Double-layer high-order network structure reconstruction method and system based on maximum likelihood estimation

Abstract

This invention discloses a method and system for reconstructing a two-layer high-order network structure based on maximum likelihood estimation, belonging to the field of network reconstruction technology. The method includes: constructing a two-layer network model based on an information propagation model and a disease propagation model; constructing a likelihood function based on node state time series, decomposing terms related to the information diffusion layer and terms related to the disease propagation layer; the former is solved by approximating the likelihood function using the state mean field approximation method, the second-order Taylor expansion method, and a two-stage reconstruction strategy; the latter is solved by approximating the likelihood function using the state mean field approximation method and the first-order Taylor expansion method; both are used for network topology reconstruction based on the solution results. This invention establishes a dynamic coupling process between the two layers through an inter-layer coupling mechanism; achieves layered optimization by decomposing the likelihood function; and finally transforms the likelihood function into a linear equation or a system of equations for solution, reducing computational complexity and ensuring reconstruction accuracy.

Description

Technical Field

[0001] This invention relates to the field of network reconstruction technology, and in particular to a method and system for reconstructing two-layer high-order network structures based on maximum likelihood estimation. Background Technology

[0002] Network structure is fundamental to understanding and controlling the dynamics of complex systems. In real-world scenarios such as the spread of infectious diseases, since only the state evolution data of nodes can be observed, the underlying network connections cannot be directly determined. Therefore, network reconstruction, which infers the underlying topology of the system from observable data, is crucial. In reality, there is a dynamic coupling effect between disease transmission and the spread of related information, and there are numerous high-order group interactions that go beyond pairwise contact. This makes the disease transmission system essentially a multi-layered, high-order complex network, further increasing the difficulty of network reconstruction.

[0003] Currently, existing methods for complex network structure identification are mainly divided into two categories: model-based methods and data-driven methods. Model-based methods embed network topology parameters into the system model by establishing dynamic equations describing the evolution of node states, and then use system identification to reconstruct the network. In contrast to model-based methods, data-driven methods rely on observed node state data, inferring network connectivity by analyzing statistical dependencies between node state sequences. Common techniques include compressed sensing, causal testing, and statistical inference. In recent years, machine learning methods have also emerged, such as unsupervised methods based on variational autoencoders (VAEs). These methods infer potential interaction graph structures from observed data through an encoder and then use a decoder to predict future node states based on the interaction graph structure.

[0004] However, traditional methods are mostly limited to single-layer networks or simplified models that only consider pairwise interactions. They fail to fully consider the dynamic coupling effect of disease transmission and information diffusion, and also ignore the high-order interactions of groups of three or more people that are common in reality. They cannot accurately characterize the coupling and propagation mechanism of multi-layer high-order networks, resulting in low credibility of reconstruction results.

[0005] Therefore, existing technologies still need improvement and development. Summary of the Invention

[0006] The technical problem to be solved by this invention is to provide a method and system for reconstructing two-layer high-order network structures based on maximum likelihood estimation, in order to address the above-mentioned deficiencies of the prior art. This aims to solve the problem that the prior art is mostly limited to single-layer networks or only considers simplified models of pairwise interactions, which fails to fully consider the dynamic coupling effect of disease transmission and information diffusion, and also ignores the high-order interactions of groups of three or more people that are common in reality. As a result, it cannot accurately characterize the coupling and propagation mechanism of multi-layer high-order networks, leading to low reliability of reconstruction results.

[0007] The technical solution adopted by this invention to solve the problem is as follows:

[0008] In a first aspect, embodiments of the present invention provide a method for reconstructing a two-layer high-order network structure based on maximum likelihood estimation, the method comprising:

[0009] A two-layer network model for co-evolution is constructed based on an information propagation model and a disease propagation model; wherein, the two-layer network model includes: an upper information diffusion layer and a lower disease propagation layer, and the upper and lower layers are connected one-to-one through nodes;

[0010] A likelihood function is constructed based on the node state time series, and then decomposed into... Item and Item; wherein, the The associated information diffusion layer structure, the The associated disease transmission layer structure;

[0011] Regarding the above The likelihood function is approximated using the state mean field approximation method and the second-order Taylor expansion method. Based on the solution, network topology reconstruction is performed. The solution process adopts a node-level parallel two-stage reconstruction strategy. The two-stage reconstruction strategy is as follows: the first stage ignores three-body interactions, calculates two-body interactions, and selects a candidate neighbor set; the second stage calculates two-body interactions and three-body interactions based on the candidate neighbor set to achieve network topology reconstruction.

[0012] Regarding the above The likelihood function is approximated using the state-mean-field approximation method and the first-order Taylor expansion method, and the network topology is reconstructed based on the solution.

[0013] Secondly, embodiments of the present invention also provide a two-layer high-order network structure reconstruction system based on maximum likelihood estimation, the system comprising:

[0014] The module is used to construct a two-layer network model that co-evolves based on the information propagation model and the disease propagation model; wherein the two-layer network model includes an upper information diffusion layer and a lower disease propagation layer, and the upper and lower layers are connected one-to-one through nodes;

[0015] The decomposition module is used to construct a likelihood function based on the node state time series and decompose it into... Item and Item; wherein, the The associated information diffusion layer structure, the The associated disease transmission layer structure;

[0016] The solver module is used for the following... The likelihood function is approximated using the state mean field approximation method and the second-order Taylor expansion method. Based on the solution, network topology reconstruction is performed. The solution process adopts a node-level parallel two-stage reconstruction strategy. The two-stage reconstruction strategy is as follows: the first stage ignores three-body interactions, calculates two-body interactions, and selects a candidate neighbor set; the second stage calculates two-body interactions and three-body interactions based on the candidate neighbor set to achieve network topology reconstruction.

[0017] Regarding the above The likelihood function is approximated using the state-mean-field approximation method and the first-order Taylor expansion method, and the network topology is reconstructed based on the solution.

