A method and system for modeling the dynamics of temporal and high-order synergistic complex network propagation

Abstract

The application relates to the technical field of complex networks and propagation dynamics, and particularly discloses a time sequence and high-order collaborative complex network propagation dynamics modeling method and system, which comprises the following steps: constructing a semi-random simplex complex network topology basic framework; defining a multi-channel propagation dynamics mechanism fusing time sequence and high-order structure based on a network model, and mapping to obtain a macroscopic dynamics equation set based on a heterogeneous mean field approximation theory; constructing a Jacobian matrix based on the macroscopic dynamics equation set, and analytically deducing a propagation critical threshold under multi-mechanism coupling by solving the spectral radius of the Jacobian matrix; and performing numerical simulation based on the propagation critical threshold and the dynamics equation set to evaluate the propagation dynamics behavior under different combinations of social influence preference and structure connection preference; the application precisely decouples the coupling mechanism of time sequence dynamics and high-order structure, quantitatively reveals the trade-off law between propagation efficiency and topological robustness, and is particularly suitable for sudden public safety events in modern urban governance.

Description

Technical Field

[0001] This invention relates to the field of complex networks and propagation dynamics, and in particular to a method and system for modeling propagation dynamics of temporal and high-order cooperative complex networks. Background Technology

[0002] In the fields of complex networks and propagation dynamics, constructing accurate mathematical models to simulate social contagion processes such as disease outbreaks, information diffusion, and behavioral transmission is crucial for understanding the evolutionary patterns of complex systems and formulating scientific intervention strategies. Classical propagation models are mostly based on simple contagion paradigms, relying on the linear superposition of influence through pairwise interactions between nodes. To characterize the diversity of real-world interactions, existing technologies have gradually introduced temporal evolution features based on activity-driven networks to capture random instantaneous connections, or introduced higher-order network structures based on simple complexes to capture group-level interactions.

[0003] However, with a deeper understanding of the complexity of real-world social systems, existing modeling techniques have gradually revealed their shortcomings in terms of realism and theoretical completeness, especially in the context of public safety emergencies in modern urban governance, such as the prevention and control of sudden epidemics and viral online rumors. Social interactions in real-world urban environments are essentially a complex connectivity structure: it includes both transient weak connections based on commuting and shopping, which are high-frequency, random, and rapidly dissipate over time, and steady-state higher-order units based on families, offices, and close-knit communities, which have strong nonlinear reinforcement effects and significant clustering characteristics. Most existing time-series models are limited to pairwise interactions and cannot characterize the nonlinear social reinforcement effects triggered by multiple channels; while existing higher-order structure schemes are generally constrained by static topology assumptions, detached from the objective fact that network structures evolve dynamically over time.

[0004] On the other hand, existing monitoring and prediction systems suffer from significant blind spots in scenario cognition. Current technical solutions often simplify crowd interaction into a single level of contact, ignoring the fundamental differences in individuals' dual preferences across different social scenarios: namely, the structural connection preference that determines how individuals allocate social capacity to form interaction channels, and the social impact preference that determines how individuals weigh the information weights of different channels. This disconnect between dynamic and static elements, coupled with confused mechanisms, prevents decision-makers from quantitatively assessing the dynamic weight fluctuations and functional game characteristics of global diffusion based on long-range paths and group reinforcement based on local structures throughout the entire system evolution cycle. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a method for modeling the propagation dynamics of temporal and higher-order collaborative complex networks. This method integrates a dual preference decoupling mechanism of social influence and structural connectivity, combines instantaneous dynamic edges with static simple complex interactions, and is based on a semi-random simple complex network architecture. This method solves the technical problems of existing models, which suffer from inaccurate calculation of propagation critical thresholds under multi-mechanism coupling and distortion in the simulation and evaluation of real heterogeneous network propagation situations because the temporal dynamics and higher-order structures are separated and cannot be decoupled and accurately characterize individual dual preferences.

[0006] On the one hand, a method for modeling the propagation dynamics of temporal and high-order cooperative complex networks is provided, including: Construct a network topology infrastructure, wherein the architecture is a semi-random simple complex network model; Based on the aforementioned semi-random simplex complex network model, a multi-channel propagation dynamic mechanism integrating temporal sequence and higher-order structure is defined; based on the aforementioned multi-channel propagation dynamic mechanism and heterogeneous mean-field approximation theory, the evolution of micro-individual states is mapped to a set of macro-dynamic equations; Based on the aforementioned macroscopic dynamic equations, linear stability analysis is performed on the disease-free equilibrium point to construct a Jacobian matrix describing the evolution of the system state. By solving the spectral radius of the Jacobian matrix, the critical threshold for propagation under multi-mechanism coupling is analytically derived. Numerical simulations were performed based on the aforementioned critical threshold for propagation and the set of dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences.

[0007] Preferably, the semi-random simplex network model includes: a static simplex skeleton for representing stable social groups, and dynamic temporal edges for representing instantaneous random interactions, which together constitute a heterogeneous topological basis for describing the complex connectivity of the real world.

[0008] Preferably, the construction of the network topology infrastructure further includes: Network generation rules are configured based on structural connectivity preference parameters to adjust the distribution ratio of individual social capacity among dynamic temporal edges, static pairwise connections, and static simple complex skeletons; The dynamic temporal connection is generated based on a node activity-driven model. In each discrete time step, pairs of weak connections are randomly generated according to the activity of the nodes and automatically disconnected in the next time step. This is used to simulate random social contact with no memory. The static simple complex skeleton is generated, which consists of paired strong connecting edges and closed triangular structures, and is used to simulate social structures with stable connections.

