A network community discovery system and method through matrix analysis

Abstract

The application relates to the technical field of network analysis, and discloses a network community discovery system and method through matrix analysis, which comprises the following modules: a network module, which converts a static network topology into an information propagation model, and establishes a node state time sequence dynamic model by defining a node information processing rule and a multi-round propagation mechanism; a phase space reconstruction module, which maps high-dimensional time sequence data to low-dimensional phase space through a nonlinear dimension reduction method to form node trajectory distribution data; a community feature module, which analyzes the convergence, oscillation mode and attractor feature of the node trajectory, and generates community structure feature data through trajectory similarity; and a community division module, which identifies a community boundary through density clustering, and constructs a hierarchical community organization through multi-scale analysis. The application can deeply mine the internal community structure of a network from the perspective of dynamic information propagation, and overcomes the limitation that traditional methods only consider static topology.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of network analysis, and discloses a network community discovery system and method based on matrix analysis. BACKGROUND

[0002] Existing network community discovery methods are mainly based on the topological structure characteristics of graphs, such as modularity optimization and spectral clustering. These methods only consider the static connection relationship between nodes, and ignore the dynamic interaction characteristics of network nodes in the information propagation process. Traditional methods have many shortcomings, for example: the method based on static topology cannot capture the functional characteristics of the network in the dynamic process of information propagation and opinion evolution, resulting in that the discovered community structure cannot truly reflect the functional module division of the network; the existing methods do not deeply describe the dynamic behavior of nodes, lack quantitative analysis of the state evolution trajectory of nodes, and cannot distinguish nodes with similar connection patterns but different dynamic behaviors; most of the traditional community discovery methods can only obtain single-scale community division results, and it is difficult to reveal the inherent hierarchical organizational structure of the network, while actual networks often have multi-scale community characteristics from micro to macro. SUMMARY

[0003] This section is intended to summarize some aspects of the embodiments of the present application and briefly introduce some preferred embodiments. Some simplifications or omissions may be made in this section and the abstract and title of the specification to avoid obscuring the purpose of this section, the abstract and the title, and such simplifications or omissions cannot be used to limit the scope of the present application.

[0004] To solve the above technical problems, the present application provides a network community discovery system and method based on matrix analysis.

[0005] In one aspect, the present application provides a network community discovery system based on matrix analysis, comprising a network module, which converts a static network topology into an information propagation model, establishes a network node state time series dynamic model by defining an information processing rule of a node and a multi-round propagation mechanism, and outputs node state time series data;

[0006] A phase space reconstruction module receives the node state time series data output by the network module, maps high-dimensional time series state sequence data to a low-dimensional phase space through a nonlinear dimension reduction method, and forms node trajectory distribution data;

[0007] A community feature module receives the node trajectory distribution data output by the phase space reconstruction module, analyzes the dynamic characteristics of the node trajectory, and generates community structure feature data through trajectory similarity;

[0008] The community division module receives the community structure feature data output by the community feature module, identifies community boundaries in the phase space through density clustering, and constructs a hierarchical community organization through multi-scale analysis.

[0009] As a preferred scheme of the network community discovery system through matrix analysis of the application, wherein:

[0010] The information propagation model is used to define an information processing rule for each node in the network, the information processing rule being a state transition function, the input being a current state of the node and a state set of all directly connected neighbor nodes in the previous propagation round, and the output being a new state of the current node in the next propagation round; the current state being a discrete or continuous numerical value, used to represent an activity level, a viewpoint tendency or an infection state of the node under the influence of information;

[0011] The multi-round propagation mechanism is used to synchronously or asynchronously apply the information processing rule to all nodes in the network; in each round of propagation, each node independently calculates a new state of itself through the state transition function according to the states of the neighbor nodes in the previous round.

[0012] As a preferred scheme of the network community discovery system through matrix analysis of the application, wherein:

[0013] The method for establishing a network node state time series dynamic model and outputting node state time series data comprises:

[0014] S101 gives an initial state to each node in the network, and sets a total number of propagation rounds or sets a state convergence judgment condition to terminate the propagation;

[0015] S102 drives the states of all nodes to be synchronously updated through the multi-round propagation mechanism in each round of propagation; after each round of propagation ends, the states of all nodes in the network form a high-dimensional system state snapshot;

[0016] S103 continuously records the states of all nodes after each round of propagation ends in the order of the propagation rounds, and forms a multi-dimensional time series data set.

[0017] As a preferred scheme of the network community discovery system through matrix analysis of the application, wherein:

[0018] The nonlinear dimension reduction assumes that the high-dimensional node state time series data is actually located on an intrinsic low-dimensional nonlinear manifold, and learns and preserves the local geometric structure of the nonlinear manifold by analyzing the proximity relationship between high-dimensional data points;

[0019] The mapping takes the high-dimensional state sequence of each node at multiple continuous time steps as a dynamic evolution path of the node in a high-dimensional state space;

[0020] The manifold learning calculates a corresponding low-dimensional coordinate for the high-dimensional state of each time point;

[0021] The entire high-dimensional state sequence of each node is converted into a continuous or discrete trajectory in the low-dimensional phase space.

[0022] As a preferred scheme of the network community discovery system through matrix analysis of the present application, wherein:

[0023] The dynamic characteristics include trajectory convergence, oscillation mode and attractor characteristics;

[0024] The coordinate points of all time steps in the low-dimensional phase space of each node are connected in time sequence to form a phase space trajectory representing the dynamic evolution of the node;

[0025] The node trajectory distribution data is a trajectory set of all nodes in the low-dimensional phase space;

[0026] The space positions of each point on each trajectory in the trajectory set are used to represent the similarity and difference of the node dynamics through the form, direction and density distribution of the trajectory.

[0027] As a preferred scheme of the network community discovery system through matrix analysis of the present application, wherein:

[0028] The convergence of the node trajectory is analyzed by matrix operation;

[0029] The matrix operation method includes:

[0030] S301 calculates the distribution covariance matrix formed by all trajectory points of each node trajectory in the low-dimensional phase space;

[0031] S302 performs eigenvalue decomposition on the distribution covariance matrix to extract a sequence of eigenvalues;

[0032] S303 quantifies the convergence degree of the node trajectory by analyzing the numerical distribution and decay rate of the sequence of eigenvalues;

[0033] If the eigenvalue rapidly decays to zero, the trajectory has little change in the corresponding dimension and converges to a point or a subspace as a whole.

