CN116577988A - Higher-order network pinning control method - Google Patents

Higher-order network pinning control method Download PDF

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CN116577988A
CN116577988A CN202310546787.0A CN202310546787A CN116577988A CN 116577988 A CN116577988 A CN 116577988A CN 202310546787 A CN202310546787 A CN 202310546787A CN 116577988 A CN116577988 A CN 116577988A
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simplex
node
pinning
network
order
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CN116577988B (en
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项林英
夏儒雅
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Northeastern University Qinhuangdao Branch
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

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Abstract

The invention provides a control method for the drag of a high-order network, and relates to the technical field of high-order network synchronization. The present invention contemplates representing higher order networks with simple complex shapes. The ultimate goal of the control-constrained higher order network is to fully synchronize the higher order network to the desired state. Based on the dynamic equation of the D-dimensional simplex complex and the purpose of controlling the higher-order network by the pinning control, the pinning controller is designed. And selecting a controlled node, applying a pinning controller to the controlled node, and performing pinning control on the high-order network. Compared with the method for controlling all nodes in the network, the method has the advantages of lower control cost, feasibility and practical application value. The technology can be applied to the fields of Internet, social network, electric power system and the like, and provides a new idea for solving a plurality of problems in actual life. The invention has important value for the research in the field of high-order network synchronization and is also significant for actual production and life.