[0009] Preferably, the multi-channel propagation dynamics mechanism includes pairwise random propagation via dynamic temporal edges, pairwise stable propagation via static skeletons, and group reinforcement propagation via simple complexes, and the effective propagation probability of each channel is controlled by the individual's social influence preference weight.

[0010] Preferably, based on the semi-random simplex complex network model, a multi-channel propagation dynamics mechanism that integrates temporal sequence and higher-order structure is defined; specifically: By introducing a social influence preference parameter, the baseline propagation probability is weighted and allocated according to the social influence preference parameter, resulting in: the effective probability of dynamic paired propagation channels, the effective probability of static paired propagation channels, and the effective propagation probability of simple complex group propagation channels. The propagation channel of the simple complex group follows a nonlinear reinforcement rule, and the joint propagation effect of this channel is only triggered when all other members within the simple complex are in an infected state.

[0011] Preferably, the critical propagation threshold under multi-mechanism coupling is analytically derived by solving for the spectral radius of the Jacobian matrix; specifically: Construct a linearized system matrix relating the infection state density and its auxiliary variables; Based on the eigenvalue of the largest real part of the linearized system matrix, which is the spectral radius, the critical threshold for global propagation bursts in the system is determined. The propagation threshold is explicitly expressed as a function of the individual's activity distribution statistical moments, the static skeleton's degree distribution statistical moments, and the social influence preference weights.

[0012] Preferably, numerical simulations are performed based on the propagation critical threshold and the set of dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences, including: The network size is set, including: the power law exponent of activity distribution, the number of edges sent in the time sequence, the edge probability and total average degree of the static backbone, the average degree of the static backbone network, the social impact preference parameter, the structural connection preference parameter, and including the baseline propagation rate and recovery rate. Initialize the network, and in each discrete time step, perform network topology updates cyclically to dynamically reconstruct edges and update individual states; Iterate until the system reaches a stable state, and then statistically analyze the proportions of infected and recovered individuals at the steady state. The simulation was repeated multiple times to eliminate randomness, and the average proportions of infected and recovered individuals were output as quantitative indicators to assess the scale and dynamic evolution of the outbreak.

[0013] On the other hand, a modeling system for propagation dynamics of temporal and high-order collaborative complex networks is provided, including: a semi-random simplex network construction module for constructing a network topology infrastructure, wherein the architecture is a semi-random simplex network model; The module for defining the multi-channel propagation mechanism and mapping macroscopic dynamic equations defines a multi-channel propagation dynamic mechanism that integrates temporal sequence and higher-order structure based on the semi-stochastic simplex network model. Based on the multi-channel propagation dynamic mechanism and heterogeneous mean-field approximation theory, the evolution of microscopic individual states is mapped to a set of macroscopic dynamic equations. The critical threshold analytical calculation module performs linear stability analysis on the disease-free equilibrium point based on the macroscopic dynamic equations, constructs a Jacobian matrix describing the evolution of the system state, and derives the critical threshold for propagation under multi-mechanism coupling analytically by solving the spectral radius of the Jacobian matrix. The synergy effect simulation evaluation module performs numerical simulations based on the propagation critical threshold and dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences.

[0014] Furthermore, an electronic device is also provided, including: Memory, used for non-transitory storage of computer-readable instructions; and Processor, for executing the computer-readable instructions, When the computer-readable instructions are executed by the processor, they perform the method described in the first aspect above.

[0015] In another aspect, a storage medium is also provided for non-transitory storage of computer-readable instructions, wherein when the non-transitory computer-readable instructions are executed by a computer, the method described in the first aspect is performed.

[0016] The above technical solution has the following advantages or beneficial effects: This invention innovatively integrates a static simple complex skeleton used to characterize strong connections in stable social groups with a dynamic temporal edge used to characterize weak connections in instantaneous random interactions. On the same topological basis, it achieves simultaneous capture of stability and randomness in social interactions, significantly improving the model's realism and completeness in depicting the propagation of complex social systems.

[0017] The multi-channel propagation dynamics mechanism that integrates temporal and higher-order structures provided by this invention achieves deep decoupling and accurate modeling of individual dual-preference mechanisms within a unified theoretical framework. This is achieved by introducing structural connectivity preference parameters. With social influence preference parameters This method can quantify the heterogeneous behavioral characteristics of individuals in allocating social capacity and receiving information influence from two dimensions: network structure dynamics and network propagation dynamics. By subdividing instantaneous edge-based random propagation, static skeleton-based stable propagation, and simple complex group nonlinear reinforcement propagation, this mechanism effectively reveals the functional differentiation and synergistic effects of different types of social interactions, such as pairwise interactions and group reinforcement, in complex contagion processes, eliminating prediction bias caused by confusion in propagation mechanisms. Attached Figure Description

[0018] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0019] Figure 1 This is a flowchart of the complex network propagation dynamics modeling method that combines temporal and higher-order structure provided in Embodiment 1 of the present invention; Figure 2 This is a schematic diagram of the topology generation and temporal evolution process of a semi-random simplex complex network provided in Embodiment 1 of the present invention; Figure 2 (a) is (b) is 1, (c) is 2, (d) is 3; Figure 3 This is a schematic diagram of the multi-channel propagation dynamics mechanism and state transition of the fusion timing and higher-order structure provided in Embodiment 1 of the present invention; Figure 3 In the diagram, (a) represents dynamic pairwise propagation, (b) represents static pairwise propagation, (c) represents reinforced group propagation, and (d) represents the recovery process. Figure 4 This is an architecture diagram of the complex network propagation dynamics modeling system that combines timing and higher-order structure, as provided in Embodiment 2 of the present invention. Detailed Implementation

[0020] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0021] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments of the invention. The terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion, for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0022] In this embodiment of the invention, "and / or" is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, and B existing alone. Furthermore, in the description of this invention, "multiple" refers to two or more.