[0034] As a preferred scheme of the network community discovery system through matrix analysis of the present application, wherein:

[0035] The oscillation mode of the node trajectory and the extraction of the attractor characteristics are analyzed by matrix operation;

[0036] calculating a transition probability matrix between consecutive state points on a node trajectory, identifying dominant eigenvalues and eigenvectors of the transition probability matrix through spectral analysis, and analyzing oscillation modes of the node trajectory;

[0037] regarding a set of trajectory points of all nodes in the network as a whole, generating a cluster label vector of the trajectory points through density clustering of the whole, and constructing a co-occurrence matrix through the cluster label vector, wherein each element represents a frequency of two nodes sharing a same cluster attractor, and the co-occurrence matrix is used for analyzing attractor characteristics.

[0038] As a preferred scheme of the network community discovery system through matrix analysis of the application, the community structure characteristic data is generated through trajectory similarity.

[0039] constructing a trajectory similarity matrix, wherein each element is used for measuring a similarity degree of overall trajectory shapes of a pair of nodes in a low-dimensional phase space;

[0040] the similarity measurement comprehensively considers a geometric shape, a dynamic evolution mode and an attributed attractor characteristic of the trajectory;

[0041] the trajectory similarity matrix is taken as the community structure characteristic data.

[0042] the trajectory similarity matrix is a symmetric semi-positive definite matrix, and a size of an element value is used for reflecting a similarity of dynamic behaviors of a corresponding node pair.

[0043] As a preferred scheme of the network community discovery system through matrix analysis of the application, the following are included:

[0044] the community boundary in the phase space is identified through density clustering, and a hierarchical community organization is constructed through multi-scale analysis, including:

[0045] the community structure characteristic data is taken as a proximity measurement of nodes in a characteristic space, and a clustering algorithm is executed.

[0046] the clustering algorithm forms a preliminary community division by identifying high-density node regions separated by low-density regions, wherein nodes located in a core of a high-density region are divided into a same community, and nodes located at edges of different high-density regions or in low-density regions are identified as community boundaries or noise points.

[0047] a community division result in different granularity scales is generated by dynamically adjusting a core parameter in the density clustering algorithm for controlling a community formation density.

[0048] The community division results of different granularity scales are organized into a hierarchical tree structure, which includes coarse-grained scales and fine-grained scales.

[0049] The community at the coarse-grained scale is taken as a parent node, contains and decomposes into multiple child communities at the fine-grained scale, and is used to show the hierarchical organization relationship of the network community structure.

[0050] The application provides a network community discovery method through matrix analysis, which comprises the following steps:

[0051] The static network topology is converted into an information propagation model, a network node state time series dynamic model is established by defining the information processing rules of nodes and the multi-round propagation mechanism, and node state time series data are outputted.

[0052] The node state time series data are received, high-dimensional time series state sequence data are mapped to low-dimensional phase space through a nonlinear dimension reduction method, and node trajectory distribution data are formed.

[0053] The node trajectory distribution data are received, the dynamic characteristics of the node trajectories are analyzed, the dynamic characteristics include trajectory convergence, oscillation mode and attractor feature, and community structure feature data are generated through trajectory similarity.

[0054] The community structure feature data are received, the community boundaries in the phase space are identified through density clustering, and hierarchical community organization is constructed through multi-scale analysis.

[0055] The application has the following beneficial effects:

[0056] The application converts the static topology into an information propagation model through a network module, establishes a node state time series dynamic model, can mine the network community structure from the dynamic perspective of information propagation, is more consistent with the operation mechanism of the actual network, maps high-dimensional time series state sequences to low-dimensional phase space through nonlinear dimension reduction processing of the phase space reconstruction module, effectively extracts the essential features of node evolution, and overcomes the dimension problem of high-dimensional data processing.

[0057] The community feature module of the application quantifies the dynamic behavior characteristics of nodes from multiple dimensions by analyzing the convergence, oscillation mode and attractor feature of the node trajectories, provides rich feature basis for community division, generates community structure feature data through trajectory similarity, defines the community relationship based on the actual evolution trajectory similarity of nodes in the phase space, avoids the influence of subjective parameter setting, can accurately distinguish core community members from boundary nodes, and improves the accuracy of community division.

[0058] The application realizes objective quantification of the node dynamic characteristics by eigenvalue decomposition and spectral analysis on the covariance matrix of node trajectories, transition probability matrix, etc., and the trajectory similarity matrix constructed by the application comprehensively considers geometric shape, dynamic evolution mode and attractor characteristics, and can fully reflect the similarity degree of nodes in dynamic behavior. BRIEF DESCRIPTION OF DRAWINGS

[0059] In order to more clearly illustrate the technical solutions of the embodiments of the application, the drawings needed in the embodiment description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the application, and other drawings can be obtained by those skilled in the art without creative labor. Among them:

[0060] Figure 1 A structural diagram of a network community discovery system through matrix analysis provided by the application;

[0061] Figure 2 A flowchart of a network community discovery method through matrix analysis provided by the application;

[0062] Figure 3 A flowchart of a network community discovery method through matrix analysis provided by the application;

[0063] Figure 4 A state matrix representation form of a network community discovery system through matrix analysis provided by the application;

[0064] Figure 5 A community structure diagram of a network community discovery system through matrix analysis provided by the application. DETAILED DESCRIPTION

[0065] In order to make the above-mentioned purposes, features and advantages of the application more obvious and easy to understand, the specific embodiments of the application will be described in detail below with reference to the drawings of the specification.

[0066] In the following description, many specific details are set forth in order to provide a thorough understanding of the application, but the application can also be implemented in other ways different from the description, and those skilled in the art can make similar generalizations without departing from the connotation of the application, therefore the application is not limited by the specific embodiments disclosed below.

[0067] Secondly, the term "an embodiment" or "embodiment" as used herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of this application. The phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it a single embodiment or an embodiment selectively excluded from other embodiments.

[0068] Example 1

[0069] like Figure 1 As shown, a network community discovery system using matrix analysis includes...

