonlinear stochastic characteristics. To effectively constrain and regulate the dynamic behavior of high-order networks, it is necessary to design control methods for high-order networks described by nonlinear stochastic differential equations, taking into account the structural characteristics of simplex complexes. This is not only significant for expanding complex network theory but also provides theoretical basis and technical support for solving control problems in high-dimensional, uncertain, and complex systems. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a constraint control method for stochastic high-order networks based on simplex complexes. This invention targets high-order networks described by nonlinear stochastic differential equations that are susceptible to noise, and achieves synchronization by designing a reasonable and effective constraint control law. Compared with existing technologies, this invention fully considers the high-order interactions and stochastic characteristics of real-world network systems. The constraint control method provided can effectively regulate stochastic high-order networks, improve system robustness, reduce control costs, and has a wider range of applications in practice.

[0005] A method for constraining and controlling stochastic high-order networks based on simple complexes includes the following steps:

[0006] Step 1: Construct a simple complex network with N nodes:

[0007] Construct a simple complex network G = (V, E) with N nodes, where V = {v1, v2, ..., v...} N} represents the node set, v N Let N be the Nth node, and E be the edge set; (i1, i2, ..., i d+1 ) represents higher-order interactions between nodes, i d+1 Let be the (d+1)th node; use simplexes to represent higher-order interactions between nodes, where 0-simplex is a node, 1-simplex is an edge, 2-simplex is a triangle, and d-simplex is the complete graph formed by d+1 nodes; the set of all simplexes in a network is called its simplex complex, and simplex complexes are used to represent higher-order networks, which are undirected, unweighted, and connected;

[0008] Step 2: Based on the high-order network topology, determine the dynamic equations of the D-dimensional stochastic simplex network:

[0009]

[0010] Where, x i (t)=[x i1 (t),x i2 (t),…,x im (t)] T ∈R m Let i = 1, 2, ..., N represent the state vector of node i at time t, and x im (t) represents the m-th component of the state vector of node i. This represents the state vector of node j1, with the superscript T indicating transpose, and R... m Denotes the set of m-dimensional real vectors; F(·):R m ×R + →R m R is a continuously differentiable function describing the dynamics of a node itself. + H represents the set of positive real numbers; (d) d = 1, 2, ..., D: R (d+1)m ×R + →R m This represents the internal coupling function between nodal state variables in a d-simplex, satisfying the condition... Where D represents the dimension of the simplex complex network, x represents the node state, and σ is a constant. d >0, d=1,2,…,D represents the coupling strength; The definition is as follows: when (i,j1,j2,…,j…) d When it belongs to the d-simplex, otherwise random term σ v g(x i dW(t) represents the uncertainty of the nodal dynamics, W(t) is a one-dimensional Brownian motion used to represent noise, and g(·):R m ×R + →R m It is the noise transfer function, σ v Indicates noise intensity;

[0011] Step 3: Design a restraint control law to enable the stochastic simplex network to achieve the synchronization goal;

[0012] The control law is designed as follows:

[0013] u i (t)=b i σ1H (1) (x i (t),x s (t)) (2)

[0014] Where i = 1, 2, ..., N, the constant σ1 > 0 represents the coupling strength, H (1) :R 2m ×R + →R m b represents the coupling function between node state variables; i This represents the control gain, b, when the control action is applied to node i. i >0, otherwise b i =0; x s (t)=[x s1 (t),x s2 (t),…,x sm (t)] T ∈R m To represent a synchronized state, the following conditions must be met:

[0015] dx s (t)=F(x s (t))+σ v g(x s (t))dW(t) (3)

[0016] Control all nodes in the entire network to synchronize state x. s (t), the constrained D-dimensional stochastic simplex network is represented as follows:

[0017]