[0023] Furthermore, to facilitate a clear description of the technical solutions of the embodiments of the present invention, the terms "first" and "second" are used in the embodiments of the present invention to distinguish identical or similar items with essentially the same function and effect. Those skilled in the art will understand that the terms "first" and "second" do not limit the quantity or execution order, and the terms "first" and "second" are not necessarily different.

[0024] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0025] All data acquisition in this embodiment is carried out in accordance with laws and regulations and with user consent, and the data is used legally.

[0026] Example 1 Existing modeling techniques are increasingly showing shortcomings in terms of realism and theoretical completeness, especially in the prevention and control of public safety emergencies in modern urban governance, such as sudden epidemics and viral online rumors. These challenges are severe: on the one hand, the topological characterization of propagation mechanisms suffers from a disconnect between temporal and higher-order aspects; on the other hand, existing monitoring and prediction systems have significant blind spots in scenario cognition. Therefore, due to the lack of a unified theoretical framework that can effectively decouple dual preferences and coordinate multi-dimensional paths, existing technologies not only fail to meticulously characterize the coexistence and bidirectional feedback of these two preferences within a unified system, but also, due to the fragmentation of theoretical frameworks and processing methods, cannot accurately assess the synergistic effect of dynamic random interactions and static local higher-order structures in complex propagation. This severely restricts the model's ability to predict the propagation trends of heterogeneous networks in the real world.

[0027] To address the technical problems existing in current technologies, such as the inaccurate calculation of propagation critical thresholds under multi-mechanism coupling and the inability to decouple and accurately characterize individual dual preferences due to the separation of temporal dynamics and higher-order structures, resulting in distortions in the simulation and evaluation of real heterogeneous network propagation situations, this invention provides a complex network propagation dynamics modeling method and system that integrates a decoupling mechanism for dual preferences of social influence and structural connectivity, combines instantaneous dynamic edge connections with static simple complex interaction, and is based on a semi-random simple complex network infrastructure with temporal and higher-order structure coordination.

[0028] Figure 1 This is a flowchart of a method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to the present invention. The method includes the following steps: S1: Construct the network topology infrastructure, which is a semi-random simplex complex network model.

[0029] The semi-random simplicial complex network model includes a static simplicial complex skeleton for representing stable social groups and dynamic temporal links for representing instantaneous random interactions. Together, they constitute a heterogeneous topological basis for describing the complex connectivity of the real world.

[0030] Building network topology infrastructure also includes: Based on structural connectivity preference parameters Configure network generation rules to adjust the distribution ratio of individual social capacity among dynamic temporal edges, static pairwise connections, and static simple complex skeletons; The dynamic temporal connection is generated based on a node activity-driven model. In each discrete time step, pairs of weak connections are randomly generated according to the activity of the nodes and automatically disconnected in the next time step. This is used to simulate random social contact with no memory. The static simplex skeleton is generated, which consists of pairs of strongly connected edges (1-simulacra) and closed triangular structures (2-simulacra) to simulate social structures with stable connections.

[0031] like Figure 2 As shown, the construction and evolution logic of the semi-random simple complex network in the embodiment of the present invention is illustrated; This refers to the network graph at time t. Subgraphs (a)-(d) are schematic diagrams illustrating the time-varying changes of the semi-random simplex complex network. (a) is... (b) is 1, (c) is 2, (d) is 3. Specifically, in the diagram, gray dots represent susceptible nodes, red dots represent infected nodes, and gray solid lines between nodes represent static paired edges (representing stable social relationships). The blue triangular area represents a higher-order simple complex structure composed of three nodes (representing group reinforcement paths). The red dashed lines in the diagram represent dynamic temporal edges generated by the activity-driven model (representing random contact behavior). Comparing subgraphs (a) to (d), it can be seen that as time step t evolves from 0 to 3, the static skeleton (gray solid lines) and higher-order structure (blue area) in the network maintain topological constancy; while the dynamic edges represented by the red dashed lines are randomly reconstructed at each time step according to the activity probability of individuals. This sequence diagram intuitively represents the semi-random characteristics of this invention, namely, the co-evolutionary mechanism of instantaneous random interaction superimposed on the persistent social skeleton.

[0032] A semi-random simplex complex network topology architecture based on temporal, static, and high-order path collaboration is constructed to establish the underlying logic describing the complex connectivity of the real world, specifically: Step S1.1: Construct a static high-order skeleton representing stable social relationships: Define persistent topological features, where 0-simplex represents individual nodes, 1-simplex represents paired strong connections, and 2-simplex represents a closed triangle structure; in this embodiment, the generation of 2-simplex follows the clique complex assumption, that is, the corresponding 2-simplex structure is constructed if and only if there are 1-simplex connections between all three nodes; Step S1.2: Constructing dynamic temporal interactions representing random contacts: for the network Each node in the node The distribution follows a power-law distribution. activity level In practical applications, nodes represent city residents or social media users, and their activity levels... Representative node The activation probability of generating random social interaction behavior within each discrete time step t; node Based on this probability, it is activated and randomly selected. Each node initiates an instantaneous connection.

