Method for ensuring successful topology identification of high-order complex dynamic network

Abstract

The invention discloses a method for ensuring successful topology identification of a high-order complex dynamic network. The method comprises the following steps: establishing a complex network dynamic model containing a high-order structure; generating a preset matrix; generating a preset signal; establishing an isolated unit auxiliary network of a preset evolution trajectory; constructing a topological parameter estimator; and constructing a self-adaptive controller. With the evolution of the network, the nodes corresponding to the identified network and the auxiliary network realize external synchronization, and in the process, the topological parameter estimator successfully identifies the topological structure of the high-order network. According to the topology identification method of the technical scheme, it can be ensured that the topological structure of the high-order complex dynamic network can be accurately identified, and research on a real complex network is facilitated.

Description

Technical Field

[0001] The present invention belongs to the field of topological structure identification, and particularly relates to a method for ensuring successful topological identification of a high-order complex dynamic network. Background Art

[0002] Complex networks widely exist in various fields such as social networks, ecological networks, and power networks. The topological structure is the basis of network science research and plays a crucial role in aspects such as network control and dynamic analysis. However, in many cases, the exact network topological structure is unknown, which makes topological identification a key task in network science.

[0003] Previous work on topological identification mainly focused on traditional networks with only pairwise interactions, but real-world networks usually involve higher-order interactions among three or more units. Compared with the previous two-body interactions, the higher-order structure can better describe the interactions between system units. So far, only a few methods have been developed to identify the topological structure of higher-order networks, mainly including data-based methods and model-based techniques. However, existing methods may not be able to successfully identify the network topological structure in some cases. Summary of the Invention

[0004] Aiming at the above technical problems, the purpose of the present invention is to provide a method for identifying the topological structure of a complex dynamic network using a preset signal. This method can ensure the successful identification of the topological structure of a high-order complex dynamic network, facilitating the research on complex networks in the real world.

[0005] The technical solution provided by the present invention is as follows:

[0006] In a first aspect, the present invention provides a method for ensuring successful topological identification of a high-order complex dynamic network, including the following steps:

[0007] (1) Establish a complex network dynamics model with a higher-order structure as the network to be identified;

[0008] (2) Generate a preset matrix: Continuously sample from a continuous distribution until a preset matrix that meets the requirements is generated;

[0009] (3) Generate a preset signal: Generate a preset signal according to the preset matrix;

[0010] (4) Establish an auxiliary network composed of isolated units whose evolution trajectories are determined by the preset signal;

[0011] (5) Construct a topological parameter estimator: According to the unit state information of the network to be identified and combined with the model prior information, construct topological parameter estimators for two-body and three-body interactions respectively;

[0012] (6) Construct an adaptive controller: Construct an adaptive controller for the coupled topology parameter estimator, and apply the corresponding adaptive controller to each unit in the network to be identified.

[0013] In a second aspect, the present invention provides an electronic device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the method for ensuring successful topology identification of a high-order complex dynamic network as described in the first aspect is implemented.

[0014] The beneficial effects of the present invention are as follows:

[0015] The present invention proposes a method for ensuring successful identification of the topology structure of a high-order complex dynamic network using pre-designed signals. The present invention is applicable to various networks, including weighted or unweighted, directed or undirected, connected or disconnected, and networks with partially or fully unknown structures. In particular, the present invention can handle both traditional networks and high-order networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a schematic flow chart for the present invention to ensure successful topology identification of a high-order complex dynamic network;

[0017] Figure 2 It is a schematic diagram of the network structure of the network to be identified for testing the topology identification method of the present invention;

[0018] Figure 3 It is a schematic diagram of the two-body interaction structure of the network to be identified for testing the topology identification method of the present invention;

[0019] Figure 4 It is a schematic diagram of the three-body interaction structure related to unit 1 of the network to be identified for testing the topology identification method of the present invention;