[0018] Where, x i (t)=[x i1 (t),x i2 (t),…,xim (t)] T ∈R m Let i = 1, 2, ..., N represent the state vector of node i at time t, and x im (t) represents the m-th component of the state vector of node i. This represents the state vector of node j1, with the superscript T indicating transpose, and R... m Denotes the set of m-dimensional real vectors; F(·):R m →R m R is a continuously differentiable function describing the dynamics of a node itself. + H represents the set of positive real numbers; (d) d = 1, 2, ..., D: R (d+1)m →R m This represents the internal coupling function between nodal state variables in a d-simplex, satisfying the condition... Where D represents the dimension of the simplex complex network, x represents the node state, and σ is a constant. d >0, d=1,2,…,D represents the coupling strength; The definition is as follows: when (i,j1,j2,…,j…) d When it belongs to the d-simplex, otherwise random term σ v g(x i dW(t) represents the uncertainty of nodal dynamics, W(t) is a one-dimensional Brownian motion used to represent noise, and g(·):R m →R m It is the noise transfer function, σ v Indicates noise intensity; u i It is the designed restraint and control law;

[0019] Step 4: Based on the master stability function method, obtain the stability criteria for constrained stochastic simplex networks;

[0020] Step 4.1: Define the nodal error as e i (t)=x i (t)-x s For a given network (t), i = 1, 2, ..., N, based on existing master stability function methods, the stochastic master stability equation for a constrained stochastic simplex network is theoretically derived as follows:

[0021]

[0022] Where y(t) is an auxiliary variable, J F and J g These are the Jacobian matrices of functions F(·) and g(·) in the synchronous state, respectively, and σ vLet W(t) represent noise intensity and W(t) represent one-dimensional Brownian motion. For the function ω (1) In the Jacobian matrix of the synchronous state, ω (1) The following conditions must be met:

[0023] H (1) (x i (t),x j (t))=ω (1) (x j (t))-ω (1) (x i (t))

[0024] H (2) (x i (t),x j (t),x k (t))=ω (2) (x j (t),x k (t))-ω (2) (x i (t),x i (t)) (6)

[0025] ω (2) (x(t),x(t))=ω (1) (x(t))

[0026] Among them, H (1) and H (2) Let ω represent the internal coupling functions between the nodal state variables in the 1-simplex and 2-simplex, respectively. (1) and ω (2) Representing H respectively (1) and H (2) Related functions;

[0027] Step 4.2: A constrained D-dimensional stochastic simplex network is locally exponentially stable in a stochastic sense if and only if the maximum Lyapunov exponent ∧ of the stochastic master stability equation (5) is less than 0, where the maximum Lyapunov exponent ∧ is related to η and σ v The relevant function, η is a matrix P D =σ1C+σ2L (2) +…+σ D L (D) Characteristic roots, C = L (1) +B, B = diag{b1,b2,….b N} is a diagonal matrix, b i ,i=1,2,…,N is the control gain of the corresponding node i, L (i)Let i = 1, 2, ..., D be the generalized Laplacian matrix of the i-simplex; σ is a constant. d >0, d=1,2,…,D represents the coupling strength; σ v The noise intensity is represented by the maximum Lyapunov exponent ∧, which is also called the stochastic master stable function. By designing a restraint control law that satisfies the above conditions, the stochastic simplex network (4) can achieve the synchronization goal.

[0028] The beneficial effects of adopting the above technical solution are as follows:

[0029] This invention provides a simplex complex-based stochastic high-order network restraint control method, which effectively solves the problem of dynamic behavior control in nonlinear stochastic high-order networks, possessing significant theoretical and practical value. By introducing a simplex complex framework, the model can accurately describe the high-order associations and complex interactions between nodes in the network, more closely reflecting the characteristics of real-world systems. Simultaneously, by employing a nonlinear Iton-type stochastic differential equation description method, the network model of this invention can realistically reproduce the nonlinear characteristics of real-world systems and the impact of modeling noise. The restraint control strategy designed based on this method has a wider range of applications and lower control costs. This method not only enhances network stability and prevents the system from falling into instability or functional degradation, but also provides strong theoretical support and technical tools for the dynamic behavior regulation of complex systems such as biological networks, social networks, and communication networks, and is widely applicable to high-dimensional, uncertain, and complex systems in modern science and technology. Attached Figure Description