[0033] At the current time step After completing the state update, remove all established transient connections from the network and then... Entering the next time step, thus forming a mechanism for the dynamic evolution of the topology over time; Step S1.3: Perform topology integration and parameter allocation: Linearly superimpose the static skeleton and dynamic interaction layer described above, and determine the total average degree based on structural connection preferences. ;in, The average degree of the static backbone network is determined by adjusting... Probability of static connection To achieve proportional control of network architectures that combine randomness and stability.

[0034] As another implementation method, network topology infrastructure can also employ hypergraph technology, which can also describe multi-group interactions that go beyond pairwise relationships. By constructing a semi-random hypergraph model, higher-order modeling of structural connection preferences can also be achieved.

[0035] As another implementation method, the activity-driven model can also use a Markov time-series evolution mechanism in which the activation and disconnection of edges follow specific transition probabilities, or a time-series random graph model to generate instantaneous dynamic connections, as long as the mechanism can capture instantaneous interactions driven by social influence preferences.

[0036] For ease of understanding, "nodes" are used in the description of network topology evolution, while "individuals" are used in the description of social contagion dynamics simulation.

[0037] This invention utilizes parameters and By quantitatively allocating the limited social capacity of individuals to three topological dimensions—dynamic temporal interaction, static pairwise connections, and static high-order group structure—a complex network architecture that combines randomness and stability is constructed. This achieves a unified representation of the coexistence characteristics of memoryless random contact and highly clustered stable groups in complex interactive environments, thereby solving the problem of the disconnect between temporal dynamics and high-order structure description in existing models from the topological level.

[0038] The semi-random simplex complex network infrastructure constructed in this invention addresses the technical deficiency of existing propagation dynamics modeling techniques, which separate temporal dynamic evolution from higher-order static topology. Traditional modeling schemes, due to their inability to handle only the dynamic time-varying characteristics of edges (such as activity-driven networks) or the higher-order interaction characteristics of static topologies (such as simplex complex networks), suffer from structural biases when simulating real-world social systems with complex interaction features. This invention organically integrates the static simplex complex skeleton used to represent strong connections in stable social groups with dynamic temporal edges used to represent weak connections in instantaneous random interactions. On the same topological foundation, it achieves simultaneous capture of stability and randomness in social interactions, significantly improving the realism and completeness of the model's characterization of the propagation dynamics of complex social systems.

[0039] S2: Based on the semi-random simplex complex network model, a multi-channel propagation dynamic mechanism integrating temporal sequence and higher-order structure is defined; based on the multi-channel propagation dynamic mechanism and heterogeneous mean field approximation theory, the evolution of micro-individual state is mapped to a set of macro-dynamic equations to describe the evolution law of population state density under different feature classifications.

[0040] like Figure 3The diagram shown illustrates the multi-channel propagation dynamics mechanism and state transitions that integrate temporal and higher-order structures. Figure 3 In the diagram, (a) represents dynamic pairwise propagation, (b) represents static pairwise propagation, (c) represents reinforced group propagation, and (d) represents the recovery process.

[0041] The multi-channel propagation dynamics mechanism includes: pairwise random propagation via dynamic temporal edges, pairwise stable propagation via static skeletons, and group reinforcement propagation via simple complexes. The effective propagation probability of each channel is controlled by the individual's social influence preference weight.

[0042] Based on the aforementioned semi-stochastic simplex network model, a multi-channel propagation dynamics mechanism integrating temporal sequence and higher-order structure is defined; specifically, it introduces a social influence preference parameter. It is used to adjust the degree of acceptance of individuals to different communication channels, that is, by adjusting the weight distribution of instantaneous dynamic contact communication and steady-state high-order group reinforcement communication; Based on the aforementioned social influence preference parameters, the baseline propagation probability is... By performing a weighted allocation, we obtain the effective probability of dynamic paired propagation channels. , Probability of effectiveness of static paired communication channels and the effective transmission probability of simple complex group transmission channels ; The simple complex group propagation channel follows a nonlinear reinforcement rule, meaning that the joint propagation effect of this channel is only triggered when all other members within the simple complex are in an infected state.

[0043] Furthermore, based on the aforementioned multi-channel propagation dynamics mechanism and heterogeneous mean-field approximation theory, the evolution of micro-level individual states is mapped to a set of macro-level dynamic equations, which also includes: classifying individuals in the network according to their activity levels. Static interactivity and simple complexity Divided into statistically equivalent feature classes ; For each feature class, construct a description of the susceptibility state ( S ), infected state ( I ) and recovery state ( R A deterministic set of differential equations relating density to time; the set of differential equations includes integral terms describing temporal dynamic interactions and nonlinear coupling terms describing the effects of static higher-order structures.