[0018] Thirdly, embodiments of the present invention also provide a terminal, the terminal including a memory and one or more processors; the memory stores one or more programs; the programs include instructions for executing the two-layer high-order network structure reconstruction method based on maximum likelihood estimation as described above; the processor is used to execute the programs.

[0019] Fourthly, embodiments of the present invention also provide a computer-readable storage medium having stored thereon a plurality of instructions adapted to be loaded and executed by a processor to implement the steps of the two-layer high-order network structure reconstruction method based on maximum likelihood estimation as described above.

[0020] The beneficial effects of this invention are as follows: This embodiment of the invention constructs a sUAU-SIS model, introduces an inter-layer coupling mechanism to bind the disease transmission layer and the information diffusion layer, and models the dynamic coupling process between the two. Secondly, it introduces higher-order interactions and identifies second-order neighbors by traversing node triples during the network reconstruction stage, fully preserving the dynamic characteristics of higher-order interactions and making the model more closely reflect the complex correlation logic of actual transmission. Furthermore, it decomposes the likelihood function into corresponding information diffusion layers. Item, corresponding disease transmission layer The system achieves hierarchical optimization; then it overcomes the computational challenge of node heterogeneity by using state-mean-field approximation, and transforms the nonlinear likelihood function into a system of linear equations by combining second-order Taylor expansion; finally, through a node-level parallel two-stage reconstruction strategy, it significantly reduces computational complexity while ensuring reconstruction accuracy. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0022] Figure 1 This is a flowchart illustrating the two-layer high-order network structure reconstruction method based on maximum likelihood estimation provided in this embodiment of the invention.

[0023] Figure 2 This is a schematic diagram of the two-layer network model structure and propagation dynamics provided in the embodiments of the present invention.

[0024] Figure 3 The fixed first-order degree provided in the embodiments of the present invention When =8, at different second-order degrees ( A schematic diagram of the reconstruction results of the information diffusion layer under (=3,4,5).

[0025] Figure 4 The fixed second degree provided in the embodiments of the present invention When =4, at different first-order degrees ( A schematic diagram of the reconstruction results of the information diffusion layer under (=10, 12, 14).

[0026] Figure 5 This is a schematic diagram of the reconstruction result of the disease transmission layer provided in an embodiment of the present invention.

[0027] Figure 6 This is a schematic diagram of the reconstruction results under different noise levels provided in the embodiments of the present invention.

[0028] Figure 7 This is a schematic diagram comparing the reconstruction performance of four real interactive networks provided in the embodiments of the present invention.

[0029] Figure 8 This is a schematic diagram of a module of a two-layer high-order network structure reconstruction system based on maximum likelihood estimation provided in an embodiment of the present invention.

[0030] Figure 9 This is a schematic diagram of the terminal provided in the embodiment of the present invention. Detailed Implementation

[0031] This invention discloses a method and system for reconstructing two-layer high-order network structures based on maximum likelihood estimation. To make the objectives, technical solutions, and effects of this invention clearer and more explicit, the invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only for explaining the invention and are not intended to limit the invention.

[0032] Those skilled in the art will understand that, unless specifically stated otherwise, the singular forms “a,” “an,” “the,” and “the” used herein may also include the plural forms. It should be further understood that the term “comprising” as used in this specification means the presence of the stated features, integers, steps, operations, elements, and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. It should be understood that when we say an element is “connected” or “coupled” to another element, it can be directly connected or coupled to the other element, or there may be intermediate elements. Furthermore, “connected” or “coupled” as used herein can include wireless connections or wireless coupling. The term “and / or” as used herein includes all or any units and all combinations of one or more associated listed items.

[0033] It will be understood by those skilled in the art that, unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. It should also be understood that terms such as those defined in general dictionaries should be understood to have the same meaning as in the context of the prior art, and should not be interpreted in an idealized or overly formal sense unless specifically defined as herein.

[0034] To address the aforementioned shortcomings of existing technologies, this invention provides a method for reconstructing a two-layer high-order network structure based on maximum likelihood estimation, such as... Figure 1 As shown, the method specifically includes the following steps:

[0035] Step S100: Construct a co-evolutionary two-layer network model based on the information propagation model and the disease propagation model; wherein, the two-layer network model includes: an upper information diffusion layer and a lower disease propagation layer, and the upper and lower layers are connected one-to-one through nodes.

[0036] Furthermore, the specific steps for constructing a co-evolutionary two-layer network model based on the information propagation model and the disease propagation model include:

[0037] The information diffusion layer is constructed based on the information propagation model; the information diffusion layer is a 2-simplex structure containing two-body interaction and three-body interaction, and is defined by the adjacency matrix and adjacency tensor corresponding to the information diffusion layer;

[0038] The disease transmission layer is constructed based on the disease transmission model; the disease transmission layer is a pairwise interactive network, defined by the adjacency matrix corresponding to the disease transmission layer.

[0039] Specifically, this embodiment models the dynamic process of disease transmission and information diffusion coupled together. It introduces inter-layer coupling and higher-order interactions into the traditional disease transmission model, constructing a two-layer network model. The ultimate goal is to reconstruct the network topology of the two-layer network model with higher-order interactions by combining the infection dynamics model with node infection state data.

[0040] The underlying principle is as follows: two distinct yet closely interconnected network layers are constructed using an information dissemination model and a disease transmission model, forming a co-evolving whole. The upper layer is the information diffusion layer, and the lower layer is the disease transmission layer. Nodes in both layers are connected in a one-to-one correspondence, jointly describing the co-evolution of disease transmission and information diffusion in the real world. This means that a node's state in the information diffusion layer—such as whether it has acquired protective information or its level of awareness of disease transmission events—will affect its state in the disease transmission layer. Conversely, the infected state of a node in the disease transmission layer may also drive it to spread relevant information in the information diffusion layer, thus realistically depicting the dynamic process of disease transmission and information diffusion intertwining and influencing each other in reality. The inputs to the two-layer network model are the time series of node states in the information diffusion layer and the time series of node states in the disease transmission layer; the outputs are the adjacency matrix and adjacency tensor of the information diffusion layer, and the adjacency matrix of the disease transmission layer.