[0030] Figure 1 A flowchart of a stochastic high-order network restraint control method based on simple complexes provided for the implementation of this invention;

[0031] Figure 2 This is a schematic diagram of a simple complex network structure in an embodiment of the present invention;

[0032] Figure 3 This is a diagram showing the state evolution of nodes in a stochastic simplex network under constraint control in the implementation of this invention. Detailed Implementation

[0033] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0034] A method for constraining control based on stochastic high-order networks using simple complexes, such as... Figure 1 As shown, it includes the following steps:

[0035] Step 1: Construct a simple complex network with N nodes:

[0036] Construct a simple complex network G = (V, E) with N nodes, where V = {v1, v2, ..., v...} N} represents the node set, v N Let N be the Nth node, and E be the edge set; (i1, i2, ..., i d+1 ) represents higher-order interactions between nodes, i d+1 Let be the (d+1)th node; use simplexes to represent higher-order interactions between nodes, where 0-simplex is a node, 1-simplex is an edge, 2-simplex is a triangle, and d-simplex is the complete graph formed by d+1 nodes; the set of all simplexes in a network is called its simplex complex, and simplex complexes are used to represent higher-order networks, which are undirected, unweighted, and connected;

[0037] like Figure 2 As shown, this embodiment constructs a network with four nodes, comprising four 0-simplexes, five 1-simplexes, and two 2-simplexes. All of these simplexes form a simplicium complex, which is a 2-dimensional simplicium complex, i.e., D = 2.

[0038] Step 2: Based on the high-order network topology, determine the dynamic equations of the D-dimensional stochastic simplex network:

[0039]

[0040] Where, x i (t)=[x i1 (t),x i2 (t),…,x im (t)] T ∈R m Let i = 1, 2, ..., N represent the state vector of node i at time t, and x im (t) represents the m-th component of the state vector of node i. This represents the state vector of node j1, with the superscript T indicating transpose, and R... m Denotes the set of m-dimensional real vectors; F(·):R m ×R + →R m R is a continuously differentiable function describing the dynamics of a node itself. + H represents the set of positive real numbers; (d) d = 1, 2, ..., D: R (d+1)m ×R + →R m This represents the internal coupling function between nodal state variables in a d-simplex, satisfying the condition... Where D represents the dimension of the simplex complex network, x represents the node state, and σ is a constant. d >0, d=1,2,…,D represents the coupling strength; The definition is as follows: when (i,j1,j2,…,j…) d When it belongs to the d-simplex, otherwise random term σ v g(x i dW(t) represents the uncertainty of the nodal dynamics, W(t) is a one-dimensional Brownian motion used to represent noise, and g(·):R m ×R + →R m It is the noise transfer function, σ v This represents the noise intensity; it should be noted that equation (1) is an Itō-type nonlinear stochastic differential equation. Type), the simple complex network described by equation (1) is affected by modeling noise;

[0041] Step 3: Design a restraint control law to enable the stochastic simplex network to achieve the synchronization goal;

[0042] The control law is designed as follows:

[0043] u i (t)=b i σ1H (1) (x i (t),x s (t)) (2)

[0044] Where i = 1, 2, ..., N, the constant σ1 > 0 represents the coupling strength, H (1) :R 2m ×R + →R m b represents the coupling function between node state variables; i This represents the control gain, b, when the control action is applied to node i. i >0, otherwise b i =0; x s (t)=[x s1 (t),x s2 (t),…,x sm (t)] T ∈R m To represent a synchronized state, the following conditions must be met:

[0045] dx s (t)=F(x s (t))+σ v g(x s (t))dW(t) (3)

[0046] Control all nodes in the entire network to synchronize state x. s (t), the constrained D-dimensional stochastic simplex network is represented as follows:

[0047]