[0044] Specifically: Step S2.1: Based on the semi-random simplex complex network model, define a multi-channel propagation dynamic mechanism that integrates temporal sequence and higher-order structure: define the set of possible states of an individual in the system as follows. ,in, For susceptible state, Infected state, To represent the recovery state; a social influence preference parameter is introduced. , for the baseline propagation probability By performing parameterized weighted allocation, the effective infection probabilities of three independent transmission channels are obtained, i.e., the multi-channel transmission dynamics mechanism: (1) Dynamic paired communication channels: ; (2) Static paired transmission channels: ; (3) Strengthen group communication channels: ; At the same time, a uniform recovery rate is set. It is used to simulate the transformation process of an individual from an infected state to a susceptible state (SIS model) or a recovered state (SIR model); Step S2.2: Deriving the individual survival probability and microstate transition rules: Individuals in susceptible state (S) At each discrete time step The probability of not being infected inside The product of the probabilities of non-infection from the three channels:

[0045] in, and Each represents an individual The set of the 1-simplex and 2-simplex neighbors, and Represents a node and In an infected state I The probability; N represents the network size; Indicates the number of edges sent in a sequential connection; Represents an individual The probability of activity; Represents an individual The activity probability; the first term represents the non-infection probability caused by dynamic pairing, the second term represents the non-infection probability caused by static pairing, and the third term represents the non-infection probability caused by higher-order simple complex structure; each sub-probability is calculated based on the effective probability determined in step S2.1 and the real-time state of the neighboring individuals; Step S2.3: Constructing a set of macroscopic dynamic equations based on the heterogeneous mean-field approximation: In order to establish the logical connection between the microscopic interaction process and the macroscopic evolution equations in discrete time, this embodiment uses a small step size. Under the assumption that the probability of propagation through various channels at the micro level will be... Mapped to the propagation rate at the macro level Based on the principle of linear approximation, the propagation probability and propagation speed are assumed to satisfy... Based on the above mapping relationship, the effective propagation probability of each channel determined in step S2.1 is... , , This translates into the effective transmission rate of each channel. , , ; Subsequently, based on individual activity levels Static interactivity and simple complexity Heterogeneity on the surface divides system nodes into statistically equivalent feature classes. Furthermore, a discrete-time iterative equation set for the evolution of the density of characteristic class populations was established.

[0046] It is important to note that under the HMF framework, the focus is no longer on individual entities. status Instead, it groups individuals in the network with the same activity level, degree, and other characteristics into the same feature class. and , "Together" is a statistical characteristic variable used to describe this type of group. (Indicated by subscript) Parameters representing the microscopic, individual level; variables without subscripts (e.g. ) represents the feature class parameters based on heterogeneous mean field theory.

[0047] With infection state density Taking the evolution of the system as an example, the system is built at time The dynamic evolution model:

[0048] in, This represents the infection state density at time t; Indicates dynamic interactive items; Indicates static paired terms; Indicates a higher-order enhancement term; This indicates the recovery rate.

[0049] The specific terms of infection flux in formula (2) are as follows: (1) Dynamic interactive items Used to depict instantaneous interactive contagion driven by activity; (3) in, This represents the activity variable in the integral term within the feature space; Represents the static degree variable within the integration space; Denotes the simple complex degree variable within the integration space; This indicates that at time t, there is a characteristic triplet ( The infection state density of characteristic individuals; This represents the effective propagation rate of dynamic paired propagation channels after adjustment by social influence preference weights.

[0050] (2) Static paired terms , used to characterize persistent propagation under a static skeleton; (4) in, This indicates that at time t, the feature class ( The density of individuals in a susceptible state; This represents the effective propagation rate of static paired propagation channels after adjustment by social influence preference weights; This represents the probability of an edge in a static network pointing to an infected individual.

[0051] Advanced Enhancement Items Used to characterize nonlinear cooperative reinforcement propagation based on triangular simplicities: (5) in, This represents the effective transmission rate of the reinforced group transmission channel (or higher-order simplex transmission channel) after adjustment by social influence preference weights, and is used to characterize the nonlinear social reinforcement infection intensity based on the 2-simplex (triangle) structure.

[0052] S3: Based on the macroscopic dynamic equations, linear stability analysis is performed on the disease-free equilibrium point to construct a Jacobian matrix describing the evolution of the system state. By solving the spectral radius of the Jacobian matrix, the critical threshold for propagation under multi-mechanism coupling is analytically derived. This includes: constructing a linearized system Jacobian matrix for the infection state density and its auxiliary variables. ; Based on the eigenvalue of the largest real part of the linearized system matrix, the spectral radius... To determine the critical threshold for a global outbreak; The propagation threshold is explicitly expressed as the statistical moments of an individual's activity distribution. Statistical moments of degree distribution of static skeleton and social influence preference weights , The function.

[0053] The specific process is as follows: Construct the linearized system Jacobian matrix for the infection state density and its auxiliary variables. To solve for the outbreak critical point, an auxiliary macroscopic variable is introduced: total infection density. Activity-weighted density Based on the linearized macroscopic evolution equations, the following 2×2 Jacobian matrix is ​​constructed. : (6) Based on the eigenvalue of the largest real part of the linearized system matrix, i.e., the spectral radius Determine the critical threshold for a global propagation outbreak in the system: solve the matrix using the power method or characteristic equation. spectral radius The critical criterion for system outbreak is set as follows:

[0054] Obtaining the analytical solution for the propagation critical threshold: By performing an algebraic solution on the above criterion, the propagation critical threshold is obtained. Explicit expression: (7) in, Indicates the recovery rate; , This represents a parameter representing social influence preferences. This represents the average degree of a static backbone network.

[0055] The epidemic threshold can be seen from formula (7). From recovery rate Social influence preference parameters The network topology and its characteristics are jointly determined by the number of edges sent by active nodes. The first moment of activity distribution and second moment Static average degree .

[0056] This invention constructs a linearized Jacobian matrix based on heterogeneous mean-field theory. and using its spectral radius The critical threshold for propagation is derived analytically. This feature establishes a quantitative mapping relationship between propagation risk and statistical moments of network activity, statistical moments of static degree distribution, and preference weights, realizing a leap from numerical evolution simulation to a theoretical analysis and prediction framework.