[0041] The construction steps are as follows: First, an information diffusion layer is constructed based on an information propagation model. Considering that information propagation in reality involves not only pairwise interactions between individuals (two-body interactions) but also group-level influences, such as the synchronous diffusion of information in groups of three or more, this layer adopts a 2-simple complex structure that includes both two-body and three-body interactions. This structure accurately captures the high-order interaction characteristics in information propagation, and the specific topological relationships are defined by the adjacency matrix and adjacency tensor corresponding to this layer. Second, a disease propagation layer is constructed based on a disease propagation model. Combining the fundamental characteristics of disease propagation, this layer adopts a traditional pairwise interaction network structure, primarily depicting the disease propagation process resulting from pairwise contact between individuals. The topological relationships are directly defined by the adjacency matrix corresponding to this layer. This design retains an accurate description of the disease propagation process while achieving coupling between the two layers' dynamic processes through a one-to-one correspondence with the nodes of the information diffusion layer.

[0042] For example, in the problem modeling phase, to depict the real-world disease transmission process, such as... Figure 2As shown, a co-evolutionary two-layer network model can be constructed based on the classic unaware-aware-unaware (UAU) information transmission model and the susceptible-infected-susceptible (SIS) disease transmission model. This two-layer network model can also be called the simple complex unaware-aware-unaware-susceptible–infected–susceptible (sUAU-SIS) model. In this two-layer network model, each node represents the same individual in both layers of the network, and edges represent the interaction relationships between individuals.

[0043] The upper layer of this two-layer network model is the information diffusion layer, with a 2-simplex topology. This structure includes not only traditional two-body interactions (edges between nodes) but also three-body interactions (2-simplexes, or triangular faces, formed by three nodes). The network structure of this layer consists of the adjacency matrix of two-body interactions. Adjacency tensor of interaction with the three-body system Common definition. Represents nodes in the information diffusion layer With nodes The connection relationship between them, Represents a node With nodes There is a border between them. Represents a node With nodes There is no boundary between them; Represents a node , and The three-body connection relationship between them, where, Represents a node , and There is a border between them. Represents a node , and There is no boundary between them.

[0044] The lower layer of this two-layer network model is the disease transmission layer, whose topology is a traditional pairwise network (pairwise is a supervised learning ranking method), containing only two-body interaction structures between nodes. The network structure of this layer consists of an adjacency matrix. definition, Represents nodes in the disease transmission layer With nodes The connection relationship between them, Represents a node With nodes There is a border between them. Represents a node With nodes There are no boundaries between them. This coupling structure realistically reflects the biological mechanism by which individuals in the real world take protective measures after obtaining epidemic information, thereby reducing the risk of infection, and provides a reliable dynamic basis for accurately reconstructing network topology.

[0045] In one implementation, the node states of the two-layer network model include: unconscious-susceptible state, conscious-susceptible state, and conscious-infected state; wherein, the unconscious-susceptible state is a composite state composed of the unconscious state of the information diffusion layer and the susceptible state of the disease transmission layer; the conscious-susceptible state is a composite state composed of the conscious state of the information diffusion layer and the susceptible state of the disease transmission layer; and the conscious-infected state is a composite state composed of the conscious state of the information diffusion layer and the infected state of the disease transmission layer.

[0046] Furthermore, the transition conditions for the node states in the two-layer network model include:

[0047] In the information diffusion layer, unconscious nodes are transformed into conscious nodes through at least one of two-body interaction, three-body interaction, or disease infection, and conscious nodes forget with the highest probability.

[0048] In the disease transmission layer, the infection probability of conscious nodes is lower than that of unconscious nodes, and infected nodes recover with a second probability.

[0049] For example, the evolution of node states in a two-layer network model follows a coupled dynamic model, which can be replaced by other propagation models, such as the Susceptible-Infected-Recovered (SIR) model.

[0050] Each node is in one of three composite states at any given time: Unconscious-Susceptible (US state), Conscious-Susceptible (AS state), or Conscious-Infected (AI state). The Unconscious-Susceptible state is a composite state consisting of the Unconscious state (U state) of the information diffusion layer and the Susceptible state (S state) of the disease transmission layer. The Conscious-Susceptible state is a composite state consisting of the Conscious state (A state) of the information diffusion layer and the Susceptible state of the disease transmission layer. The Conscious-Infected state is a composite state consisting of the Conscious state of the information diffusion layer and the Infected state (I state) of the disease transmission layer.

[0051] The transition conditions between each state are as follows:

[0052] In the information diffusion layer, unconscious nodes transform into conscious nodes through three pathways: First, through two-body interaction, they are probabilistically influenced by a conscious neighbor node. Successfully notified; secondly, through three-body interaction, it is probabilistically communicated by two neighboring nodes that are simultaneously in a conscious state. The three factors are: first, the combined effect; and second, if a node is infected at the epidemic layer (i.e., in an infected state), it will automatically and immediately become conscious at the information diffusion layer. Furthermore, conscious nodes will probabilistically... The information is spontaneously forgotten, and the person returns to an unconscious state.

[0053] In the disease transmission layer, whether a susceptible node is infected depends on its state in the information diffusion layer. If it is in an unconscious state, it will be infected by an infected neighboring node with probability. Infection; if the individual is conscious, the probability of infection is... And satisfy This reflects the protective behaviors of conscious individuals. Regardless of their information state, nodes in the infected state are classified with probability. Recovery leads to a susceptible state.

[0054] The aforementioned probabilities collectively constitute the parameter set of the dynamic model, specifically manifested as follows: .in, This represents the set of parameters for the dynamic model. These parameters collectively control the co-evolution of information and epidemics in the two-layer network model.

[0055] Step S200: Construct a likelihood function based on the node state time series and decompose it into... Item and Item; wherein, the The associated information diffusion layer structure, the The associated disease transmission layer structure.

