A method for predicting the dynamics of complex networks based on BKAN

Abstract

This invention relates to the field of data processing technology and provides a method for predicting the dynamics of complex networks based on Bayesian-Kanyanine (BKAN). The method includes the following steps: obtaining the initial node states and the network's adjacency matrix; constructing a graph topology based on the adjacency matrix and obtaining the connections between nodes; learning and modeling the dynamic laws based on the initial node states and the connections between nodes, combining Bayes' theorem and the KAN method to simultaneously characterize nonlinear relationships and uncertainties; and using numerical integration to perform time evolution on the model and predict the state. This invention can systematically model noise in observed data, quantify the uncertainty of node states through the Bayesian module, thereby effectively improving the stability and reliability of the prediction results. Simultaneously, the KAN module can capture the complex nonlinear relationships between nodes in the network, accurately describe the system's dynamic evolution laws, and further improve prediction accuracy.

Description

Technical Field

[0001] This invention belongs to the field of data processing technology, and in particular relates to a method for predicting the dynamics of complex networks based on BKAN. Background Technology

[0002] Complex dynamical systems consist of multiple interacting subsystems or components, and their overall evolution often exhibits highly nonlinear and emergent characteristics. Such systems are ubiquitous in nature and society, such as thermal diffusion processes and typical Lorentz chaotic systems. The goal of dynamic prediction of complex networks is to characterize and predict the dynamic behavior of complex systems in a data-driven manner.

[0003] Accurate acquisition of complex network dynamics models can more precisely predict the dynamic behavior of complex systems, revealing their underlying patterns. For example, modeling and predicting traffic flow can reveal the dynamic evolution of traffic conditions in urban road systems, thereby optimizing urban traffic; predicting epidemics can facilitate their prevention and control. However, in practical applications, the data acquired often suffers from noise and missing information, significantly increasing the difficulty of prediction.

[0004] Therefore, the core issue in predicting the dynamics of complex networks lies in how to model noise in the data and accurately predict the behavior of complex systems.

[0005] Currently, research on complex network dynamics can be mainly divided into two categories. The first category is modeling methods based on static graph structures, whose core assumption is that the topological relationships of the system remain unchanged during the evolution process. On a fixed graph structure, this type of method simulates the continuous-time evolution of node states by embedding a neural differential equation framework (such as NDCN). The second category is modeling methods based on dynamic graph structures, whose key feature is that it can adaptively characterize the evolution of the system topology over time, enabling the model to capture both temporal dependencies and time-varying topological features (such as DNND).

[0006] Modeling methods based on static graph structures are built on the assumption that the topological relationships of a system remain relatively constant during its evolution. However, the topological relationships of complex systems in reality often change over time, and the observation data usually contains a lot of noise, which makes modeling methods based on static topology prone to bias.

[0007] Modeling methods based on dynamic graph structures can capture both temporal dependencies and time-varying topological features. However, these methods still have certain limitations in characterizing complex nonlinear relationships in systems. Furthermore, due to the lack of explicit modeling of noise in the observed data, these methods may produce biases in predicting system behavior.

[0008] Bayes' theorem can guide models to model data noise, but complex network dynamics prediction methods based on Bayes' theorem lack modeling of complex nonlinear relationships in the system. Summary of the Invention

[0009] The purpose of this invention is to provide a method for predicting the dynamics of complex networks based on BKAN, which aims to solve the problems mentioned in the background art.

[0010] The present invention is implemented as follows: a method for predicting the dynamics of complex networks based on BKAN, comprising the following steps:

[0011] Obtain the initial node states and the adjacency matrix of the network;

[0012] Graph topology is constructed based on adjacency matrix to obtain the connection relationships between nodes;

[0013] Based on the initial node state and the connection relationship between nodes, the dynamic laws are learned and modeled by combining Bayes' theorem and the KAN method, which is used to simultaneously characterize nonlinear relationships and uncertainties.

[0014] Numerical integration methods are used to perform time evolution on the model and predict the state.