[0057] As an alternative implementation, the Microscopic Markov Chain Approach (MMCA) or a pairwise approximation method can be used instead of HMF. For smaller networks or networks with high isomorphism, MMCA can provide accurate state transition probabilities at the node level, and the critical propagation threshold under multi-mechanism coupling can also be derived. This invention utilizes social influence preference parameters and By concretizing an individual’s social influence preferences into three independent and parallel propagation paths, the parameters of dynamic pairwise interactions, static pairwise interactions, and high-order group reinforcement interactions are decoupled, thereby quantifying the functional differences of heterogeneous topological paths in the propagation process and characterizing the nonlinear reinforcement effect in complex contagion.

[0058] The multi-channel propagation dynamics mechanism that integrates temporal and higher-order structures provided by this invention achieves deep decoupling and accurate modeling of individual dual-preference mechanisms within a unified theoretical framework. This is achieved by introducing structural connectivity preference parameters (…). ) and social influence preference parameters ( This invention quantifies the heterogeneous behavioral characteristics of individuals in allocating social capacity and receiving information influence from two dimensions: network structure dynamics and network propagation dynamics. By subdividing instantaneous edge-based random propagation, static skeleton-based stable propagation, and simple complex group nonlinear reinforcement propagation, this mechanism effectively reveals the functional differentiation and synergistic effects of different types of social interactions, such as pairwise interactions and group reinforcement, in complex contagion processes, eliminating prediction bias caused by confusion in propagation mechanisms.

[0059] This invention, based on heterogeneous mean-field theory, maps microscopic discrete transitions into a system of macroscopic deterministic differential equations through multidimensional heterogeneous feature classification. This technique constructs a logical bridge from microscopic topology to macroscopic evolution, enabling a leap in system description from the "individual level" to the "feature class level," significantly reducing the computational dimensionality of the model and thus providing solid theoretical support for subsequent propagation threshold analysis.

[0060] This invention, based on heterogeneous mean-field (HMF) theory and linear stability analysis, achieves an explicit analytical solution for the propagation critical threshold under multi-mechanism coupling. By mapping microscopic discrete-time evolution rules to macroscopic Jacobian matrices and solving for their spectral radii, this invention establishes a direct quantitative mapping relationship between the propagation critical threshold and underlying topological statistical moments and individual behavioral preferences. Based on this analytical framework, this invention can quantitatively reveal the functional differentiation and synergistic effects of different types of social interactions, such as pairwise interactions and group reinforcement, in complex infectious processes, providing scientific theoretical criteria and decision support for formulating targeted intervention strategies.

[0061] S4: Based on the aforementioned critical threshold for propagation and the set of dynamic equations, numerical simulations are performed to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences, and to quantify the synergistic effect of temporal dynamic interaction and static higher-order structure on the scale of propagation outbreaks.

[0062] The assessment of propagation dynamics under different combinations of social influence preferences and structural connectivity preferences also includes: Set network size N Power-law exponent of activity distribution The number of edges sent in a time-series connection Edge connection probability of static skeleton and overall average degree Average degree of static backbone network Network parameters, social influence preference parameters Structural connection preference parameters and including the reference propagation rate Recovery rate Equal propagation parameters; Initialize the network, and in each discrete time step, perform network topology update (dynamic edge reconstruction) and individual state update cyclically; Iterate until the system reaches a steady state, and then count the proportion of infected individuals at steady state. and the proportion of recovered individuals ; Repeat the simulation multiple times to eliminate randomness and output the proportion of infected individuals. and the proportion of recovered individuals The average value is used as a quantitative indicator to assess the scale and dynamic evolution of the outbreak. Furthermore, by simulating the dynamic evolution of social contagion, the propagation threshold and outbreak scale under different combinations of social influence preferences and structural connectivity preferences are evaluated, specifically: Step S4.1: Set individual dual preference parameters, including weight parameters representing structural connectivity preferences. and weighting parameters that characterize social influence preferences ; Step S4.2: Set the parameters of the semi-random simplex complex network model, including the network size. N Power-law exponent of activity distribution The number of edges sent in a time-series connection Edge connection probability of static skeleton and overall average degree Average degree of static backbone network .

[0063] Step S4.3: Set the propagation dynamics reference parameters, including the reference propagation rate. Recovery rate and the total number of simulation repetitions Ave ; Step S4.4: Initialize individual states by randomly selecting a preset proportion of individuals in the network as infected states. I The remaining individuals were set to a susceptible state. S ; Step S4.5: Statistically analyze the current state distribution and calculate the proportion of infected individuals in the network at the current time. and its auxiliary variables ; Step S4.6: Perform propagation probability weighted allocation, and calculate the dynamic pairwise effective rate using the allocation logic in step S2.1. Static Paired Effective Rate Strengthen the effective rate of groups ; Step S4.7: Update the timing topology in each... At any given moment, instantaneous dynamic edges are generated according to the activity-driven rules specified in step S1.2, and superimposed on the static simple complex skeleton to construct the complex network topology of the current step; Step S4.8: Perform microstate evolution calculations for susceptible individuals. The probability that a device is not infected by dynamic neighbors, static neighbors, or higher-order neighbors at the current time is calculated according to formula (1). And update whether it has transitioned to an infected state based on this; Step S4.9: Perform recovery process calculations. For individuals in an infected state, calculate based on the recovery rate. Update its transition state to the susceptible or recovering state; Step S4.10: Reset dynamic connections, remove transient dynamic edges established in the current time step, maintain the static skeleton structure, and let... Entering the next evolutionary cycle; Step S4.11: Determine the evolutionary steady state, and repeat steps S4.5 to S4.10 until the proportion of infected individuals is reached. and the proportion of recovered individuals If the fluctuation range is less than the preset threshold, the propagation process is determined to have reached a steady state and the current iteration ends; Step S4.12: Statistically analyze steady-state indicators and record the proportion of infected individuals at steady state. The proportion of individuals in a restorative state and parsing threshold Deviation; Step S4.13: Eliminate random errors and repeat steps S4.4 to S4.12 for a total of [number missing]. AveNext, calculate Ave The proportion of infected individuals in the simulation results and the proportion of recovered individuals The average proportion is used to output the simulation results of social transmission dynamics.