[0056] Specifically, the network structure inversion problem is transformed into an optimizable mathematical problem through probabilistic modeling. Node state time series records the binary states of nodes at each time point in the information diffusion layer and the disease transmission layer. Based on these observation data, a likelihood function is constructed to realize the inversion of multi-layer high-order network structures through a maximum likelihood estimation framework.

[0057] In one implementation, a likelihood function is constructed based on the node state time series, and then decomposed into... Item and The specific steps of this project include: constructing a likelihood function based on the node state time series, transforming the likelihood function from a product form to a summation form through logarithmic transformation, and decomposing it into... Item and The item; among which, before the step of constructing the likelihood function based on the node state time series, it also includes: determining the calculation method of three state maintenance probabilities, which are respectively used to describe the state maintenance probability of unconscious nodes in the information diffusion layer, the state maintenance probability of unconscious-susceptible nodes in the disease transmission layer, and the state maintenance probability of conscious-susceptible nodes in the disease transmission layer.

[0058] Specifically, the node state time series in this embodiment includes: the node state time series of the information diffusion layer and the node state time series of the disease transmission layer. The reconstruction method relies solely on these node state time series and does not presuppose any prior knowledge of the network topology to completely infer the topology of the two-layer network model.

[0059] Node state time series (also known as observation time series) records the binary state of nodes at each time point in the information diffusion layer and the disease transmission layer. Node state time series specifically include two types: node state time series in the information diffusion layer. , Represents a node exist Always in state A Represents a node exist Always in U state Indicates the length of the time series. Represents the number of nodes in the network; time series of node states in the disease transmission layer. , Represents a node exist Always in state I, Represents a node exist It is always in state S.

[0060] Next, a likelihood function is constructed based on the observed node state time series. The aim is to transform the network reconstruction problem into a likelihood function optimization problem based on the node state time series through the maximum likelihood estimation framework, thereby enabling the inversion of the two-layer network model structure using the maximum likelihood estimation framework.

[0061] Since the original likelihood function is typically a product of probabilities at each time step, its computation and optimization are extremely difficult. Therefore, this embodiment transforms it into a summation form using a logarithmic transformation to ensure the effectiveness of subsequent optimization. Based on the summation form after the logarithmic transformation, it can be further decomposed into... item, Items and Item. Among them, The item only includes two recovery rates and does not involve network structure-related parameters; A dedicated information diffusion layer structure; This item is specifically associated with the disease transmission layer structure. Through this decomposition, subsequent actions can be taken to target… Item and The network structures of the information diffusion layer and the disease transmission layer are optimized separately and accurately inverted, ultimately achieving the inversion of the entire two-layer network model structure. Before realizing the overall network reconstruction, three state maintenance probabilities need to be defined to describe the probability that a node will not undergo a state transition within a unit of time. Finally, through probabilistic modeling, the network structure inversion problem is transformed into an optimizable mathematical problem.

[0062] For example, we define three state maintenance probabilities to describe the likelihood that a node will not undergo a state transition within a unit of time:

[0063] ;

[0064] ;

[0065] .

[0066] in, This represents a node in state U within the information diffusion layer. exist The probability of remaining in the U state without being successfully notified by any neighboring node at any given moment; This represents a node in the US state within the disease transmission layer. exist The probability of remaining in the S state without being infected at any time; This represents a node in the AS state within the disease transmission layer. exist The probability of remaining in the S state without being infected at any time.

[0067] Considering the state transitions of all nodes at all time steps, the likelihood function is expressed as:

[0068] ;

[0069] in, Represents the likelihood function; Indicates the total number of time steps; Represents all time steps from arrive The product of.

[0070] By transforming the likelihood function from a product form to a summation form using logarithmic transformation, the log-likelihood function can be decomposed into three parts:

[0071] ;

[0072] in The item contains only two recovery rates. and It is independent of network structure. The item contains network structure information of the information diffusion layer. (node (first-order edge neighbors within the information diffusion layer) and (node (second-order edge neighbors within the information diffusion layer) The item contains network structure information of the disease transmission layer. (node (Neighbors within the disease transmission layer).

[0073] Step S300, for the above The likelihood function is approximated using the state mean field approximation method and the second-order Taylor expansion method. Based on the solution, the network topology is reconstructed. The solution process adopts a node-level parallel two-stage reconstruction strategy. The two-stage reconstruction strategy is as follows: the first stage ignores three-body interactions, calculates two-body interactions, and selects a candidate neighbor set; the second stage calculates two-body interactions and three-body interactions based on the candidate neighbor set to realize network topology reconstruction.

[0074] Step S400, for the above The likelihood function is approximated using the state-mean-field approximation method and the first-order Taylor expansion method, and the network topology is reconstructed based on the solution.

[0075] Specifically, since the likelihood function is complex and difficult to solve directly, this embodiment adopts an approximate solution scheme. First, the state mean field approximation method is introduced, which estimates the number of infected neighbors by weighted averaging of the node's historical states. This effectively overcomes the computational difficulties caused by the heterogeneity of node behavior and provides expansion points for subsequent Taylor expansions.

[0076] Building upon this, to fit the nonlinear effects of higher-order interactions, a second-order Taylor expansion strategy is adopted. Approximate calculations are performed at the expansion points determined by the state mean field, ensuring computational feasibility while fully preserving the dynamic characteristics of higher-order interactions. Through this series of mathematical transformations, the original nonconvex optimization problem is ultimately transformed into a problem concerning the adjacency matrix corresponding to the information diffusion layer. The adjacency tensor corresponding to the information diffusion layer and the adjacency matrix of the disease transmission layer Solving a system of linear equations.

[0077] For example, regarding the process of solving the likelihood function, this embodiment provides an overview of the method. The terms are expanded and solved to reconstruct the information diffusion layer.

[0078] The formula for expressing the term is as follows:

[0079] ;

[0080] in:

[0081] ;

[0082] ;

[0083] .

[0084] In the formula, It is a binary value variable used to indicate nodes. At any moment and Both are in US status ( =1 indicates that the node is in state U at both of these times). Represents a logarithmic function; Represents a node The probability of not being informed by first-order neighbors; Represents a node The probability of not being informed by second-order neighbors; It is a binary value variable. =1 indicates a node At any moment For the US state, at time It is in AS state.