[0015] This invention provides a method for predicting the dynamics of complex networks based on BKAN, which can systematically model noise in observation data and quantify the uncertainty of node states through Bayesian modules, thereby effectively improving the stability and reliability of prediction results. At the same time, the KAN module can capture the complex nonlinear relationships between nodes in the network, accurately describe the dynamic evolution of the system, and further improve the prediction accuracy. Attached Figure Description

[0016] Figure 1 A flowchart illustrating a complex network dynamics prediction method based on BKAN provided in an embodiment of the present invention;

[0017] Figure 2 A flowchart of the modeling dynamics rules provided for embodiments of the present invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0019] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0020] like Figure 1The flowchart shown is a method for predicting complex network dynamics based on BKAN, provided in an embodiment of the present invention, including the following steps:

[0021] S1. Obtain the initial node state and the adjacency matrix of the network;

[0022] S2. Construct a graph topology based on the adjacency matrix and obtain the connection relationships between nodes:

[0023] S3. Based on the initial node state and the connection relationship between nodes, the dynamic laws are learned and modeled by combining Bayes' theorem and the KAN method to simultaneously characterize nonlinear relationships and uncertainties.

[0024] S4. Use numerical integration methods to perform time evolution on the model and predict the state.

[0025] Specifically, this embodiment of the invention first constructs the network topology based on observation data and uses the state information of each node as input for dynamic modeling. Based on this, BKAN (Bayesian Kolmogorov-Arnold Network) learns and models the individual dynamics of each node and the interaction dynamics between nodes, thus simultaneously characterizing individual evolution and network coupling effects within a unified framework. Because BKAN has the ability to explicitly model uncertainty, it can not only effectively capture highly nonlinear interactions in complex systems but also reasonably represent and process noise in the data, improving the robustness and prediction accuracy of the model. Finally, the trained BKAN is used as a surrogate model, combined with numerical integration methods to simulate the system's time evolution, thereby obtaining the predicted trajectory of the system state at future moments. The implementation scheme is described in detail below:

[0026] 1. Establish the graph topology:

[0027] In the modeling and prediction of complex network dynamics systems, the topological information between nodes is the core of characterizing the system's evolution. The interaction between different nodes is usually described by the adjacency matrix, which can reflect the connection relationship and weight information between each node in the system. However, this type of modeling method based on the adjacency matrix usually assumes that the network topology is a static structure, that is, the connection relationship between nodes remains unchanged in time. Although this static topology assumption is reasonable in some steady-state systems, it has obvious limitations when describing complex networks with significant time-varying characteristics or non-equilibrium dynamic characteristics (such as biochemical reaction networks, social propagation networks or neurodynamic systems).

[0028] Therefore, in this embodiment of the invention, the adjacency matrix needs to be processed again after receiving the data, retaining only the connection relationships between each node while ignoring their relationship strength. The specific method is as follows:

[0029] ;

[0030] in, The adjacency matrix represents the reconstructed graph topology. Indicates the number of nodes. Indicates the first The node and the first Does each node have a connection?

[0031] Based on this adjacency matrix, a surrogate model is further introduced to learn the interaction patterns between nodes. By jointly modeling the state of each node and its neighborhood structure, the surrogate model can adaptively capture the nonlinear correlation features between nodes, thereby enabling the modeling and prediction of the system evolution process in complex networks.

[0032] 2. Modeling dynamic rules, the process is as follows: Figure 2 As shown, the specific process of breast cancer:

[0033] 2.1 Structural principles and functional representation of KAN:

[0034] KAN (Kolmogorov-Arnold Network) is based on the Kolmogorov-Arnold representation theorem. Its nonlinear activation function design fully utilizes the theorem's ability to decompose multidimensional continuous functions, theoretically ensuring the network's ability to effectively fit highly nonlinear and complex functions. Leveraging this characteristic, KAN can accurately capture the nonlinear relationships and coupling effects between nodes in complex dynamic system modeling, providing powerful expressive power and modeling flexibility for system state prediction. KAN can be represented as:

[0035] ;

[0036] ;

[0037] in, Indicates the network in the 1st... The layer's input vector, Then used to depict the first The first in the layer The node to the first The nonlinear mapping relationship between nodes can be viewed as the activation function form of this layer. Constructed using basis function expansion, where the parameters This represents the corresponding coefficient, while Then it is the B-spline basis function. Since the B-spline function has good smoothness and local support properties, it can effectively fit complex nonlinear function relationships while ensuring numerical stability, thus enabling the network to have stronger expressive power and flexibility when modeling high-dimensional inputs.