[0048] Where, x i (t)=[x i1 (t),x i2 (t),…,x im (t)] T ∈R m Let i = 1, 2, ..., N represent the state vector of node i at time t, and x im (t) represents the m-th component of the state vector of node i. This represents the state vector of node j1, with the superscript T indicating transpose, and R... m Denotes the set of m-dimensional real vectors; F(·):R m →R m R is a continuously differentiable function describing the dynamics of a node itself. + H represents the set of positive real numbers; (d) d = 1, 2, ..., D: R (d+1)m →R m This represents the internal coupling function between nodal state variables in a d-simplex, satisfying the condition... Where D represents the dimension of the simplex complex network, x represents the node state, and σ is a constant. d >0, d=1,2,…,D represents the coupling strength; The definition is as follows: when (i,j1,j2,…,j…) d When it belongs to the d-simplex, otherwise random term σ v g(x i dW(t) represents the uncertainty of nodal dynamics, W(t) is a one-dimensional Brownian motion used to represent noise, and g(·):R m →R m It is the noise transfer function, σ v Indicates noise intensity; u i The designed restraint control law is shown in equation (2). It should be noted that the restraint control law designed in this invention only needs to apply control to a small number of nodes in the network, which greatly reduces the control cost.

[0049] Step 4: Based on the master stability function method, obtain the stability criteria for constrained stochastic simplex networks;

[0050] Step 4.1: Define the nodal error as e i (t)=x i(t)-x s For a given network (t) i = 1, 2, ..., N, based on the existing Master Stability Function (MSF) method, the stochastic master stability equation for a constrained stochastic simplex network is theoretically derived, as follows:

[0051]

[0052] Where y(t) is an auxiliary variable, J F and J g These are the Jacobian matrices of functions F(·) and g(·) in the synchronous state, respectively, and σ v Let W(t) represent noise intensity and W(t) represent one-dimensional Brownian motion. For the function ω (1) In the Jacobian matrix of the synchronous state, ω (1) The following conditions must be met:

[0053] H (1) (x i (t),x j (t))=ω (1) (x j (t))-ω (1) (x i (t))

[0054] H (2) (x i (t),x j (t),x k (t))=ω (2) (x j (t),x k (t))-ω (2) (x i (t),x i (t)) (6)

[0055] ω (2) (x(t),x(t))=ω (1) (x(t))

[0056] Among them, H (1) and H (2) Let ω represent the internal coupling functions between the nodal state variables in the 1-simplex and 2-simplex, respectively. (1) and ω (2) Representing H respectively (1) and H (2) Related functions;

[0057] Step 4.2: A constrained D-dimensional stochastic simplex network is locally exponentially stable in a stochastic sense if and only if the maximum Lyapunov exponent ∧ of the stochastic master stability equation (5) is less than 0, where the maximum Lyapunov exponent ∧ is related to η and σ v The relevant function, η is a matrix P D =σ1C+σ2L (2) +…+σ D L (D) Characteristic roots, C = L (1) +B, B = diag{b1,b2,….b N} is a diagonal matrix, b i ,i=1,2,…,N is the control gain of the corresponding node i, L (i) Let i = 1, 2, ..., D be the generalized Laplacian matrix of the i-simplex; σ is a constant. d >0, d=1,2,…,D represents the coupling strength; σ v The noise intensity is represented by the maximum Lyapunov exponent ∧, which is also called the Stochastic Master Stability Function (SMSF). By designing a restraint control law that satisfies the above conditions, the stochastic simplex network (4) can achieve the synchronization goal.

[0058] In this embodiment, the node's self-dynamics is set as system. The system's own dynamic equations are:

[0059]

[0060] The parameters are set as a = β = 0.2 and c = 9.

[0061] The goal of this embodiment is to bring all nodes in the network to a synchronized state x. s (t)=[x s1 (t),x s2 (t),x s3 (t)] T .

[0062] This example sets the function. The coupling strength is σ1 = 0.5, σ2 = 0.01; the noise intensity is set as σ v =3; the noise transfer function is g(x) i (t))=x i (t); control is applied to node 1, with a control gain of b1 = 2; simulation is performed using MATLAB to obtain the evolution diagram of the state of the four nodes over time, as shown below. Figure 3 As shown. By Figure 3It can be seen that after the random simplex network is subjected to restraint control, the states of all nodes in the system gradually tend to be synchronized, thus achieving the synchronization goal, which shows that the restraint control method proposed in this invention is effective.