[0070] The network module transforms the static network topology into an information propagation model. By defining the information processing rules of nodes and the multi-round propagation mechanism, it establishes a dynamic model of the network node state sequence and outputs the node state time sequence data.

[0071] The process of transforming a static network topology into an information propagation model defines an information processing rule for each node in the network. The information processing rule is a state transition function. The input is the node's current state and the set of states of all its directly connected neighboring nodes in the previous propagation round. The output is the new state of the current node in the next propagation round. The current state is a discrete or continuous numerical value used to characterize the node's activity level, opinion bias, or infection state under the influence of information.

[0072] The multi-round propagation mechanism applies the information processing rules synchronously or asynchronously to all nodes in the network; in each round of propagation, each node independently calculates its new state based on the state of its neighboring nodes in the previous round through the state transition function.

[0073] In this application, a preferred method for dynamic propagation network modeling transforms a static network topology into a dynamic model with temporal characteristics. The specific implementation method includes:

[0074] Each node in the network is assigned an initial state value, which can be a discrete category identifier or a continuous intensity value, used to characterize the node's activity level, opinion tendency, or infection status at the beginning of information propagation; termination conditions for the propagation process are set, including a preset maximum propagation round threshold or a defined criterion for state convergence, to ensure that the propagation process can be completed within a reasonable range.

[0075] Further, an independent state transition function is defined for each node, which takes the current state value of the node itself and the state value set of all directly connected neighbor nodes in the previous propagation round as input parameters. The state transition function contains specific logical judgment mechanisms, which calculate the new state value of the node in the next propagation round by comprehensively considering the state maintenance tendency and neighbor state influence weight. This rule design enables each node to have autonomous information processing capability and adjust its state according to local environmental changes.

[0076] Further, the defined information processing rule is applied to all nodes in the network in a synchronous or asynchronous manner during each propagation round. In the synchronous update mode, all nodes calculate the new state in parallel according to the neighbor state in the previous round. In the asynchronous update mode, nodes update the state in a specific order. When calculating the new state, each node only depends on the state set of its neighbor nodes at the end of the previous round, and independently completes the calculation through the state transition function, ensuring the locality and autonomy of node state update.

[0077] Further, by comparing the amplitude of node state change between adjacent rounds, it is determined whether the network as a whole has reached a stable state. If the state change is below a preset threshold or reaches the maximum propagation round, the propagation process is terminated.

[0078] Further, after each round of propagation, the current state value of all nodes is recorded, and these data are organized in chronological order to form a complete node state time series dataset. This dataset completely records the state evolution trajectory of each node in the network during the entire propagation process, providing a data basis for subsequent analysis.

[0079] Through the above technical process, the system successfully converts the static network connection relationship into a dynamic node state evolution process, establishing a time series dynamic model that can reflect the information propagation characteristics of the network.

[0080] The method for establishing a network node state time series dynamic model and outputting a node state time series data comprises:

[0081] S101 assigns an initial state to each node in the network and sets the total number of propagation rounds or sets a state convergence judgment condition to terminate the propagation.

[0082] Specifically, each node in the network is assigned an initial state value, which is generated by a random initialization method or a deterministic initialization method based on the node topology characteristics.

[0083] Further, the termination condition of the propagation process is set, including two optional termination mechanisms: one is to set a fixed total number of propagation rounds, and the automatic termination is performed when the preset number of rounds is reached; the other is to set a judgment condition of state convergence, and the convergence is determined and the propagation is terminated when the total amplitude of state change between consecutive rounds is lower than a set threshold

[0084] In each round of propagation, the state of all nodes is driven to be synchronously updated through the multi-round propagation mechanism; after each round of propagation, the state of all nodes in the network constitutes a high-dimensional system state snapshot;

[0085] In each round of propagation, the system adopts a synchronous update mechanism to drive all nodes to update the state. The specific execution process is as follows: first, the state values of all nodes in the current round are collected, then based on the connection relationship between the nodes, the state set of the neighbor nodes of each node in the current round is obtained, each node calculates the new state value of the next round through a pre-defined state transition function according to its current state and the neighbor state set, and after the new state of all nodes is calculated, the state of the entire network is uniformly updated to form a system state snapshot after the end of the round. This state snapshot contains the state values of all nodes in the network at that moment, and constitutes a high-dimensional system state vector.

[0086] S103, in order of the propagation rounds, continuously records the state of all nodes after each round of propagation, forming a multi-dimensional time series data set.

[0087] The system automatically collects and stores the state values of all nodes after each round of propagation. The data recording process is performed in the order of the propagation rounds to ensure the continuity of the time series. The system arranges the state evolution trajectory of each node according to the time dimension to form an independent time series data sequence for each node. At the same time, the system integrates the time series data of all nodes into a unified multi-dimensional time series data set, which records the dynamic evolution behavior of the entire network system in the information propagation process. The finally generated data set can be directly used for subsequent phase space reconstruction analysis to provide a complete dynamic behavior data basis for community discovery.

[0088] The phase space reconstruction module receives the node state time series data output by the network module, and maps the high-dimensional time series state sequence data to a low-dimensional phase space through a nonlinear dimension reduction method to form node trajectory distribution data;

[0089] The nonlinear dimension reduction assumes that the high-dimensional node state time series data is actually located on an intrinsic low-dimensional nonlinear manifold, and the local geometric structure of the nonlinear manifold is learned and maintained by analyzing the proximity relationship between high-dimensional data points;

[0090] A preferred example is shown in Figure 4 The specific form of matrix analysis includes:

[0091] The matrix systematically records the state evolution history of all nodes in the entire network during the information propagation process in the form of a two-dimensional table.

[0092] Among them, the row direction represents the m nodes in the network, from node 1 to node m, each node occupies a row, covering all entities in the network.

[0093] The column direction represents the continuous n time steps from t1 to tn, that is, the multiple iteration rounds of information propagation, which constitutes the time sequence dimension of dynamic analysis.

[0094] Each cell in the matrix is marked as a state value, which is used to store the state value of a specific node at a specific time.

[0095] The state value is a discrete or continuous numerical value used to quantify the activity, opinion tendency or infection state of the node under the influence of information.

[0096] As shown in Figure 4 The matrix is the output of the network module, that is, the structured representation of the high-dimensional system state snapshot, which is used to capture the system-level dynamic behavior of the static network topology driven by the information propagation model, and to convert the static connection relationship into dynamic time sequence data.