[0085] For the likelihood function Items respectively about and Taking the partial derivatives and setting them to 0, we obtain the first system of equations (also known as the linear system of equations):

[0086] ;

[0087] As can be seen from the above equation, both equations contain a common unknown term. Therefore, we define the unknown term... as follows:

[0088] ;

[0089] In the formula, the independent variable and This represents the number of infected neighbors. It also couples unknown network structure parameters ( , ) and dynamic parameters ( ), and the independent variable and It also depends on the network structure to be solved, making the equations unsolvable directly. Therefore, for The handling of this issue becomes the core of the problem.

[0090] To address this issue, the following estimation method combines the state-mean-field (SMF) approximation with a second-order Taylor expansion:

[0091] First, the reference point for the Taylor expansion is determined using the state mean-field approximation method. The state mean-field approximation method refers to using statistical information about a node's historical states to estimate the collective influence of its neighbors, thereby minimizing the unknown number of infected neighbors. , It is associated with computable statistics.

[0092] Specifically, it is estimated using the following formula. and approximate points and :

[0093] ;

[0094] ;

[0095] in:

[0096] ;

[0097] ;

[0098] It is a node exist The weighted average of the neighbor states at time 1. and These are the first-order and second-order degrees to be estimated, respectively.

[0099] Based on the above state-mean-field approximation method, we can and Approximately expressed as: , .

[0100] Substituting the approximate result into the log-likelihood function Item, obtain information about parameters and Approximate function Through the analysis of Regarding respectively and By taking the derivative and setting it to zero, we can solve for the solution. and The value of can then be used to calculate the expansion point. and and the function value at the expansion point .

[0101] Secondly, at the point of expansion to A second-order Taylor expansion is performed. The choice of a second-order expansion over the traditional first-order expansion is based on the inherent requirements of high-order interaction dynamics. In high-order networks, group interactions (such as three-body interactions) introduce inherently nonlinear effects, and the first-order linear approximation cannot fully capture their dynamic characteristics. The second-order Taylor expansion, by introducing quadratic terms, can more accurately describe... The curvature variation near the reference point significantly improves reconstruction accuracy, especially in applications that reconstruct complex three-body interactive structures.

[0102] objective function The corresponding second-order Taylor expansion approximation is as follows:

[0103] ;

[0104] in:

[0105] ;

[0106] ;

[0107] .

[0108] In the formula, the coefficients , as well as All are derived from the expansion point The value at a given point is determined by whether it is a known or computable value.

[0109] Finally, substituting the approximate expression from the second-order Taylor expansion back into the aforementioned first system of equations yields a new second system of equations:

[0110] ;

[0111] Solving the second system of equations will yield the answer. and By traversing all possible node pairs and triplet The nodes can be obtained. All first-order and second-order neighbors.

[0112] The final obtained second system of equations is applied to all nodes. The processes are executed in parallel, ultimately enabling a complete reconstruction of the overall network structure of the information diffusion layer.

[0113] In practical applications, the process of solving a system of equations can be done using a nonlinear least squares solver or by other optimization algorithms, such as gradient descent, expectation-maximization, or quasi-Newton methods.

[0114] Furthermore, to alleviate the computational complexity caused by the complex propagation model and improve the computational efficiency and accuracy of network reconstruction, this embodiment adopts a two-stage reconstruction strategy based on node-level parallelism. The two-stage reconstruction strategy refers to using the structural characteristics of simple complexes to decompose the complex joint reconstruction problem into two sequentially executed sub-problems with lower computational complexity. This method is particularly suitable for the reconstruction of large-scale networks, effectively reducing the computational burden and improving reconstruction accuracy.

[0115] The two-stage reconstruction strategy is implemented by temporarily ignoring the effects of three-body interactions and identifying the candidate neighbor node set by solving the simplified likelihood function, i.e., two-body interaction calculation, thereby effectively compressing the solution space. Specifically, in the first stage, the 2-simplex structure of the information diffusion layer is simplified into a regular pairwise network, i.e., all three-body interactions are temporarily ignored. The item can be rewritten as:

[0116] ;

[0117] The process of solving this simplified likelihood function is consistent with the derivation of the complete model mentioned above: first, the expansion point is obtained by applying the state mean field (SMF) approximation. Subsequently, the nonlinear term Taylor expansion transforms the maximum likelihood estimation problem into a linear problem. Since the model contains no higher-order interaction terms, the problem is significantly simplified. By solving the resulting system of linear equations and binarizing them using Otsu's algorithm, the node equations can finally be obtained. This process identifies all first-order neighbors, thus determining the candidate neighbor set and laying the foundation for precise reconstruction in the second stage. The key advantage of this step is that it reduces a high-dimensional nonlinear optimization problem to a relatively simple linear problem by ignoring the nonlinear terms of higher-order interactions. Although this step cannot identify three-body interactions, it can quickly and roughly filter out possible neighbors related to the nodes. There are connected neighbor nodes. The resulting set of candidate neighbor nodes significantly narrows the scope of the subsequent precise search, laying an important foundation for the fine-grained reconstruction in the second step.

[0118] The second stage calculates the interactions between two-body and three-body nodes on the selected candidate neighbor set to jointly infer the complete network structure. Finally, a smart grayscale thresholding method is used for binarization to achieve network topology inference. The entire computation process employs a node-level parallel architecture to realize the topology reconstruction of the two-layer network model, significantly improving the reconstruction efficiency of large-scale networks. In practical applications, in addition to the grayscale thresholding method, other binarization techniques can also be used, such as the maximum entropy thresholding method or optimization methods based on sparse regularization.

[0119] For the disease transmission layer, which only contains two-body interactions, the reconstruction method is similar to that of the information diffusion layer, except that the disease transmission layer corresponds to... The term solution uses only the first-order Taylor expansion method and does not require a two-stage reconstruction strategy.