Description

Higher-order network pinning control method
Technical Field
The invention relates to the technical field of high-order network synchronization, in particular to a control method for controlling the drag of a high-order network.
Background
In complex networks, interactions between multiple nodes are referred to as higher-order interactions, which are on the higher-order structure of the network. In reality most systems contain high-order interactions, for example, predation relationships among multiple populations in an ecosystem can be expressed as high-order interactions; in chemical systems, interactions between multiple compounds can also be represented as higher order interactions. Researchers have introduced simplex to describe higher-order structures in networks. In an undirected graph, a d-simplex (d-simplex) is a complete graph formed by d+1 nodes in the network, and may represent a higher-order structure formed by d+1 nodes in the network. The topology of the simplex components of the different orders is called simplex (simplicial complex). A simplex complex of all simplex combinations in an undirected graph can be used to represent a higher order network. Past studies have given a general kinetic model of simplex, while demonstrating that simplex is an effective mathematical tool for representing higher order networks. In recent research, researchers have placed more and more attention in the field of higher order network synchronization. When the network itself cannot synchronize to the desired state, the network is typically synchronized to the desired state by adding some means of external control. The control is to control a part of nodes in the network, so that all the nodes in the network tend to be in the same state.
In practical production applications, one would want the higher order network to be fully synchronized to the desired state, but the higher order network itself is difficult to achieve this goal. For this reason, the pinning control method is considered to be applied to the higher order network.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a control method for controlling the drag of a high-order network.
A control method for the high-order network is provided, which comprises the following steps:
step 1: constructing an undirected graph with N nodes:
constructing an unowned undirected graph g= (V, E) with N nodes, wherein v= { V 1 ，v 2 ，...，v N "is a node set, v N Represents the nth node, e= { (v) i ，v j ) [ V ] is the edge set i and vj Respectively representing an ith node and a jth node; (i) 1 ，i 2 ，...，i d+1 ) Representing higher-order interactions between nodes, i d+1 Represents the (d+1) th node in the higher order interaction; high-order interaction between nodes is represented by using simplex, wherein 0-simplex is set as a node, 1-simplex is an edge, 2-simplex is a triangle, and d-simplex is formed by d+1 nodesA completed graph; the collection of all simplex in the undirected graph is called simplex, and the simplex is used for representing a higher-order network;
step 2: determining a dynamic equation of the D-dimensional simplex complex according to a network topology structure:
wherein ,a state vector representing node i, and a superscript T represents a transpose of the vector; f R m →R m Is a continuously differentiable function describing the self dynamics of the node; h is a (d) ，d＝1,2,…,D：R (d+1)×m →R m Representing a coupling function, h, between multiple node state variables with synchronous non-invasiveness under d-simplex (d) (x,x,…,x)≡0，/>Constant sigma d >0, d=1, 2, …, D representing the coupling strength; />Is defined as follows: when (i, j) 1 ,j 2 ,…,j d ) Belongs to d-simplex->Otherwise->f(x i ) Is the self-kinetic equation of the node; j (j) D Represents the (D+1) th node in the D-simplex, where D is the order of the highest order simplex in the simplex.
Step 3: designing a pinning controller according to the purpose of pinning control of the higher-order network; selecting a controlled node, and controlling the control of the higher order network in a pinning way:
step 3.1: the establishment requirement is as follows:fully synchronize equation (1) to the desired state x s When t → infinity, the following conditions are satisfied:
x s ＝x 1 ＝…＝x N (2)
wherein ,at the same time satisfy f (x) s )＝0。
Step 3.2: designing a hold-down controller; adding control input to the controlled node, designing a hold-down controller u i The specific expression is:
u i ＝b i σ 1 h (1) (x i ,x s ) (3)
wherein the constant b i >0 represents a control gain;representing an expected state at network synchronization;
when x is s ＝x i When h (1) (x i ,x s ) ≡0, control input is zero; at the same time by adjusting b i To adjust the magnitude of the control input;
step 3.3: the control method comprises the steps of applying a pinning controller to a high-order network, and controlling q nodes in the network to obtain a pinning-controlled D-dimensional simplex complex dynamics equation, wherein the pinning-controlled D-dimensional simplex complex dynamics equation is expressed as:
wherein ,ξi For binary variable, applying a pinning control on node i, then ζ i =1, otherwise ζ i =0; then
Step 3.4: judging whether the equation (4) is synchronous to a desired state by utilizing the error; the error is defined as follows:
wherein ,representing vector x i The first item of (a); />Representing the desired state x s In (3).
The beneficial effects of adopting above-mentioned technical scheme to produce lie in:
the invention provides a control method for the drag of a high-order network. The control method overcomes the defect that the higher-order network is difficult to synchronize with the desired state. Compared with the method for controlling all nodes, the cost of the control is lower, and the method has feasibility and practical application value. Meanwhile, compared with the complex network interacted with each other, the high-order network has universality. The invention not only provides a new thought for solving the synchronization problem of the higher-order network, but also has benefits for practical application. The research result of the invention can be applied to a plurality of fields, such as power systems, social networks, the Internet and the like, and provides a new method for solving some problems in the production and life of people.
Drawings
FIG. 1 is a schematic diagram of a drag control method according to an embodiment of the present invention;
FIG. 2 is a graph illustrating higher order network errors and node states under control of the present invention;