[0020] Figure 5 It is a schematic diagram of the three-body interaction structure related to unit 2 of the network to be identified for testing the topology identification method of the present invention;

[0021] Figure 6 It is an evolution diagram of the two-body interaction topology parameter estimation during the identification process of this embodiment;

[0022] Figure 7 It is an evolution diagram of the three-body interaction topology parameter estimation during the identification process of this embodiment;

[0023] Figure 8 It is an evolution diagram of the average identification error of the identified two-body interaction during the identification process of this embodiment;

[0024] Figure 9This is the evolution diagram of the average recognition error of the three-body interaction recognized during the identification process of this embodiment. Detailed implementation manners

[0025] The following makes a detailed description of the specific implementation manners of the present invention in conjunction with the accompanying drawings of the specification. The content of the present invention is not limited thereto.

[0026] Embodiment

[0027] Figure 1 The method flow chart of the successful topological identification of the high-order complex dynamic network in the embodiment of the present invention is shown as follows:

[0028] Step 1: Establish a complex network dynamics model with a high-order structure as the network to be identified:

[0029]

[0030] Among them, is the state vector of unit i (i = 1,..., n), d is the dimension of the unit state vector; f i : is the self-dynamics of unit i; is the coupling strength, is the coupling equation of the interaction of k units (k = 2, 3). is the adjacency tensor, representing the interaction relationship between units, indicates whether there is a two-body interaction between unit j and unit i and the corresponding interaction strength, indicates whether there is a three-body interaction between unit j and k and unit i and the corresponding interaction strength. For example, represents that there is a three-body interaction between unit j and k and unit i, otherwise not.

[0031] Furthermore, it is assumed that there is no self-loop in the network, that is, if i = j, then If i = j or j = k or i = k, then

[0032] For the convenience of description, in the present invention, a complex dynamic network containing only two-body and three-body interactions is taken as an example to introduce the present invention. In fact, for a network with a higher order, the present invention is still applicable only by making corresponding modifications to the present invention.

[0033] The identification objective is to identify and

[0034] Considering that only part of the topology needs to be identified in the actual situation, it may be assumed that the units whose topologies need to be identified are 1,..., l, that is, it is necessary to identify and where \(i = 1,\ldots,l\), \(j = 1,\ldots,n\), \(k = 1,\ldots,n\) and \(i\neq j\), \(i\neq k\), \(j\neq k\).

[0035] Step 2: Generate a preset matrix:

[0036] For a \((d\beta'\times n)\) - dimensional matrix, sample from a continuous distribution such as a normal distribution or a uniform distribution to obtain each element of the matrix. If the obtained matrix meets certain conditions, output the matrix; otherwise, resample until a matrix that meets the requirements is generated as the preset matrix, denoted as where

[0037] The requirement to be met is that substituting into \(G i (t)\) gives a \((d\beta'\times\beta)\) - dimensional matrix is full - rank for any unit \(i\) (\(i = 1,\ldots,l\)) of interest.

[0038] where is the state vector of unit \(i\) (\(i = 1,\ldots,n\)) in the auxiliary network, and \(x j (t)\), \(x k (t)\) are the state vectors of unit \(j\) (\(i = 1,\ldots,n\)) and unit \(k\) (\(i = 1,\ldots,n\)) in the auxiliary network respectively.

[0039] Substituting into \(G i (t)\) to obtain is as follows:

[0040] Define a \((d\times n)\) - dimensional sampling matrix where is the state obtained by sampling (the auxiliary network constructed in Step 4) at time \(t ε . In fact, is composed of \(\beta'\) - form matrices. The difference from is that is preset instead of being sampled. Substituting

[0041] into \(G (t)\) to obtain i (t)\) is the same as the way of substituting into \(G i (t)\) to obtain(t)\) to obtain . Substitute into \(Gi (t), obtain Exemplary: If n = 3, l = 1, then:

[0042]

[0043] Using the same method, substitute into G i (t), obtain

[0044] Step 3. Generate a preset signal according to a preset matrix:

[0045] Define the scalar-valued basis function z(t) as:

[0046]

[0047] where t is a time variable.