[0038] 2.2 Bayesian extension of BKAN and uncertainty modeling:

[0039] Building upon KAN, BKAN introduces a Bayesian inference framework, combining Bayes' theorem with the power of function decomposition. This allows for explicit modeling of noise and uncertainty in the data. Specifically, BKAN no longer treats network parameters as fixed values ​​but models them as probability distributions, thus characterizing the model's uncertainty in the parameter space. Through approximate Bayesian inference, BKAN can effectively estimate the posterior distribution of the parameters and generate probabilistic predictions. This mechanism not only improves the model's robustness to noise disturbances and limited data but also enables the quantification of uncertainty in the predictions, providing a more reliable and interpretable tool for modeling and inferring complex systems. The formula is as follows:

[0040] ;

[0041] in, This represents the optimal set of parameters obtained through optimization. Indicates the parameter Solve to minimize the subsequent objective function; This represents the Kullback–Leibler divergence, used to measure the difference between two distributions; These are the learnable parameters in BKAN. To use adjustable parameters The approximate distribution represented For given data Given the true posterior distribution, this equation aims to minimize the KL divergence to make the approximate distribution as close as possible to the true posterior, thereby obtaining the optimal parameters. In order to achieve reliable probabilistic inferences about the system or model;

[0042] Due to the true posterior distribution The unknown can be expressed as a problem of maximizing the Evidence Lower Bound (ELBO):

[0043] ;

[0044] in, The lower bound of the evidence is the objective function for optimization. It is the likelihood function, representing the likelihood of a given latent variable. In this case, data The probability of occurrence; It is the prior distribution of a latent variable, representing the distribution of the variable before the data is observed. The prior assumptions; Let be the expected log-likelihood term, representing the value of the log-likelihood under the current variational distribution. Below, the average interpretability of the observed data; This is the Kullback–Leibler divergence term, used to measure the approximate posterior distribution. With prior distribution The differences between them;

[0045] 2.3 Decomposition Modeling of Dynamic Systems:

[0046] After training BKAN into a surrogate model of the system, the surrogate model can be combined with numerical integration methods (such as Neural ODE) to make continuous-time predictions of the state of complex systems.

[0047] The overall evolution of a dynamic system can be decoupled into two parts: one part describes the intrinsic dynamic behavior of each node, and the other part characterizes the interactions between nodes and network coupling effects. A surrogate model is used to model these two parts. Used for learning its own dynamics Used to learn interaction dynamics, where Indicates time Changing system variables Represents the topology of the system;

[0048] Based on the above decomposition idea, a surrogate model is constructed to describe the state evolution of continuous-time systems. Its form can simultaneously capture individual evolution and network coupling effects, providing a unified and interpretable mathematical framework for high-precision prediction of complex dynamic systems. Therefore, this surrogate model is used to describe the dynamic evolution of continuous-time systems. :

[0049] ;

[0050] in, Indicates time Changing system variables Represents the system's topology. This represents the parameter set of the proxy model BKAN.

[0051] 3. Continuous-time prediction of systems based on numerical integration:

[0052] 3.1. State update and prediction mechanism based on numerical integration:

[0053] To predict the system state over time, it is necessary to perform time integration on the above differential equations. This process can be viewed as using BKAN to replace the numerical integration of the right-hand side of the real system equations, thereby performing state deduction of the system in the continuous time domain. Its basic form is as follows:

[0054] ;

[0055] in, For time step;

[0056] because Since there is no analytical integral solution, it is necessary to approximate the solution through numerical integration algorithms. To this end, this embodiment of the invention combines the dynamic equation with the framework of Neural Ordinary Differential Equation (Neural ODE). By introducing a high-precision numerical integration method, the system state is calculated step by step, thereby achieving efficient approximation and accurate prediction of complex nonlinear dynamic processes while maintaining the differentiability of the model.

[0057] 3.2 Discretization Update Mechanism and Iterative Prediction Process:

[0058] The solution process of Neural ODE transforms continuous-time differential equations into discrete-time step state update forms through numerical integration methods (such as Runge-Kutta (RK4), Adams-Bashforth-Moulton (ABM), etc.).