[0097] Further, this high-dimensional time sequence will be input to the phase space reconstruction module, and through nonlinear dimension reduction method, it is mapped to low-dimensional phase space to form node trajectory distribution data.

[0098] The mapping regards the high-dimensional state sequence of each node at multiple consecutive time steps as a dynamic evolution path of the node in the high-dimensional state space.

[0099] Through the manifold learning, a corresponding low-dimensional coordinate is calculated for each high-dimensional state at each time point.

[0100] The entire high-dimensional state sequence of each node is converted into a continuous or discrete trajectory in the low-dimensional phase space.

[0101] In this application, a preferred specific implementation method for converting high-dimensional time sequence data to low-dimensional phase space through the phase space reconstruction module includes:

[0102] S201 receives node state time sequence data from the network module, which contains the state values of all nodes in the network at multiple consecutive time steps, parses and verifies the input data to ensure the integrity of the data format and the correctness of the time sequence;

[0103] Further, the dimension characteristics and sample size of the evaluation data are assessed.

[0104] In S202, according to the basic principle of manifold learning, it is assumed that the high-dimensional time series data is actually distributed on a potential low-dimensional nonlinear manifold. By analyzing the neighborhood relationship between high-dimensional data points, the local neighborhood structure of the data points is constructed.

[0105] Specifically, the distance or similarity between data points is calculated to determine the nearest neighbor set of each data point, and the local geometric characteristics of the original high-dimensional space are learned.

[0106] Further, an optimal mapping from the high-dimensional space to the low-dimensional space is found by an optimization algorithm. The optimal mapping process aims to keep the data points in the low-dimensional space the same neighborhood relationship as in the high-dimensional space, i.e. the adjacent points in the high-dimensional space are still adjacent in the low-dimensional space. The corresponding low-dimensional coordinate representation is calculated for each high-dimensional data point, and the dimension compression from the high-dimensional state space to the low-dimensional phase space is completed.

[0107] It should be noted that those skilled in the art can learn how to realize the conversion of high-dimensional time series data to low-dimensional phase space according to the technical means of manifold learning disclosed in the present application combined with the basic principle of manifold learning known in the art.

[0108] In S203, for each node, the high-dimensional state sequence of the node at multiple consecutive time steps is regarded as the complete evolution path of the node in the high-dimensional state space. The dimension reduction mapping relationship obtained by S202 is used to convert the high-dimensional state of each time step of the node into a coordinate point in the low-dimensional phase space.

[0109] Further, all coordinate points of the node in the low-dimensional phase space are connected in time sequence to form the evolution trajectory of the current node in the phase space. The evolution trajectory is in the form of a continuous curve or a discrete point sequence, which accurately reflects the motion law and evolution trend of the node state in the phase space.

[0110] In S204, the phase space trajectories of all nodes are integrated to form a complete node trajectory distribution data set.

[0111] The node trajectory distribution data set records the spatial position, morphological characteristics and dynamic properties of each node trajectory.

[0112] Further, the trajectory data is standardized and organized to ensure data consistency and accessibility.

[0113] The dynamic characteristics include trajectory convergence, oscillation mode and attractor characteristics.

[0114] All coordinate points of each node in the low-dimensional phase space at all time steps are connected in time sequence to form a phase space trajectory representing the dynamic evolution of the node.

[0115] The node trajectory distribution data is a set of trajectories of all nodes in the low-dimensional phase space;

[0116] The trajectory set contains the spatial positions of each point on each trajectory, and represents the similarities and differences of node dynamics through the shape, direction and density distribution of the trajectory.

[0117] In this application, a preferred node trajectory dynamic characteristic analysis and trajectory distribution data generation method includes:

[0118] S2001 evaluates the convergence characteristics of each node trajectory in the phase space, and judges whether the node state tends to be stable by calculating the convergence degree of the end point of the trajectory and the contraction trend of the trajectory path.

[0119] Specifically, for convergent trajectories, record their convergence speed and final convergence position; for divergent or continuously changing trajectories, quantify their change amplitude and fluctuation range. The convergence analysis of the trajectory reflects the ability of the node to reach a steady state in the information propagation process.

[0120] S2002 detects periodic or quasi-periodic oscillation behavior existing in the current node trajectory.

[0121] Further, by analyzing the fluctuation frequency, amplitude change and period of the trajectory curve, different oscillation modes are identified, and the oscillation modes are divided into regular oscillation, chaotic oscillation and random fluctuation, and the characteristic parameters of each mode are recorded.

[0122] S2003 identifies the specific region of the trajectory trend in the phase space, i.e. the attractor feature.

[0123] Further, by analyzing the density distribution and aggregation of the trajectory points, high-density regions in the phase space are found, which correspond to stable states or typical behavior patterns;

[0124] Further, record the spatial position, influence range and attraction strength of each attractor, and determine the attribution relationship of each node trajectory to different attractors.

[0125] S2004 integrates all node phase space trajectories into a complete trajectory distribution data set, each trajectory completely records all coordinate point sequences in the phase space, and extracts the shape features of the trajectory, including trajectory length, curvature change, direction trend and other geometric properties.

[0126] Further, by analyzing the spatial distribution density and relative position relationship of the trajectory in the phase space, the system establishes a dynamic similarity measure between nodes.

[0127] The trajectory trend reflects the evolution direction of the node state, and the densely distributed trajectory area represents a node cluster with similar dynamic behavior. The similarity and difference of the nodes in the dynamic characteristics are quantified by the spatial distance and the morphological similarity between the trajectories.

[0128] The application realizes the conversion from the original trajectory data to the deep dynamic characteristic understanding, and provides data information and different features of different communities for community discovery based on the dynamic characteristics.

[0129] The community feature module receives the node trajectory distribution data output by the phase space reconstruction module, analyzes the dynamic characteristics of the node trajectory, and generates community structure feature data through trajectory similarity;

[0130] The analysis of the convergence of the node trajectory is performed by matrix operation;

[0131] The matrix operation method includes:

[0132] S301, for each node trajectory, a distribution covariance matrix formed by all trajectory points in the low-dimensional phase space is calculated;

[0133] Specifically, for each node phase space trajectory, first, the center position of all trajectory points, i.e. the mean point of the trajectory in the low-dimensional phase space, is calculated.