wherein, graph (a) -high-order network topology, (b) -error curve, graph (c) -node state curve.
Detailed Description
The following describes in further detail the embodiments of the present invention with reference to the drawings and examples. The following examples are illustrative of the invention and are not intended to limit the scope of the invention.
A control method for the high-order network is shown in FIG. 1, which comprises the following specific steps:
step 1: constructing an undirected graph with N nodes:
constructing an unowned undirected graph g= (V, E) with N nodes, wherein v= { V 1 ,v 2 ,…,v N "is a node set, v N Represents the nth node, e= { (v) i ,v j ) [ V ] is the edge set i and vj Respectively representing an ith node and a jth node; (i) 1 ,i 2 ,…,i d+1 ) Representing higher-order interactions between nodes, i d+1 Represents the (d+1) th node in the higher order interaction; the method comprises the steps of representing high-order interaction among nodes by using simplex, wherein 0-simplex is set as a node, 1-simplex is an edge, 2-simplex is a triangle, and d-simplex is a complete graph formed by d+1 nodes; the collection of all simplex in the undirected graph is called simplex, and the simplex is used for representing a higher-order network;
in this embodiment, as shown in FIG. 2 (a), a network with six nodes is constructed, comprising six 0-simplex, seven 1-simplex and two 2-simplex. All of the above simplex forms a simplex complex, which is a 2-dimensional simplex complex, i.e. d=2.
Step 2: determining a dynamic equation of the D-dimensional simplex complex according to a network topology structure:
wherein ,a state vector representing node i, and a superscript T represents a transpose of the vector; f R m →R m Is a continuously differentiable function describing the self dynamics of the node; h is a (d) ，d＝1,2,…,D：R (d+1)×m →R m Representing d-simplex downtoolsCoupling function between multiple node state variables with synchronous non-invasiveness, h (d) (x,x,…,x)≡0，/>Constant sigma d >0, d=1, 2, …, D representing the coupling strength; />Is defined as follows: when (i, j) 1 ,j 2 ,…,j d ) Belongs to d-simplex->Otherwise->f(x i ) Is the self-kinetic equation of the node; j (j) D Represents the (D+1) th node in the D-simplex, where D is the order of the highest order simplex in the simplex.
Step 3: designing a pinning controller according to the purpose of pinning control of the higher-order network; selecting a controlled node, and controlling the control of the higher order network in a pinning way:
step 3.1: the establishment requirement is as follows: fully synchronize equation (1) to the desired state x s When t → infinity, the following conditions are satisfied:
x s ＝x 1 ＝…＝x N (2)
wherein ,at the same time satisfy f (x) s )＝0。
Step 3.2: to meet the requirements in step 3.1, designing a hold-down controller;
the invention controls a part of nodes in the network, namely, adds control input to the controlled node, designs a control unit u i The specific expression is:
u i ＝b i σ 1 h (1) (x i ,x s ) (3)
wherein,constant b i >0 represents a control gain;representing an expected state at network synchronization;
from equation (3), the controller provides a control gain b i Coupling function h (1) (x i ,x s ) And a coupling coefficient sigma 1 A control input obtained by multiplication; when x is s ＝x i When h (1) (x i ,x s ) ≡0, control input is zero; at the same time by adjusting b i To adjust the magnitude of the control input;
step 3.3: the control method comprises the steps of applying a pinning controller to a high-order network, and controlling q nodes in the network to obtain a pinning-controlled D-dimensional simplex complex dynamics equation, wherein the pinning-controlled D-dimensional simplex complex dynamics equation is expressed as:
wherein ,ξi For binary variable, applying a pinning control on node i, then ζ i =1, otherwise ζ i =0; then
Step 3.4: judging whether the equation (4) is synchronous to a desired state by utilizing the error; the error is defined as follows:
wherein ,representing vector x i The first item of (a); />Representing the desired state x s In (3).
In the present embodiment, the self-kinetic equation f (x i ) Considered to beThe system. />The system's own kinetic equation is:
where a=β=0.2 and c=9 are set.
In this case, one balance point of the chaotic system is x s ＝[0.0044,-0.0222,0.0222] T Satisfies the following conditionsAnd satisfy f (x) s ) =0. The aim of this embodiment is to control the state of all nodes in the network to this equilibrium point.
The present example sets the coupling function to Further write out controller u i Is the dynamic equation of (2):
substituting the formulas (6) and (7) and the coupling function into the formula (4) to obtain the D-dimensional simplex complex dynamics equation controlled by the drag. The kinetic equation of the node to which the pinning control is not applied is described as:
the kinetic equation of the node to which the pinning control is applied is described as:
this embodiment controls nodes 1,2,3 in fig. 2 (a), i.e. q=3. Set control gain b 1 ＝b 2 ＝b 3 =b=2; coupling strength sigma 1 ＝1，σ 2 =0.1. Simulation using MATLAB for state x of six nodes i I=1, 2, …,6 and systematic errors are tracked. In this example, the differential equations in equations (8), (9) are solved using a fourth-order fixed-step Runge-Kutta algorithm, where the step δt=1×10 -3 Step number t=100000. The simulation diagrams are shown in fig. 2 (b) and (c).
As can be seen from fig. 2 (b) and (c), after the higher-order network is controlled in a pinning manner, the states of all nodes in the system gradually tend to the equilibrium point x s The error tends to 0, the control target is realized, and the control method is effective.
The foregoing description is only of the preferred embodiments of the present disclosure and description of the principles of the technology being employed. It will be appreciated by those skilled in the art that the scope of the invention in the embodiments of the present disclosure is not limited to the specific combination of the above technical features, but encompasses other technical features formed by any combination of the above technical features or their equivalents without departing from the spirit of the invention. Such as the above-described features, are mutually substituted with (but not limited to) the features having similar functions disclosed in the embodiments of the present disclosure.