[0059] ;

[0060] The formula uses BKAN to calculate the rate of change of the system's state at each time step, and then uses the step size... The state is iteratively updated, starting from time step. Update to time step The process is regarded as an iteration. Through step-by-step iteration, multi-step prediction is achieved to obtain the state estimate of the system at multiple future time points.

[0061] Using the above methods, BKAN can not only learn the nonlinear evolution of nodes in complex systems, but also characterize the impact of interactions between nodes on the system state, providing a feasible and efficient technical means for continuous-time prediction and uncertainty quantification.

[0062] Performance evaluation:

[0063] Three representative network dynamics models were selected as experimental subjects to comprehensively evaluate the applicability and predictive performance of the proposed method under different dynamic laws. These three models are: thermal diffusion model, biochemical reaction model and birth-death model.

[0064] Heat diffusion model: The heat diffusion model is a classic physical dynamic model used to describe the diffusion process of physical quantities (such as temperature or substance concentration) in space or between network nodes over time. Its dynamic laws can be expressed as:

[0065] ;

[0066] in Indicates the first The node to the first The relationship between the nodes The thermal diffusivity is used to characterize the intensity of diffusion of state (temperature) between nodes. Represents a node exist The state (temperature) at any given time, setting ;

[0067] Biochemical reaction models (Bio): Biochemical reaction models are used to describe the nonlinear interactions between proteins in complex biological systems. Their kinetic laws can be expressed as follows:

[0068] ;

[0069] in Indicates protein exist Concentration at time, It indicates that it is a protein. and The effective rate constant of the interaction between them, where a and b represent the linear decay coefficient and the interaction strength coefficient, respectively, are set. , ;

[0070] Birth-Death Model: The birth-death model is a type of stochastic dynamic model used to characterize the random change in the number of individuals in a system over time. Its dynamic laws can be expressed as:

[0071] ;

[0072] in express exist The number of individuals at any given moment. Indicates the birth rate. Indicates mortality rate, Indicates the strength of the birth item, set , , .

[0073] For each dynamic, a network with 100 nodes was designed according to its laws, and time series data of length 200 was generated;

[0074] Two common noise types in the data were selected for testing: Gaussian noise and t-distributed noise. The former can simulate the random disturbances that are common in real data, while the latter is more suitable for characterizing the data distribution with heavy tails and outliers, so as to comprehensively evaluate the robustness and adaptability of the model under different noise environments.

[0075] Gaussian noise is a type of random noise that follows a normal distribution. It is commonly used to describe random disturbances that are prevalent in simulated data. Its probability density formula is as follows:

[0076] ;

[0077] in The mean, To set the variance, , ;

[0078] t-distribution noise: This is a type of random noise with heavy-tailed characteristics, used to simulate data distributions containing outliers or extreme disturbances. Its probability density formula is as follows:

[0079] ;

[0080] in For degrees of freedom, Set for gamma function ;

[0081] To verify the predictive advantages of the embodiments of the present invention under different network dynamics scenarios, the predictive performance of the proposed method was systematically compared with the existing methods DNND and NDCN. Specifically, under the above three types of network dynamics models (thermal diffusion model, biochemical reaction model, and birth-death model), two types of typical data noise (Gaussian noise and t-distribution noise) were introduced respectively. The mean absolute percentage error (MAPE) was selected as the comparison index to quantify the accuracy of each method in predicting node states. All experiments were completed on an Intel(R) Xeon(R) Gold 5120 @ 2.20GHz CPU and a server GPU with four NVIDIA TITAN RTX processors. The server operating system was Ubuntu 20.04.6.

[0082] The specific results of the comparison after adding Gaussian noise are shown in Table 1:

[0083] Table 1

[0084]

[0085] The specific results of the comparison after adding t-distributed noise are shown in Table 2:

[0086] Table 2

[0087]

[0088] Experimental results show that the method proposed in this embodiment of the invention can more accurately characterize the noise characteristics in complex network dynamic systems when modeling and predicting them, and effectively capture the complex nonlinear relationships between nodes and within the nodes themselves. Compared with existing traditional methods, the method in this embodiment of the invention shows significant advantages in prediction accuracy, noise robustness, and nonlinear dynamic feature expression, and is effective and applicable in complex system modeling and prediction.