[0120] The reconstruction process of the disease transmission layer involves solving the corresponding likelihood function. Item implementation, The formula for expressing the term is as follows: ;

[0121] in:

[0122] ;

[0123] ;

[0124] ;

[0125] In the formula, It is a binary value variable. =1 indicates a node At any moment For the US state, at time Currently in US or AS status; It is a binary value variable. =1 indicates a node At any moment In AS state, at time Currently in US or AS status; Represents a logarithmic function; Represents a node The probability of not being infected by neighbors in state U; Represents a node The probability of being infected by a neighbor in state A; It is a binary value variable. =1 indicates a node At any moment For the US state, at time In AI mode; It is a binary value variable. =1 indicates a node At any moment In state AS, at time It is in AI mode.

[0126] For the likelihood function Item about Taking the derivative and setting it to 0, we obtain the first equation:

[0127] ;

[0128] Define unknowns and as follows:

[0129] ;

[0130] Similarly, the state mean field is used for approximate estimation. Approximate point :

[0131] ;

[0132] in:

[0133] .

[0134] It is a node exist The weighted average of the neighbor states at time 1. This is the first-order degree of the disease transmission layer that needs to be estimated.

[0135] Based on the above state-mean-field approximation method, we can and Approximately expressed as: , .

[0136] Substituting the approximate result into the log-likelihood function Item, obtain information about parameters and Approximate function .

[0137] Through the Regarding respectively and By taking the partial derivative and setting it to zero, we can solve for the solution. and The value of can then be used to calculate the expansion point. and the function value at the expansion point and .

[0138] Since the network structure in the disease transmission layer consists only of pairwise edges and does not involve higher-order interactions, therefore at the unfolding point... to Perform a first-order Taylor expansion.

[0139] objective function The corresponding first-order Taylor expansion approximation is as follows:

[0140] ;

[0141] ;

[0142] in:

[0143] ;

[0144] ;

[0145] Substituting the approximate expression from the first Taylor expansion back into the first equation, we obtain the new second equation:

[0146] ;

[0147] Solving the second equation will yield the answer. By traversing all possible node pairs The nodes can be obtained. All first-order and second-order neighbors.

[0148] The final obtained second equation is applied to all nodes. By executing these operations in parallel, the overall network structure of the disease transmission layer can be completely reconstructed. Ultimately, the network structures of both the information diffusion layer and the disease transmission layer can be obtained.

[0149] The beneficial effects of this invention are as follows:

[0150] 1) This invention proposes a multi-layer high-order network reconstruction method, which combines the prior knowledge of the infection model with the data-driven method to infer the pairwise interaction and high-order group interaction structure in the multi-layer network from the infection data of the nodes.

[0151] 2) The state-mean-field approximation method in this invention estimates the number of infected neighbors by weighted averaging of the historical states of each node, models the mutual influence between nodes transmitted through topological links, and provides a benchmark for subsequent network reconstruction. The maximum likelihood estimation framework has both a rigorous mathematical theoretical foundation and good engineering applicability.

[0152] 3) The second-order Taylor expansion method in this invention, by introducing a nonlinear term to fit the complex dynamic characteristics brought about by higher-order interactions, is particularly suitable for network reconstruction scenarios involving higher-order interactions. Through the organic combination of state mean field approximation and second-order Taylor expansion, it maintains the rigor of statistical inference methods while successfully solving the nonlinear estimation problem brought about by higher-order interactions.

[0153] 4) This invention employs a two-step reconstruction strategy, optimizing the reconstruction process by first screening candidate neighbors and then jointly inferring the complete structure. It also utilizes a node-level parallel computing architecture to achieve efficient reconstruction of large-scale networks. This enables the reconstruction of two-layer high-order networks (including two-body and three-body interactions) from simple binary infectious data, making it possible to analyze the coupling mechanisms of information propagation and disease transmission in the real world, and providing an analytical tool for studying multi-level dynamic processes in complex systems.

[0154] To verify the technical effects of this invention, the following experimental data are provided. The experiment specifically includes the following three parts:

[0155] (1) To evaluate the applicability of the method of the present invention under different data conditions and network scales, three classic artificial networks were tested: random networks (Erdene networks) and random networks (Erdene networks). Performance tests were conducted on scale-free networks (Barabási-Albert) and small-world networks (Watts-Strogatz), with 100 nodes per layer. The average degree (including first-order average degree) was adjusted. With second-order average degree The study evaluated the impact of network type and different average degrees on reconstruction performance.

[0156] (2) By introducing Gaussian white noise of different intensities to simulate interference factors in actual data acquisition, the anti-interference performance of the method of the present invention was evaluated.

[0157] (3) Validation was performed on four real networks: Hypertext2009, Thiers12, InVS15 and LyonSchool. The number of nodes in these networks ranged from 85 to 222, representing real interaction structures in different scenarios.

[0158] All experiments used F1 (a classification model evaluation metric used in machine learning and information retrieval) as the quantitative evaluation metric. This is because network reconstruction is essentially a binary classification problem with extremely imbalanced classes (real edges are fewer than non-edges), and F1 comprehensively considers precision and recall, objectively reflecting the method's ability to identify the minority class (real connections). To generate the binary node state time series, the propagation dynamics parameters were fixed at... In the experiment, the network used a first-order average degree. With second-order average degree As a key structural parameter, the lower disease transmission layer contains only paired interactions, and its corresponding second-order average degree 0.

[0159] The test results are as follows:

[0160] The experiment evaluated the reconstruction performance under different network types and degree settings. Two-body interaction represents the interaction between nodes, while three-body interaction corresponds to the group interaction involving three nodes. Together, they form an information diffusion layer (2-simplex complex).

[0161] Figure 3 This demonstrates a random network (corresponding to) Figure 3 The leftmost image inside), scale-free network (corresponding to) Figure 3 (intermediate image within), small-world network (corresponding) Figure 3 The reconstruction performance of the rightmost image (within the image) under three network types, specifically the fixed first-order degree in the information diffusion layer. At 8 o'clock, at different second-order degrees ( The reconstruction results under the conditions of 3, 4, 5). Among them, two-body interaction represents the interaction relationship between two nodes, and three-body interaction represents the group interaction of three nodes. Together, they constitute the 2-simplex structure of the information diffusion layer.