[0048] Let β′≥2. Define Define the switching rules for the state indices ξ1 and ξ2:

[0049]

[0050] Then design the preset signal as:

[0051]

[0052] where i = 1, …, n.

[0053] Step 4. Establish an isolated unit auxiliary network whose evolution trajectory is the preset signal:

[0054] This auxiliary network is composed of n isolated units, where each unit corresponds one-to-one to each unit in the corresponding network to be identified (1). However, there is no interaction relationship between the nodes in this auxiliary network. Denote the dynamic evolution equation of the i-th (i = 1, …, n) unit as:

[0055]

[0056] where is the state vector of the i-th unit in the auxiliary network, is the derivative of x i with respect to time. ψ i (t) is the evolution equation of the i-th unit in the auxiliary network, and its expression is:

[0057]

[0058] where

[0059]

[0060] It should be noted that the evolution trajectory of the preset auxiliary network is the preset signal given in Step 3.

[0061] Step 5: Construct a topological parameter estimator:

[0062] In order to estimate the topology related to the unit of interest, according to the unit state information of the network to be identified and combined with the prior information of the model such as the coupling method, the following topological parameter estimators are designed for two-body and three-body interactions respectively

[0063]

[0064] where is the estimate of the parameter characterizing the two-body interaction topology ; is the derivative with respect to time; is the update intensity of the two-body interaction topology parameter estimator; is the estimate of the parameter characterizing the three-body interaction topology ; is the derivative with respect to time; is the update intensity of the three-body interaction topology parameter estimator; e i = x i - y i (i = 1, …, n) is the state error between the corresponding units of the network to be identified and the auxiliary network.

[0065] Step 6: Construct an adaptive controller:

[0066] Define the transformation equation S as:

[0067]

[0068]

[0069] Apply the adaptive controller u i (i = 1, …, n) of the coupling topology parameter estimator to each unit in the network to be identified (1)

[0070]

[0071] where is the state vector of unit i in the auxiliary network, and t is the time variable; f i : is the self-dynamics of unit i in the network to be identified, and f i (x i ) is to substitute the state of unit i in the auxiliary network into f i, the output is a d-dimensional vector; k i is the adaptive intensity, is the derivative of k i with respect to time; d i is the proportionality coefficient that adjusts the growth rate of k i

[0072] Subsequently, the evolution equation of the error system between the identified network (1) and the auxiliary network (2) is constructed. As the network evolves, based on the Lyapunov analysis method, it can be shown that the corresponding nodes of network (1) and network (2) achieve outer synchronization, and it can be verified that the present invention can successfully identify the topological structure during this process.

[0073] The following will introduce in detail how to use the method of the present invention to successfully identify the topological structure of a high-order complex dynamic network step by step according to a typical example. The specific steps are as follows:

[0074] (1) Consider a high-order network composed of ten units, and its network topological structure is as shown in the appendix Figure 2 shown. Among them, the schematic diagram of the two-body interaction structure is as shown in the appendix Figure 3 shown. Without loss of generality, set the units with unknown topology (the units of interest) as unit 1 and unit 2, and the schematic diagrams of the corresponding three-body interaction structures are respectively as shown in the appendix Figure 4 , appendix Figure 5 shown. Use the unified chaotic system as the unit's own dynamics,

[0075]

[0076] where, α i ∈[0, 1]. When α i ∈[0, 0.8), the system belongs to the generalized Lorenz system; when α i ∈(0.8, 1], the system belongs to the generalized Chen system; when α i = 0.8, the system belongs to the system. For all units in network (1), a unified chaotic system with α i = 0 is set as its own dynamics, and the two-body and three-body interaction equations are respectively set as and The coupling strengths σ2 = σ3 = 0.1, and the initial state of the unit is set as y i (0) = [3i - 2, 3i - 1, 3i] T (i = 1,..., n).