Claims (4)

1. A method for controlling the pinning of a higher order network, comprising the steps of:
step 1: constructing an unowned undirected graph with N nodes;
step 2: determining a dynamic equation of the D-dimensional simplex complex according to the network topology structure;
step 3: designing a pinning controller according to the purpose of pinning control of the higher-order network; and selecting a controlled node to carry out control of the higher-order network.
2. The method for controlling the pinning of the higher order network according to claim 1, wherein the step 1 specifically comprises: constructing an unowned undirected graph g= (V, E) with N nodes, wherein v= { V 1 ,v 2 ,…,v N "is a node set, v N Represents the nth node, e= { (v) i ,v j ) [ V ] is the edge set i and vj Respectively representing an ith node and a jth node; (i) 1 ,i 2 ,…,i d+1 ) Representing higher-order interactions between nodes, i d+1 Represents the (d+1) th node in the higher order interaction; the method comprises the steps of representing high-order interaction among nodes by using simplex, wherein 0-simplex is set as a node, 1-simplex is an edge, 2-simplex is a triangle, and d-simplex is a complete graph formed by d+1 nodes; the collection of all simplex in the undirected graph is called simplex, and the simplex is used to represent higher order networks.
3. The method of drag control of a higher order network according to claim 1, wherein the dynamic equation of the D-dimensional simplex complex in step 2:
wherein ,a state vector representing node i, and a superscript T represents a transpose of the vector; f R m →R m Is a continuously differentiable function describing the self dynamics of the node; h is a (d) ，d＝1,2,…,D：R (d+1)×m →R m Representing multiple nodes with synchronous non-invasiveness under d-simplexCoupling function between state variables->Constant sigma d >0, d=1, 2, …, D representing the coupling strength; />Is defined as follows: when (i, j) 1 ,j 2 ,…,j d ) Belongs to d-simplex->Otherwise->f(x i ) Is the self-kinetic equation of the node; j (j) D Represents the (D+1) th node in the D-simplex, where D is the order of the highest order simplex in the simplex.
4. The method for controlling the pinning of the higher order network according to claim 1, wherein the step 3 specifically comprises the steps of:
step 3.1: the establishment requirement is as follows: fully synchronize equation (1) to the desired state x s When t → infinity, the following conditions are satisfied:
x s ＝x 1 ＝…＝x N (2)
wherein ,at the same time satisfy f (x) s )＝0；
Step 3.2: designing a hold-down controller; adding control input to the controlled node, designing a hold-down controller u i The specific expression is:
u i ＝b i σ 1 h (1) (x i ,x s ) (3)
wherein the constant b i >0 represents a control gain;representing an expected state at network synchronization;
when x is s ＝x i When h (1) (x i ,x s ) ≡0, control input is zero; at the same time by adjusting b i To adjust the magnitude of the control input;
step 3.3: the control method comprises the steps of applying a pinning controller to a high-order network, and controlling q nodes in the network to obtain a pinning-controlled D-dimensional simplex complex dynamics equation, wherein the pinning-controlled D-dimensional simplex complex dynamics equation is expressed as:
wherein ,ξi For binary variable, applying a pinning control on node i, then ζ i =1, otherwise ζ i =0; then
Step 3.4: judging whether the equation (4) is synchronous to a desired state by utilizing the error; the error is defined as follows:
wherein ,representing vector x i The first item of (a); />Representing the desired state x s In (3).
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JPH03192209A (en) * 1989-12-21 1991-08-22 Showa Electric Wire & Cable Co Ltd Manufacture of jelly sealed cable
CN111884849A (en) * 2020-07-23 2020-11-03 东北大学秦皇岛分校 Random network system containment synchronization stability analysis method based on spectrum moment
CN112311589A (en) * 2020-09-09 2021-02-02 重庆邮电大学 Grouping consistency control method of multi-agent under Markov switching topology
CN113467242A (en) * 2021-07-06 2021-10-01 东北大学秦皇岛分校 Method for analyzing divergence of synchronous domain of time-lag coupling network system under control of constraint

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