[0064] This embodiment verifies the propagation critical threshold through the above simulation process. The accuracy of this study reveals how individual preferences reshape the social contagion pattern of complex systems by regulating multidimensional interaction paths.

[0065] As another implementation, in step 4.1, a weight parameter representing the structural connectivity preference is given. and the weighting parameters that characterize social influence preferences ( The value changes from 0 to 1. This implementation analyzes the joint effects of structural connectivity preference and representational social influence preference on social communication dynamics when other parameters are fixed.

[0066] As another implementation, in step 4.1, weight parameters characterizing structural connectivity preferences are set. Given weight parameters representing social influence preferences ( Change from 0 to 1; In step 4.3, the propagation dynamics baseline parameters are set, including the recovery rate. and the total number of simulation repetitions Ave, Given a reference propagation rate Change from 0 to 1; This implementation analyzes social influence preferences and baseline propagation rate when other parameters are fixed. The combined effects on propagation dynamics.

[0067] As another implementation, in step 4.1, a weight parameter representing social influence preference is set ( Given weight parameters representing structural connectivity preferences Change from 0 to 1; In step 4.3, the propagation dynamics baseline parameters are set, including the recovery rate. and the total number of simulation repetitions Ave Given a reference propagation rate ; This embodiment analyzes structural connectivity preference and baseline propagation rate when other parameters are fixed. The combined impact on the dynamics of communication.

[0068] In summary, the complex network propagation dynamics modeling method based on the synergy of temporal and higher-order structures provided in this embodiment first constructs a semi-random simple complex network architecture based on the synergy of temporal, static, and higher-order paths. By quantifying individuals' preferences for structural connections across different interaction channels, a dynamic topological evolutionary basis characterizing the complex connectivity of real social systems is established. Second, a multi-channel propagation dynamics mechanism integrating social influence preferences is defined. A set of dynamic equations describing the macroscopic state evolution of the system is constructed using heterogeneous mean-field theory. Through eigenvalue analysis of the linearized Jacobian matrix, a quantitative mapping relationship between the propagation critical threshold and the statistical moments of the underlying network and the weights of dual preferences is obtained. Finally, by simulating the dynamic evolution of social contagion, the regulatory effect of different preference combinations on the outbreak scale is quantitatively evaluated, and quantitative indicators for assessing the social contagion situation and outbreak risk are output.

[0069] The method proposed in this invention not only achieves precise decoupling of the coupling mechanism between temporal dynamics and high-order structure, but also quantitatively reveals the trade-off between propagation efficiency and topological robustness. It effectively overcomes the simulation bias of traditional models when dealing with multi-path concurrent interactions, and can provide a scientific and technical means for accurate early warning of propagation risks and the formulation of hierarchical strategies in complex social network environments.

[0070] This invention, by configuring different combinations of preference parameters and cyclically updating network topology and evolving node states after initialization, can realistically simulate the dynamic trajectory of social contagion under diverse preference scenarios. This allows for a deeper understanding of the functional dichotomy and trade-off mechanism between dynamic interaction and static higher-order structures during propagation. Furthermore, by using the average steady-state outbreak size from multiple rounds of simulation, the influence of random disturbances on the calculation results can be effectively eliminated, thus accurately quantifying and reliably comparing the impact of individual dual preferences on the evolutionary characteristics and outbreak size of social contagion.

[0071] Example 2 This embodiment provides a modeling system for propagation dynamics of temporal and high-order cooperative complex networks, such as... Figure 4 As shown, it includes: a semi-random simplex network construction module, which constructs the network topology infrastructure, wherein the architecture is a semi-random simplex network model; The module for defining the multi-channel propagation mechanism and mapping macroscopic dynamic equations defines a multi-channel propagation dynamic mechanism that integrates temporal sequence and higher-order structure based on the semi-stochastic simplex network model. Based on the multi-channel propagation dynamic mechanism and heterogeneous mean-field approximation theory, the evolution of microscopic individual states is mapped to a set of macroscopic dynamic equations. The critical threshold analytical calculation module performs linear stability analysis on the disease-free equilibrium point based on the macroscopic dynamic equations, constructs a Jacobian matrix describing the evolution of the system state, and derives the critical threshold for propagation under multi-mechanism coupling analytically by solving the spectral radius of the Jacobian matrix. The synergy effect simulation evaluation module performs numerical simulations based on the propagation critical threshold and dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences.

[0072] Example 3 This embodiment also provides an electronic device, including: one or more processors, one or more memories, and one or more computer programs; wherein, the processor is connected to the memory, and the one or more computer programs are stored in the memory. When the electronic device is running, the processor executes the one or more computer programs stored in the memory to cause the electronic device to perform the method described in Embodiment 1.