[0134] Further, the system constructs the distribution covariance matrix of the node trajectory by matrix operation method. One preferred method for constructing the distribution covariance matrix of the node trajectory is as follows: the deviation vector of each trajectory point from the center position is calculated, and the covariance matrix is obtained by the outer product operation of the deviation vector. The covariance matrix is used to describe the change amplitude of the node trajectory in each dimension and the correlation between different dimension changes.

[0135] S302, eigenvalue decomposition is performed on the covariance matrix to extract the eigenvalue sequence;

[0136] Specifically, the eigenvalue decomposition operation is performed on the calculated covariance matrix. The decomposition process represents the covariance matrix as a combination of eigenvectors and eigenvalues, where the eigenvectors represent the main directions of the trajectory distribution, and the eigenvalues represent the change intensity of the trajectory in these directions. The system sorts the eigenvalues in descending order according to the numerical value, and extracts the ordered eigenvalue sequence. This sequence quantitatively reveals the energy distribution of the node trajectory in different dimension directions.

[0137] S303, the convergence degree of the node trajectory is quantified by analyzing the numerical distribution and decay rate of the eigenvalue sequence;

[0138] The convergence degree of the node trajectory is quantified by analyzing the numerical distribution and decay rate of the eigenvalue sequence.

[0139] A preferred evaluation method includes observing the concentration of larger eigenvalues in the eigenvalue sequence, analyzing the decay curve shape of the eigenvalues from large to small, and calculating the proportion of the first few main eigenvalues in the total energy.

[0140] If the eigenvalue sequence presents a rapid decay characteristic, that is, the first few eigenvalues occupy most of the energy, and the subsequent eigenvalues quickly approach zero, it is determined that the trajectory of the node has little change in the corresponding dimension direction, and the overall trajectory shows a strong convergence trend. If the eigenvalues decay slowly and the energy distribution in each dimension is relatively uniform, it indicates that the trajectory has significant changes in multiple dimensions, and the convergence degree is lower.

[0141] If the eigenvalues quickly decay to zero, the trajectory has little change in the corresponding dimension, and the overall trajectory converges to a point or subspace.

[0142] The oscillation mode of the node trajectory and the attractor feature are analyzed by matrix operation;

[0143] A transition probability matrix between consecutive state points on the node trajectory is calculated, the dominant eigenvalues and eigenvectors of the transition probability matrix are identified by spectral analysis, and the oscillation mode of the node trajectory is analyzed;

[0144] The set of trajectory points of all nodes in the entire network is regarded as a whole, a clustering label vector of the trajectory points is generated by density clustering on the whole, and a co-occurrence matrix is constructed by the clustering label vector, wherein each element represents the frequency of two nodes sharing the same clustering attractor, which is used to analyze the attractor feature.

[0145] In the present application, a preferred matrix analysis method of the oscillation mode of the node trajectory and the attractor feature includes:

[0146] S401 constructing a transition probability matrix between consecutive state points on the node trajectory;

[0147] Specifically, a probability distribution model of state transition is constructed by counting the frequency of the node trajectory from one state to another state in the phase space;

[0148] Further, the phase space is divided into several state regions, the probability of the trajectory point from the current state region to the next state region is calculated, and a complete transition probability matrix is formed.

[0149] The eigenvalues and eigenvectors of the matrix are extracted by spectral analysis of the transition probability matrix, the spectral analysis process includes calculating the eigenvalue decomposition of the matrix, identifying the dominant eigenvalues and the corresponding eigenvectors, the amplitude and phase characteristics of the dominant eigenvalues are used to reflect the strength and period of the trajectory oscillation, and the spatial distribution of the eigenvectors is used to reflect the main mode direction of the oscillation.

[0150] Further, by analyzing the distribution pattern of dominant eigenvalues, a preferred oscillation behavior includes: periodic oscillation corresponding to eigenvalues close to the unit circle, quasi-periodic oscillation formed by multiple eigenvalues combination, and chaotic oscillation characteristics exhibited by eigenvalue distribution.

[0151] S402 takes the set of trajectory points of all nodes in the entire network as a whole data space, and can perform clustering analysis on the trajectory points by using a density clustering algorithm. The density clustering algorithm automatically discovers natural clusters in the phase space by identifying high-density regions, and each cluster center corresponds to a potential attractor.

[0152] It should be noted that, based on the density clustering algorithm disclosed in the present application, a person skilled in the art can combine the technical means of trajectory point and reverse clustering analysis with the technical content of the density clustering algorithm known in the art to realize how to identify high-density regions and discover natural clusters in the phase space by using the density clustering algorithm.

[0153] The clustering process generates a clustering label for each trajectory point, and the system organizes these labels into a clustering label vector. Based on the clustering label vector, a co-occurrence matrix between nodes is constructed, wherein each matrix element represents the frequency of trajectory points of two nodes appearing in the same cluster. The frequency value quantifies the correlation strength of nodes at the attractor level, reflecting the degree of tendency of nodes to the same stable state at the attractor level.

[0154] A preferred example of feature analysis of the co-occurrence matrix includes: a highly aggregated diagonal block structure indicating the presence of multiple strong attractors, and a uniform distribution pattern reflecting the presence of a single dominant attractor. By analyzing the co-occurrence matrix, core attractor regions, transition regions and edge nodes are identified, and community structures are accurately divided.

[0155] The community structure feature data is generated by trajectory similarity;

[0156] A trajectory similarity matrix is constructed, wherein each element is used to measure the similarity of the overall trajectory shape of a pair of nodes in the low-dimensional phase space;

[0157] The similarity measure takes into account the geometric shape of the trajectory, the dynamic evolution pattern and the attractor characteristics to which it belongs;

[0158] The trajectory similarity matrix is used as community structure feature data;

[0159] The trajectory similarity matrix is a symmetric semi-positive definite matrix, and the size of the element value is used to reflect the similarity of the corresponding node pair in the dynamic behavior.

[0160] A preferred trajectory similarity matrix construction and community structure feature data generation method in the application comprises:

[0161] S501 first establishes a comprehensive trajectory similarity evaluation system, which includes set shape dimension, dynamic evolution mode dimension and attractor attribution dimension.