[0063] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.

Claims

1. A simplicial complex-based random high-order network pinning control method, characterized in that, The method comprises the following steps: Step 1: constructing a simple complex network with N nodes: Construct a simplicial complex network G = (V, E) with N nodes, where, V is the node set, is the Nth node, and E is the edge set; represents the high-order interaction between nodes, i d+1 is the d+1th node; a simplicial complex is used to represent the high-order interaction between nodes, where a 0-simplex is a node, a 1-simplex is an edge, a 2-simplex is a triangle, and a d-simplex is a complete graph composed of d+1 nodes; the set of all simplices in the network is called its simplicial complex, which is used to represent the high-order network, and the high-order network is undirected, unweighted, and connected; Step 2: determining a dynamic equation of a D-dimensional random simple complex network according to a high-order network topology structure: The dynamic equation is as follows: （1）； wherein, denotes the state vector of node i at time t, denotes the m-th component of the state vector of node i, denotes the state vector of node , the superscript T denotes the transpose, denotes a set of m-dimensional real vectors; is a continuously differentiable function describing the node's own dynamics, denotes a set of positive real numbers; denotes an internal coupling function between the state variables of nodes in a d-simplex, satisfying the condition , where D denotes the dimension of the simplicial complex network and x denotes the state of the nodes; the constant denotes the coupling strength; is defined as follows: when belongs to a d-simplex, , otherwise ; the stochastic term represents the uncertainty of the node dynamics, is a one-dimensional Brownian motion used to represent noise, is a noise transfer function, denotes the noise intensity; Step 3: designing a pinning control law to enable the random simple complex network to achieve a synchronization target; The pinning control law is designed as follows: （2）； wherein, , a constant represents the coupling strength, represents a coupling function between the node state variables; b i represents a control gain when a control action is added to node i, , otherwise ; represents a synchronized state, satisfying the following conditions: （3）； Controlling all nodes of the entire network to a synchronous state The D-dimensional random simplicial complex network under the control of constraint is represented as follows: （4）； where, denotes the state vector of node i at time t, denotes the m-th component of the state vector of node i, denotes the state vector of node , the superscript T denotes the transpose, denotes a set of m-dimensional real vectors; is a continuously differentiable function describing the node's own dynamics, denotes a set of positive real numbers; denotes an inner coupling function between the state variables of nodes in a d-simplex, satisfying the condition , where D denotes the dimension of the simplicial complex network, x denotes the state of nodes; the constant denotes the coupling strength; is defined as follows: when belongs to a d-simplex, , otherwise ; the random term represents the uncertainty of the node dynamics, is a one-dimensional Brownian motion used to represent noise, is a noise transfer function, denotes the noise intensity; is the designed damping control law; Step 4: obtaining a stability discrimination condition of the random simple complex network subjected to the pinning control according to a master stability function method; The step 4 comprises the following steps: Step 4.1: Define the node error as According to the existing master stability function method, the stochastic master stability equation of the stochastic simple complex network under pinning control is derived theoretically, which is shown as follows: （5）； where y(t) is an auxiliary variable, and are functions and is the Jacobian matrix of the synchronous state, is the noise intensity, is a one-dimensional Brownian motion, is a function is the Jacobian matrix of the synchronous state, satisfies the following conditions: ； （6）； ； wherein, and denote the internal coupling functions between the node state variables in the 1-simplex and 2-simplex, respectively, and denote the functions related to and and respectively. Step 4.2: The D-dimensional stochastic simple-complex network under the pinning control is locally exponentially stable in the sense of randomness if and only if the largest Lyapunov exponent of the stochastic master stability equation (5) where the largest Lyapunov exponent is a function related to and , is the eigenvalue of the matrix , , is a diagonal matrix, is the control gain of the corresponding node i, is the generalized Laplacian matrix of the i-simplex; the constant denotes the coupling strength; denotes the noise strength; the largest Lyapunov exponent here is also called the stochastic master stability function; by designing the pinning control law satisfying the above conditions, the stochastic simple-complex network (4) achieves the synchronization goal.

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