[0162] Figure 4 This demonstrates a random network (corresponding to) Figure 4 The leftmost image inside), scale-free network (corresponding to) Figure 4 (intermediate image within), small-world network (corresponding) Figure 4 The reconstruction performance of the rightmost image (within the image) under three network types, specifically the fixed second degree in the information diffusion layer. At 4, at different first-order degrees ( Reconstruction results under (10, 12, 14).

[0163] Figure 5 This demonstrates a random network (corresponding to) Figure 5 The leftmost image inside), scale-free network (corresponding to) Figure 5 (intermediate image within), small-world network (corresponding) Figure 5 The reconstruction performance of the three network types (the rightmost image in the image) is shown, specifically the reconstruction results under different degree combinations in the disease transmission layer.

[0164] according to Figures 3 to 5 As can be seen, this invention not only achieves effective reconstruction in three types of classic networks, but also maintains stable performance under different combinations of degree parameters, making it suitable for reconstruction tasks in various practical network scenarios. Furthermore, the reconstruction performance of three-body interactions systematically improves with increasing second-order average degree, effectively verifying the advantages of this invention in capturing complex interaction structures.

[0165] Figure 6 This demonstrates a random network (corresponding to) Figure 6 The leftmost image inside), scale-free network (corresponding to) Figure 6 (intermediate image within), small-world network (corresponding) Figure 6 (The rightmost image within) Reconstruction results at different noise levels. Each network layer contains 100 nodes, fixed. , , .in, Figure 6 The data performance corresponding to the inner disease layer reflects the reconstruction effect of the disease transmission layer.

[0166] This experiment introduces noise by randomly flipping a certain proportion of the node states. For example... Figure 6 As shown, small-world networks exhibit the strongest noise resistance, thanks to the combination of high clustering and short average paths in small-world topology, which effectively resists local disturbances caused by noise. At low noise levels, the disease propagation layer, due to its simpler topology, generally demonstrates better reconstruction robustness than the information diffusion layer; however, its performance drops sharply as noise intensifies and data becomes severely distorted. In summary, even under 15% strong noise conditions, the method of this invention maintains high reconstruction performance across all three network types, fully demonstrating the effectiveness of this invention in processing incomplete or noisy data.

[0167] Furthermore, the practicality and applicability of the method were further verified on real network datasets. Four real-world interactive networks were selected for the experiment: Hypertext2009, Thiers12, InVS15, and LyonSchool. The network characteristics of the four real-world interactive networks are detailed in Table 1.

[0168] Table 1. Network characteristics of four real-world networks

[0169]

[0170] Figure 7 The presentation compares reconstruction performance on four real-world networks. The first row shows the network topology, and the second row shows the corresponding reconstruction results. Figure 7 As shown, in real-world network environments, reconstruction performance continuously improves with increasing time series length. The disease propagation layer involves only simple pairwise topologies, achieving high and stable reconstruction accuracy even with short time series, before leveling off. For the more complex high-order interactions in the information diffusion layer, reconstruction performance continuously improves with increasing observation duration, demonstrating the ability of the method to capture complex topologies in real-world scenarios and its practical application value.

[0171] Based on the above embodiments, the present invention also provides a two-layer high-order network structure reconstruction system based on maximum likelihood estimation, such as... Figure 8 As shown, the system includes:

[0172] Module 01 is used to construct a two-layer network model that co-evolves based on the information propagation model and the disease propagation model; wherein, the two-layer network model includes: an upper information diffusion layer and a lower disease propagation layer, and the upper and lower layers are connected one-to-one through nodes;

[0173] Decomposition module 02 is used to construct a likelihood function based on the node state time series and decompose it into... Item and Item; wherein, the The associated information diffusion layer structure, the The associated disease transmission layer structure;

[0174] Solver module 03 is used for the following Items and the above For each term, the likelihood function is approximated using the state mean field approximation method and the second-order Taylor expansion method. Based on the solution results, the network topology is reconstructed. The solution process adopts a node-level parallel two-stage reconstruction strategy. The two-stage reconstruction strategy is as follows: the first stage ignores three-body interactions, calculates two-body interactions, and selects a candidate neighbor set; the second stage calculates two-body interactions and three-body interactions based on the candidate neighbor set to achieve network topology reconstruction.

[0175] Based on the above embodiments, the present invention also provides a terminal, the principle block diagram of which can be as follows: Figure 9As shown, the terminal includes a processor, memory, network interface, and display screen connected via a system bus. The processor provides computing and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system and computer programs. The internal memory provides the environment for the operation of the operating system and computer programs in the non-volatile storage media. The network interface is used for communication with external terminals via a network connection. When the computer program is executed by the processor, it implements a two-layer high-order network structure reconstruction method based on maximum likelihood estimation. The display screen can be a liquid crystal display (LCD) or an e-ink display.

[0176] Those skilled in the art will understand that Figure 9 The schematic diagram shown is merely a partial structural diagram related to the present invention and does not constitute a limitation on the terminal to which the present invention is applied. A specific terminal may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0177] In one implementation, the terminal's memory stores one or more programs, and these programs are configured to be executed by one or more processors. The programs contain instructions for performing a two-layer high-order network structure reconstruction method based on maximum likelihood estimation.

[0178] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided by this invention can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0179] It should be understood that the application of the present invention is not limited to the examples above. Those skilled in the art can make improvements or modifications based on the above description, and all such improvements and modifications should fall within the protection scope of the appended claims.