[0162] Specifically, in the geometric shape dimension, the similarity of trajectories in shape is evaluated by comparing the spatial form features of trajectory curves, including the overall trend of the trajectory, the curvature change trend and the spatial distribution range.

[0163] Specifically, in the dynamic evolution mode dimension, the time sequence evolution law of the trajectory is analyzed, including the rate, rhythm and periodicity of state change, and trajectories with similar dynamic behavior are identified.

[0164] Specifically, in the attractor attribution dimension, whether the trajectories tend to the same attractor is judged according to the final convergence position or main distribution area of the trajectories in the phase space.

[0165] S502 performs similarity calculation operation on each pair of nodes in the network, and a preferred calculation process comprises: generating independent similarity scores in three dimensions respectively, wherein the geometric shape similarity is obtained by matching the trajectory space features, the dynamic evolution similarity is obtained by comparing the time sequence change mode, and the attractor attribution similarity is obtained by analyzing the distribution relationship of the trajectories in the attractor space.

[0166] Further, the similarity scores in the three dimensions are integrated into a comprehensive similarity value by using the weighted integration or feature fusion method, and the similarity value is used to reflect the overall similarity degree of the two nodes in dynamic behavior.

[0167] S503 uses the similarity values of all node pairs to construct a complete trajectory similarity matrix.

[0168] Specifically, the trajectory similarity matrix is a symmetric matrix structure, the rows and columns of the matrix correspond to the nodes in the network, and the element value at each position in the matrix represents the comprehensive trajectory similarity between the corresponding two nodes. The system ensures the mathematical consistency of the matrix through symmetry processing, and ensures the reasonable range of similarity value through numerical normalization processing.

[0169] S504 takes the constructed trajectory similarity matrix as the core community structure feature data, and the mathematical properties of the community structure feature matrix are used to ensure its effectiveness as the basis for community division: symmetry ensures the undirected nature of node relationship, and semi-positive definite property ensures the stability of feature extraction.

[0170] Further, the quality of the data is verified by matrix eigenvalue analysis, including checking the rank of the matrix, eigenvalue distribution and other mathematical properties, to ensure that the generated feature data accurately reflects the dynamic similarity relationship between nodes.

[0171] The community division module receives the community structure feature data output by the community feature module, identifies the community boundary in the phase space through density clustering, and constructs a hierarchical community organization through multi-scale analysis.

[0172] As shown in Figure 5 A preferred hierarchical community organization implementation method includes:

[0173] The top layer is a coarse-grained layer for displaying macro-scale community division results, including two large communities, community A and community B,

[0174] Specifically, the top layer corresponds to the division results when the density clustering parameters are relatively loose, reflecting the global community pattern of the network.

[0175] The middle layer is a fine-grained layer, which is used to display that community A is decomposed into community A1 and community A2, and community B is decomposed into community B1 and community B2, reflecting that the coarse-grained community is finely decomposed by adjusting the density clustering parameters.

[0176] The bottom layer is a micro layer, which is used to display the most fine-scale community division, for example, community A1 is further decomposed into community A11 and community A12.

[0177] The system described in the present application can automatically discover and present the inherent and nested community organization relationship of the network, while traditional methods can usually only provide single-scale division.

[0178] The density clustering is used to identify the community boundary in the phase space, and the multi-scale analysis is used to construct a hierarchical community organization, which includes:

[0179] The community structure feature data is used as the proximity measure of nodes in the feature space, and a clustering algorithm is executed;

[0180] The clustering algorithm forms a preliminary community division by identifying high-density node regions separated by low-density regions, wherein nodes located at the core of the high-density region are divided into the same community, and nodes located at the edge of different high-density regions or low-density regions are identified as community boundaries or noise points;

[0181] By dynamically adjusting the core parameter in the density clustering algorithm for controlling the community formation density, community division results at different granularity scales are generated;

[0182] The community division results at different granularity scales are organized into a hierarchical tree structure, and the tree structure includes coarse-grained scales and fine-grained scales.

[0183] The community at a coarse granularity scale is taken as a parent node, contains and is decomposed into multiple sub-communities at a fine granularity scale, and is used to show the hierarchical organization relationship of the network community structure.

[0184] In the present application, a preferred community boundary identification and hierarchical community organization construction method specifically includes:

[0185] S601 receives a trajectory similarity matrix generated by a community feature module as input data, and takes the trajectory similarity matrix as a basis for measuring the proximity of nodes in a feature space.

[0186] Specifically, first, the feature data is standardized for preprocessing to ensure that the feature values of different dimensions are comparable.

[0187] Further, the system initializes the core parameters of the density clustering algorithm according to the distribution characteristics of the feature data, including the neighborhood radius threshold and the minimum density point requirement.

[0188] S602 executes the density clustering algorithm to identify the community structure by scanning the distribution of data points in the feature space.

[0189] The density clustering algorithm first locates the core nodes of high-density areas, and the core nodes contain a sufficient number of adjacent nodes within a given neighborhood radius around them, forming a dense cluster.

[0190] Further, starting from these core nodes, the community range is gradually expanded through density connection relationships, and nodes directly density reachable are included in the same community.

[0191] During the formation of the community, the system pays special attention to the identification of boundary nodes. Nodes located at the edge of different high-density areas are marked as community boundary points, which are simultaneously affected by multiple communities, and isolated nodes in low-density areas are identified as noise points, and these nodes are specially marked, so that the core area and boundary range of each community can be clearly defined through the marking system.

[0192] S603 generates multi-scale community division results by dynamically adjusting the core parameters of the density clustering algorithm. When a larger neighborhood radius and a higher minimum density threshold are selected, coarse-grained community division is produced, and only the most dense areas are identified as independent communities, and smaller or sparser communities are merged into large communities; if a smaller neighborhood radius and a lower minimum density threshold are selected, fine-grained community division is produced, and more local dense areas are identified as independent communities, and large communities are decomposed into multiple sub-communities.

[0193] By continuously adjusting the parameter values, a complete community division spectrum from the coarsest granularity to the finest granularity is generated.

[0194] S604 organizes the community division results of different granularity scales into a hierarchical tree structure.