[0077] ​(2) Next, use the method of step two of the present invention to generate a suitable preset matrix. In this embodiment, each matrix element is sampled from a uniform distribution in (-5, 5) to form a preset matrix, and the method of step two is used to check whether the sampled matrix meets the requirements until a preset matrix that meets the requirements is generated. It should be noted that because and only the first dimension interacts with other units, so the preset matrix has dimensions (dβ×n). Specifically, n = 10, d = 3, β = n 2 -2n + 1 = 81.

[0078] (3) Based on the obtained preset matrix, generate a preset signal according to the method of step three of the present invention.

[0079] (4) According to the preset signal, use the method of step four of the present invention to construct an auxiliary network composed of isolated units.

[0080] (5) According to step five, construct two-body and three-body topological parameter estimators for the units of interest. The specific form of the parameter estimator is shown in step five. In this embodiment, the units of interest are unit 1 and unit 2. Set the initial value of the topological estimator to be obtained by sampling a uniform distribution on [0, 1], and set

[0081] (6) Apply the adaptive controller u of the coupling topological parameter estimator to each unit in the network to be identified (1) i , i = 1, …, n, set d i = 1, k i (0)= 1 (i = 1, …, n). The specific form of the adaptive controller is shown in step six. Subsequently, construct the evolution equation of the error system between the network to be identified (1) and the auxiliary network (2). As the network evolves, based on the Lyapunov analysis method, it can be proved that the corresponding nodes of network (1) and network (2) achieve outer synchronization, and it can be verified that the present invention can successfully identify the topological structure of high-order complex dynamic networks during this process.

[0082] Simulation verification:

[0083] Use MATLAB to simulate this embodiment. Denote the average recognition errors of the two-body and three-body topological structures as:

[0084]

[0085]

[0086] Set the evolution time to 30,000. The simulation results are as Figures 6 to 9 shown. FromFigure 6 , Figure 7 It can be seen that both the two-body and three-body topological parameter estimators gradually converge. From Figure 8 , Figure 9 it can be seen that the average recognition errors of the two-body and three-body topological structures both gradually become 0, which indicates that the method of this embodiment ensures the successful identification of the topological structure of a high-order complex dynamic network.

[0087] This embodiment also provides an electronic device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, it implements the foregoing method for ensuring the successful topological identification of a high-order complex dynamic network.

[0088] As described above, only the preferred specific embodiments of the present invention are provided, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by those skilled in the art within the technical scope disclosed by the present invention shall be included within the scope of protection of the present invention.

Claims

1. A method for ensuring successful topological identification of a high-order complex dynamic network, characterized in that, It includes the following steps: (1) Establish a complex network dynamics model with a high-order structure as the network to be identified; (2) Generate a preset matrix: Continuously sample from a continuous distribution until a preset matrix that meets the requirements is generated; (3) Generate a preset signal: Generate a preset signal according to the preset matrix; (4) Establish an auxiliary network composed of isolated units whose evolution trajectories are determined by the preset signal; (5) Construct a topological parameter estimator: According to the unit state information of the network to be identified and combined with the prior information of the model, construct topological parameter estimators for two-body and three-body interactions respectively; (6) Construct an adaptive controller: Construct an adaptive controller that couples the topological parameter estimator and apply the corresponding adaptive controller to each unit in the network to be identified.

2. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 1, characterized in that, The state equation of the established complex network dynamics model with a high-order structure is as follows: Among them, there are n units in the network, is the state vector of unit i (i = 1, …, n), and d is the dimension of the unit state vector; is the derivative of y i with respect to time; f i : is the self-dynamics of unit i; is the coupling strength, is the coupling equation for the interaction of k units, k = 2, 3; is the adjacency tensor, representing the interaction relationship between units, indicates whether there is a two-body interaction between unit j and unit i and the corresponding interaction strength, indicates whether there is a three-body interaction between unit j and k and unit i and the corresponding interaction strength.

3. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 2, characterized in that The method for generating the preset matrix is as follows: For a (dβ′×n)-dimensional matrix, sample each element of the matrix from a continuous distribution such as a normal distribution or a uniform distribution. If the obtained matrix meets certain conditions, output the matrix; otherwise, resample until a matrix that meets the requirements is generated as the preset matrix, denoted as where j is an index, taking values from 1 to β′, and β = n 2 -2n + 1, n is the number of units in the network, and d is the dimension of the unit state vector.

4. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 3, characterized in that, In step (2), the requirements that the preset matrix needs to meet are as follows: Substitute the preset matrix into G i (t) to obtain a (dβ′×β)-dimensional matrix that is full column rank for any unit i (i = 1, …, l) of interest; Among them, are the coupling equations of the interaction of k units, where k = 2, 3; l is the number of units of interest; is the state vector of unit i (i = 1, …, n) in the auxiliary network, x j (t), x k (t) are the state vectors of unit j (i = 1, …, n) and unit k (i = 1, …, n) in the auxiliary network respectively.

5. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 4, wherein, In step (2), substituting into G i (t) gives by the following method: Define a (d×n)-dimensional sampling matrix where is the state obtained from the auxiliary network constructed in sampling step (4) at time t ε ; The state; wherein is formally the same as and the elements are preset; substituting into G i (t) gives in the same way as substituting into G i (t) gives ; substituting into G i (t) respectively gives 6. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 3, characterized in that, In step (3), the method for generating a preset signal according to the preset matrix is as follows: Define the scalar-valued basis function z(t) as: Among them, t is a time variable; Let β′≥2 and define Define the switching rules for the state indices ξ1 and ξ2: Then design the preset signal as: where, i = 1, …, n.

7. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 1, wherein The method for establishing an auxiliary network composed of isolated units whose evolution trajectories are preset signals: The auxiliary network is composed of n isolated units, where each unit corresponds one-to-one to each unit in the corresponding network to be identified; there is no interaction relationship between the nodes in the auxiliary network.

8. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 1, characterized in that, The method for constructing a topological parameter estimator: According to the unit state information of the network to be identified and combined with the prior information of the model, design the following topological parameter estimators for two-body and three-body interactions respectively Among them, is the estimation of the parameter characterizing the two-body interaction topology, is the derivative with respect to time, and is the update intensity of the two-body interaction topology parameter estimator; is the estimation of the parameter characterizing the three-body interaction topology, is the derivative with respect to time, and is the update intensity of the three-body interaction topology parameter estimator; e i = x i - y i (i = 1, …, n) is the state error between the corresponding units of the identified network and the auxiliary network.

9. The method for ensuring successful topological identification of a high-order complex dynamic network according to claim 8, characterized in that, The method for constructing an adaptive controller: Define the transformation equation S as: Apply the adaptive controller \(u\) of the coupling topology parameter estimator to each unit in the identified network respectively i (where \(i = 1,\ldots,n\)) Among them, the dynamic evolution equation of the \(i\)th (\(i = 1,\cdots,n\)) unit is is the state vector of unit \(i\) in the auxiliary network, and \(t\) is the time variable; \(f\) i : is the self-dynamics of unit \(i\) in the network to be identified, and \(f\) i (\(x\) i ) is to substitute the state of unit \(i\) in the auxiliary network into \(f\) i , and the output is a \(d\)-dimensional vector; \(k\) i is the adaptive intensity, is the derivative of \(k\) i with respect to time; \(d\) i is the proportionality coefficient that regulates the growth rate of \(k\) i .

10. An electronic device, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, wherein, When the processor executes the program, it implements the method for ensuring the successful topological identification of a high-order complex dynamic network as described in any one of claims 1 to 9.

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