[0073] Example 4 This embodiment also provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the method described in Embodiment 1.

[0074] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for modeling the propagation dynamics of temporal and high-order cooperative complex networks, characterized in that, include: Construct a network topology infrastructure, wherein the architecture is a semi-random simple complex network model; Based on the aforementioned semi-random simplex complex network model, a multi-channel propagation dynamic mechanism integrating temporal sequence and higher-order structure is defined; based on the aforementioned multi-channel propagation dynamic mechanism and heterogeneous mean-field approximation theory, the evolution of micro-individual states is mapped to a set of macro-dynamic equations; Based on the aforementioned macroscopic dynamic equations, linear stability analysis is performed on the disease-free equilibrium point to construct a Jacobian matrix describing the evolution of the system state. By solving the spectral radius of the Jacobian matrix, the critical threshold for propagation under multi-mechanism coupling is analytically derived. Numerical simulations were performed based on the aforementioned critical threshold for propagation and the set of dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences.

2. The method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to claim 1, characterized in that, The semi-random simplex network model includes a static simplex skeleton for representing stable social groups and dynamic temporal edges for representing instantaneous random interactions. Together, they constitute a heterogeneous topological basis for describing the complex connectivity of the real world.

3. The method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to claim 1, characterized in that, The construction of the network topology infrastructure also includes: Network generation rules are configured based on structural connectivity preference parameters to adjust the distribution ratio of individual social capacity among dynamic temporal edges, static pairwise connections, and static simple complex skeletons; The dynamic temporal connection is generated based on a node activity-driven model. In each discrete time step, pairs of weak connections are randomly generated according to the activity of the nodes and automatically disconnected in the next time step. This is used to simulate random social contact with no memory. The static simple complex skeleton is generated, which consists of paired strong connecting edges and closed triangular structures, and is used to simulate social structures with stable connections.

4. The method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to claim 1, characterized in that, The multi-channel propagation dynamics mechanism includes pairwise random propagation via dynamic temporal edges, pairwise stable propagation via static skeletons, and group reinforcement propagation via simple complexes. The effective propagation probability of each channel is controlled by the individual's social influence preference weight.

5. The method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to claim 1, characterized in that, Based on the aforementioned semi-stochastic simplex network model, a multi-channel propagation dynamics mechanism integrating temporal sequence and higher-order structure is defined; specifically: By introducing a social influence preference parameter, the baseline propagation probability is weighted and allocated according to the social influence preference parameter, resulting in: the effective probability of dynamic paired propagation channels, the effective probability of static paired propagation channels, and the effective propagation probability of simple complex group propagation channels. The propagation channel of the simple complex group follows a nonlinear reinforcement rule, and the joint propagation effect of this channel is only triggered when all other members within the simple complex are in an infected state.

6. The method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to claim 1, characterized in that, By solving for the spectral radius of the Jacobian matrix, the critical propagation threshold under multi-mechanism coupling is analytically derived; specifically: Construct a linearized system matrix relating the infection state density and its auxiliary variables; Based on the eigenvalue of the largest real part of the linearized system matrix, which is the spectral radius, the critical threshold for global propagation bursts in the system is determined. The propagation threshold is explicitly expressed as a function of the individual's activity distribution statistical moments, the static skeleton's degree distribution statistical moments, and the social influence preference weights.

7. The method for modeling propagation dynamics of temporal and high-order cooperative complex networks according to claim 1, characterized in that, Numerical simulations were performed based on the aforementioned propagation critical threshold and dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences, including: The network size is set, including: the power law exponent of activity distribution, the number of edges sent in the time sequence, the edge probability and total average degree of the static backbone, the average degree of the static backbone network, the social impact preference parameter, the structural connection preference parameter, and including the baseline propagation rate and recovery rate. Initialize the network, and in each discrete time step, perform network topology updates cyclically to dynamically reconstruct edges and update individual states; Iterate until the system reaches a stable state, and then statistically analyze the proportions of infected and recovered individuals at the steady state. The simulation was repeated multiple times to eliminate randomness, and the average proportions of infected and recovered individuals were output as quantitative indicators to assess the scale and dynamic evolution of the outbreak.

8. A modeling system for propagation dynamics of temporal and high-order cooperative complex networks, characterized in that, include: A semi-random simple complex network construction module is used to construct the basic network topology architecture, wherein the architecture is a semi-random simple complex network model. The module for defining the multi-channel propagation mechanism and mapping macroscopic dynamic equations defines a multi-channel propagation dynamic mechanism that integrates temporal sequence and higher-order structure based on the semi-stochastic simplex network model. Based on the multi-channel propagation dynamic mechanism and heterogeneous mean-field approximation theory, the evolution of microscopic individual states is mapped to a set of macroscopic dynamic equations. The critical threshold analytical calculation module performs linear stability analysis on the disease-free equilibrium point based on the macroscopic dynamic equations, constructs a Jacobian matrix describing the evolution of the system state, and derives the critical threshold for propagation under multi-mechanism coupling analytically by solving the spectral radius of the Jacobian matrix. The synergy effect simulation evaluation module performs numerical simulations based on the propagation critical threshold and dynamic equations to evaluate the propagation dynamics under different combinations of social influence preferences and structural connectivity preferences.

9. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that: When the processor executes the computer program, it implements the steps of the method according to any one of claims 1-7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, it implements the steps of the method described in any one of claims 1-7.

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