[0195] Specifically, the community at the coarse-grained scale is taken as a parent node, which contains and decomposes into multiple child communities at the fine-grained scale. The system establishes a containing relationship mapping between the parent and child communities to ensure the consistency of community members in the hierarchical structure.

[0196] Further, the construction process of the hierarchical tree adopts a bottom-up or top-down method: the bottom-up method gradually merges fine-grained communities into coarse-grained communities, while the top-down method gradually splits coarse-grained communities into fine-grained communities.

[0197] Further, the system records the location relationship, containing relationship and cross-scale evolution trajectory of each community in the hierarchical structure, forming a complete community hierarchical organization model.

[0198] Embodiment 2

[0199] As shown in Figure 2 A network community discovery method through matrix analysis includes: converting a static network topology into an information propagation model, establishing a network node state time series dynamic model by defining a node information processing rule and a multi-round propagation mechanism, and outputting node state time series data;

[0200] Receiving the node state time series data, mapping high-dimensional time series state sequence data to low-dimensional phase space through a nonlinear dimension reduction method to form node trajectory distribution data;

[0201] Receiving the node trajectory distribution data, analyzing the dynamic characteristics of the node trajectory, the dynamic characteristics including trajectory convergence, oscillation mode and attractor characteristics, and generating community structure feature data through trajectory similarity;

[0202] Receiving the community structure feature data, identifying community boundaries in the phase space through density clustering, and constructing a hierarchical community organization through multi-scale analysis.

[0203] In this application, a preferred embodiment of a network community discovery method through matrix analysis includes as shown in Figure 3

[0204] Converting a static network topology into an information propagation model, establishing a network node state time series dynamic model by defining a node information processing rule and a multi-round propagation mechanism, and outputting node state time series data;

[0205] ​The static network topology is converted into a dynamic information propagation model, and information processing rules are defined for each node in the network. The information processing rules determine the state value of the node at the next time through a preset logical judgment mechanism based on the current state of the node and the state set of its neighbor nodes. The node state is represented by a numerical value, which can represent the information activity, opinion tendency or infection state of the node.

[0206] Further, in each round of propagation process, all nodes update their own state in parallel or series according to the state information of their neighbor nodes in the previous round. The multi-round propagation mechanism simulates the multi-hop propagation characteristics of information in the real network, so that the state change of local nodes can gradually spread to the global through network connection. By setting the propagation round or state convergence condition, the propagation process is automatically terminated to ensure the efficiency and stability of the model running.

[0207] During the model running process, the system records the state value of each node after the end of each round of propagation in chronological order to form a complete node state time series data set. The phase state time series data set comprehensively records the state evolution trajectory of all nodes in the network during the information propagation process, providing a data basis for subsequent dynamic analysis.

[0208] The node state time series data is received, and the high-dimensional time series state sequence data is mapped to a low-dimensional phase space by a nonlinear dimension reduction method to form node trajectory distribution data.

[0209] After receiving the node state time series data output by the network module, the high-dimensional time series data is mapped to a low-dimensional phase space by a nonlinear dimension reduction technique. Through manifold learning, it is assumed that the high-dimensional node state data is actually distributed on an intrinsic low-dimensional nonlinear manifold. By analyzing the proximity relationship between high-dimensional data points, the local geometric structure characteristics of the original data space are learned and maintained.

[0210] During the mapping process, the high-dimensional state sequence of each node at multiple consecutive time steps is regarded as a dynamic evolution path of the node in the high-dimensional state space. Through a nonlinear dimension reduction algorithm, the low-dimensional coordinate value corresponding to the high-dimensional state of each time point is calculated, and the entire high-dimensional state sequence is converted into a continuous or discrete trajectory in the low-dimensional phase space.

[0211] Further, the trajectory set of all nodes in the low-dimensional phase space forms node trajectory distribution data. Each trajectory is characterized by its morphological features, trend and density distribution in the phase space.

[0212] The node trajectory distribution data is received, and the dynamic characteristics of the node trajectory are analyzed. The dynamic characteristics include trajectory convergence, oscillation mode and attractor characteristics. Community structure feature data is generated by trajectory similarity.

[0213] Trajectory convergence analysis assesses the degree of trajectory aggregation in the phase space to determine whether the state evolution of a node tends to be stable. The system quantifies the convergence degree by calculating the distribution characteristics of trajectory points, and highly convergent trajectories indicate that the node state tends to be stable, while divergent trajectories reflect the continuous dynamic changes of the node.

[0214] Oscillation pattern recognition focuses on whether the node trajectory presents periodic or quasi-periodic change rules. The system identifies different oscillation behaviors by analyzing the fluctuation characteristics and cycle patterns of the trajectory, which reflect the regular dynamic characteristics of the node in the information propagation process.

[0215] Attractor feature extraction aims to find specific areas or states to which the trajectory tends to converge in the phase space. By density analysis, the trajectory concentration area corresponding to the stable state or typical behavior pattern in the network system is identified.

[0216] Community structure feature data is generated by calculating the similarity between node trajectories. The similarity measure takes into account the geometric shape similarity, dynamic evolution pattern consistency, and attractor attribution of the trajectory, and constructs a feature matrix that comprehensively reflects the similarity of node dynamic behavior.

[0217] The community structure feature data is received, and the community boundaries in the phase space are identified by density clustering. Through multi-scale analysis, a hierarchical community organization is constructed.

[0218] After receiving the community structure feature data, the system uses a density clustering algorithm to identify the community boundaries in the phase space. By analyzing the density distribution characteristics of data points in the feature space, the core nodes in high-density areas are divided into the same community, and the nodes located at the edge of different high-density areas or in low-density areas are identified as community boundaries or noise points.

[0219] Further, by dynamically adjusting the core parameters of the density clustering algorithm, community division results of different granularity scales are generated. Parameter adjustment changes the density threshold for community formation, thereby controlling the size and granularity of the community.

[0220] Further, the community division results under different scales are organized into a hierarchical tree structure, in which the community under a coarse-grained scale serves as a parent node, containing and decomposing into multiple sub-communities under a fine-grained scale.