Claims

1. A method for reconstructing a two-layer high-order network structure based on maximum likelihood estimation, characterized in that, The method includes: A two-layer network model based on a co-evolutionary information propagation model and a disease propagation model is constructed, comprising: an information diffusion layer constructed based on the information propagation model; the information diffusion layer is a 2-simplex structure containing two-body interactions and three-body interactions, defined by the adjacency matrix and adjacency tensor corresponding to the information diffusion layer; and a disease propagation layer constructed based on the disease propagation model; the disease propagation layer is a pairwise interaction network, defined by the adjacency matrix corresponding to the disease propagation layer; wherein, the two-layer network model includes: an upper information diffusion layer and a lower disease propagation layer, the upper and lower layers being connected one-to-one by nodes; the node state types of the two-layer network model include: unconscious-susceptible state, conscious-susceptible state, and conscious-infected state; wherein, the unconscious state... The conscious-susceptible state is a composite state composed of the unconscious state of the information diffusion layer and the susceptible state of the disease transmission layer; the conscious-susceptible state is a composite state composed of the conscious state of the information diffusion layer and the susceptible state of the disease transmission layer; the conscious-infected state is a composite state composed of the conscious state of the information diffusion layer and the infected state of the disease transmission layer; the node state transition conditions of the two-layer network model include: in the information diffusion layer, unconscious nodes are transformed into conscious states through at least one of two-body interaction, three-body interaction, or disease infection, and conscious nodes forget with a first probability; in the disease transmission layer, the infection probability of conscious nodes is lower than that of unconscious nodes, and infected nodes recover with a second probability; A likelihood function is constructed based on the node state time series, and then decomposed into... Item and Item; wherein, the The associated information diffusion layer structure, the The associated disease transmission layer structure; Regarding the above The likelihood function is approximated using the state mean field approximation method and the second-order Taylor expansion method. Based on the solution, network topology reconstruction is performed. The solution process adopts a node-level parallel two-stage reconstruction strategy. The two-stage reconstruction strategy is as follows: the first stage ignores three-body interactions, calculates two-body interactions, and selects a candidate neighbor set; the second stage calculates two-body interactions and three-body interactions based on the candidate neighbor set to achieve network topology reconstruction. Regarding the above The likelihood function is approximated using the state-mean-field approximation method and the first-order Taylor expansion method, and the network topology is reconstructed based on the solution.

2. The method for reconstructing a two-layer high-order network structure based on maximum likelihood estimation according to claim 1, characterized in that, A likelihood function is constructed based on the node state time series, and then decomposed into... Item and The steps of the item include: A likelihood function is constructed based on the node state time series. The likelihood function is then transformed from a product form to a summation form using a logarithmic transformation, and further decomposed into... item, item, Item; wherein, the The item contains only two recovery rates.

3. The method for reconstructing a two-layer high-order network structure based on maximum likelihood estimation according to claim 2, characterized in that, The node state time series includes: the node state time series of the information diffusion layer and the node state time series of the disease transmission layer.

4. The method for reconstructing a two-layer high-order network structure based on maximum likelihood estimation according to claim 1, characterized in that, Before the step of constructing the likelihood function based on the node state time series, the following also includes: The calculation methods for three state maintenance probabilities are determined, which are used to describe the state maintenance probability of unconscious nodes in the information diffusion layer, the state maintenance probability of unconscious-susceptible nodes in the disease transmission layer, and the state maintenance probability of conscious-susceptible nodes in the disease transmission layer, respectively.

5. A two-layer high-order network structure reconstruction system based on maximum likelihood estimation, characterized in that, The system includes: A construction module is used to build a co-evolving two-layer network model based on an information propagation model and a disease propagation model. This includes: constructing an information diffusion layer based on the information propagation model; the information diffusion layer is a 2-simplex structure containing two-body and three-body interactions, defined by the adjacency matrix and adjacency tensor corresponding to the information diffusion layer; constructing a disease propagation layer based on the disease propagation model; the disease propagation layer is a pairwise interaction network, defined by the adjacency matrix corresponding to the disease propagation layer; wherein, the two-layer network model includes: an upper information diffusion layer and a lower disease propagation layer, with the upper and lower layers connected one-to-one by nodes; the node state types of the two-layer network model include: unconscious-susceptible state, conscious-susceptible state, and conscious-infected state; wherein, The unconscious-susceptible state is a composite state composed of the unconscious state of the information diffusion layer and the susceptible state of the disease transmission layer; the conscious-susceptible state is a composite state composed of the conscious state of the information diffusion layer and the susceptible state of the disease transmission layer; the conscious-infected state is a composite state composed of the conscious state of the information diffusion layer and the infected state of the disease transmission layer; the node state transition conditions of the two-layer network model include: in the information diffusion layer, unconscious nodes are transformed into conscious states through at least one of two-body interaction, three-body interaction, or disease infection, and conscious nodes forget with a first probability; in the disease transmission layer, the infection probability of conscious nodes is lower than that of unconscious nodes, and infected nodes recover with a second probability; The decomposition module is used to construct a likelihood function based on the node state time series and decompose it into... Item and Item; wherein, the The associated information diffusion layer structure, the The associated disease transmission layer structure; The solver module is used for the following... The likelihood function is approximated using the state mean field approximation method and the second-order Taylor expansion method. Based on the solution, network topology reconstruction is performed. The solution process adopts a node-level parallel two-stage reconstruction strategy. The two-stage reconstruction strategy is as follows: the first stage ignores three-body interactions, calculates two-body interactions, and selects a candidate neighbor set; the second stage calculates two-body interactions and three-body interactions based on the candidate neighbor set to achieve network topology reconstruction. Regarding the above The likelihood function is approximated using the state-mean-field approximation method and the first-order Taylor expansion method, and the network topology is reconstructed based on the solution.

6. A terminal, characterized in that, The terminal includes a memory and one or more processors; the memory stores one or more programs; the programs contain instructions for executing the two-layer high-order network structure reconstruction method based on maximum likelihood estimation as described in any one of claims 1 to 4; the processors are used to execute the programs.

7. A computer-readable storage medium storing a plurality of instructions thereon, characterized in that, The instructions are applicable to be loaded and executed by a processor to implement the steps of the two-layer high-order network structure reconstruction method based on maximum likelihood estimation as described in any one of claims 1 to 4.

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