[0089] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting complex network dynamics based on BKAN, characterized in that, Includes the following steps: Obtain the initial node states and the adjacency matrix of the network; Graph topology is constructed based on adjacency matrix to obtain the connection relationships between nodes; Based on the initial node state and the connection relationship between nodes, the dynamic laws are learned and modeled by combining Bayes' theorem and the KAN method, which is used to simultaneously characterize nonlinear relationships and uncertainties. Numerical integration methods are used to perform time evolution of the model and predict the state. The step of learning and modeling dynamic laws based on the initial node state and the connection relationship between nodes, combined with Bayes' theorem and the KAN method, to simultaneously characterize nonlinear relationships and uncertainties, includes: introducing a Bayesian inference framework on the basis of KAN, combining Bayes' theorem with the ability of function decomposition to obtain BKAN, and explicitly modeling noise and uncertainty in the data. This step specifically involves: The formula is as follows: ； in, This represents the optimal set of parameters obtained through optimization. Indicates the parameter Solve to minimize the subsequent objective function; This represents the Kullback–Leibler divergence, used to measure the difference between two distributions; These are the learnable parameters in BKAN. To use adjustable parameters The approximate distribution represented For given data The true posterior distribution under the given conditions; Due to the true posterior distribution The unknown can be expressed as a problem of maximizing the lower bound of the evidence: ； in, The lower bound of the evidence is the objective function for optimization. It is the likelihood function, representing the likelihood of a given latent variable. In this case, data The probability of occurrence; It is the prior distribution of a latent variable, representing the distribution of the variable before the data is observed. The prior assumptions; Let be the expected log-likelihood term, representing the value of the log-likelihood under the current variational distribution. Below, the average interpretability of the observed data; This is the Kullback–Leibler divergence term, used to measure the approximate posterior distribution. With prior distribution The differences between them.

2. The method for predicting complex network dynamics based on BKAN according to claim 1, characterized in that, The steps for constructing a graph topology based on an adjacency matrix and obtaining the connection relationships between nodes are as follows: The complementarity of the obtained adjacency matrix is ​​processed by retaining only the connection relationships between nodes while ignoring their strength. The method is as follows: ； in, The adjacency matrix represents the reconstructed graph topology. Indicates the number of nodes. Indicates the first The node and the first Does each node have a connection? 3. The method for predicting complex network dynamics based on BKAN according to claim 2, characterized in that, The step of learning and modeling the dynamic laws based on the initial node states and the connections between nodes, combined with Bayes' theorem and the KAN method, to simultaneously characterize nonlinear relationships and uncertainties, is as follows: KAN is represented as: ; ； in, Indicates the network in the 1st... The layer's input vector, Then used to depict the first The first in the layer The node to the first The nonlinear mapping relationship between nodes can be viewed as the activation function form of this layer. Constructed using basis function expansion, where the parameters This represents the corresponding coefficient, while Then it is a B-spline basis function; Based on KAN, a Bayesian inference framework is introduced, combining Bayes' theorem with the ability of function decomposition to obtain BKAN, which explicitly models noise and uncertainty in the data. BKAN is trained as a surrogate model, which is then used to model the two parts of the dynamic system.

4. The method for predicting complex network dynamics based on BKAN according to claim 3, characterized in that, The steps of training BKAN into a surrogate model and using the surrogate model to model the two parts of the dynamic system are as follows: Used for learning its own dynamics Used to learn interaction dynamics, where Indicates time Changing system variables Represents the topology of the system; Construct a proxy model to describe the state evolution of a continuous-time system. : ； in, Indicates time Changing system variables Represents the system's topology. This represents the parameter set of the proxy model BKAN.

5. The method for predicting complex network dynamics based on BKAN according to claim 4, characterized in that, The step of using numerical integration to perform time evolution on the model and obtain the predicted state is as follows: right Integrating the differential equation over time: ； in, For time step; The continuous-time differential equation is transformed into a discrete-time-step state update form using numerical integration methods: ； The formula uses BKAN to calculate the rate of change of the system's state at each time step, and then uses the step size... The state is iteratively updated, starting from time step. Update to time step The process can be viewed as an iteration. By iterating step by step, multi-step prediction can be achieved, and the state estimate of the system at multiple future time points can be obtained.

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