[0221] It is important to note that the construction and arrangement of the application shown in the various exemplary embodiments is illustrative only. Although only two embodiments have been described in detail in this disclosure, persons of ordinary skill in the art having the benefit of this disclosure will readily appreciate that many modifications are possible without making substantial deviations from the novel teachings and advantages of the subject matter described in this application. For example, the size, shape, arrangement, and relative proportions of the various elements, as well as the values of parameters such as temperature, pressure, etc., the mounting arrangements, the use of materials, colors, orientations, etc., can be changed as desired. For example, an element shown as a single configuration can be divided into two or more separate elements, the position of an element can be reversed or otherwise changed, and the nature or number of elements or positions can be modified or changed. Accordingly, all such modifications are intended to be included within the scope of the present application. The order or sequence of any process or method steps can be changed unless otherwise specified or clearly implied by the circumstances. Any reference signs in the description shall not be construed as limiting the scope of the application. Any "device plus function" clauses are intended to cover the structures described herein as performing their claimed function, and also material equivalent thereof. Other substitutions, modifications, changes and omissions can be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present application. Accordingly, the present application is not limited to the particular embodiments described herein, but extends to various modifications that still fall within the scope of the appended claims.

[0222] Furthermore, in order to provide a concise description of exemplary embodiments, not all features of an actual implementation can be described (i.e., those not necessary to enable one to practice the present application, or those not related to the current consideration of the best mode for practicing the present application).

[0223] It is to be understood that the development of the exemplary embodiments of this application can not be limited to the precise constructional arrangements shown in the drawings, and that various modifications can be made to the exemplary embodiments and the arrangement and functioning of the various elements without departing from the scope of the application as set forth in the claims.

[0224] It should be noted that the above examples are intended to be illustrative only and not limiting of the present application. Although the present application has been described in detail with reference to the preferred embodiments, those with ordinary skill in the art will appreciate that various modifications can be made without departing from the spirit and scope of the present application, which is defined by the claims.

Claims

1. A network community discovery system using matrix analysis, characterized in that, include: The network module transforms the static network topology into an information propagation model. By defining the information processing rules of nodes and the multi-round propagation mechanism, it establishes a dynamic model of the network node state sequence and outputs the node state time sequence data. The process of transforming a static network topology into an information propagation model defines an information processing rule for each node in the network. The information processing rule is a state transition function. The input is the node's current state and the set of states of all its directly connected neighboring nodes in the previous propagation round. The output is the new state of the current node in the next propagation round. The current state is a discrete or continuous numerical value used to characterize the node's activity level, opinion bias, or infection state under the influence of information. The multi-round propagation mechanism applies the information processing rules synchronously or asynchronously to all nodes in the network; in each round of propagation, each node independently calculates its new state based on the state of its neighboring nodes in the previous round through the state transition function. The method for establishing a network node state temporal dynamic model and outputting node state temporal data is as follows: S101 assigns an initial state to each node in the network and sets the total number of rounds of propagation or sets a condition for state convergence to terminate propagation. In each round of propagation, S102 drives the state of all nodes to be updated synchronously through the multi-round propagation mechanism; after each round of propagation, the state of all nodes in the network constitutes a high-dimensional system state snapshot. S103 continuously records the state of all nodes after each round of propagation in the order of propagation rounds, forming a multidimensional time series dataset; The phase space reconstruction module receives the node state time series data output by the network module and maps the high-dimensional time series state sequence data to the low-dimensional phase space through a nonlinear dimensionality reduction method to form node trajectory distribution data. The community feature module receives the node trajectory distribution data output by the phase space reconstruction module, analyzes the dynamic characteristics of the node trajectories, and generates community structure feature data through trajectory similarity. The dynamic characteristics include trajectory convergence, oscillation modes, and attractor features; Connect the coordinates of all time steps in the low-dimensional phase space of each node in chronological order to form a phase space trajectory that characterizes the dynamic evolution of the node. The node trajectory distribution data is the set of trajectories of all nodes in the low-dimensional phase space; The trajectory set includes the spatial location of each point on each trajectory, and uses the shape, direction and density distribution of the trajectory to represent the similarity and difference of the dynamics of the nodes. The community segmentation module receives community structural feature data output by the community feature module, identifies community boundaries in phase space through density clustering, and constructs hierarchical community organizations through multi-scale analysis.

2. The network community discovery system based on matrix analysis as described in claim 1, characterized in that: The nonlinear dimensionality reduction assumes that the high-dimensional node state time series data actually resides on an intrinsic low-dimensional nonlinear manifold. By analyzing the proximity relationships between high-dimensional data points, the local geometric structure of the nonlinear manifold is learned and preserved. The mapping treats the high-dimensional state sequence of each node at multiple consecutive time steps as a dynamic evolution path of the node in the high-dimensional state space. Through the manifold learning, a corresponding low-dimensional coordinate is calculated for the high-dimensional state at each time point; The entire high-dimensional state sequence of each node is transformed into a continuous or discrete trajectory in a low-dimensional phase space.

3. The network community discovery system based on matrix analysis as described in claim 1, characterized in that: The convergence of the analyzed node trajectories is determined through matrix operations; The matrix operation methods include: S301 calculates the distribution covariance matrix of all trajectory points in the low-dimensional phase space for each node's trajectory. S302 performs eigenvalue decomposition on the distribution covariance matrix and extracts its eigenvalue sequence; S303 quantifies the convergence degree of the node trajectory by analyzing the numerical distribution and decay rate of the feature value sequence; If the eigenvalues ​​decay rapidly to zero, the trajectory changes only slightly in the corresponding dimension and converges as a whole to a point or subspace.

4. The network community discovery system based on matrix analysis as described in claim 3, characterized in that: The oscillation patterns of the analyzed node trajectories and attractor features are extracted through matrix operations. Calculate the transition probability matrix between consecutive state points on the node trajectory, and identify its dominant eigenvalues ​​and eigenvectors by performing spectral analysis on the transition probability matrix to analyze the oscillation mode of the node trajectory. The set of trajectory points of all nodes in the entire network is regarded as a whole. Density clustering is performed on the whole to generate cluster label vectors of trajectory points. A co-occurrence matrix is ​​constructed using the cluster label vectors, where each element represents the frequency at which two nodes share the same cluster attractor, which is used to analyze attractor features.

5. The network community discovery system based on matrix analysis as described in claim 4, characterized in that: The method generates community structure feature data through trajectory similarity; Construct a trajectory similarity matrix, where each element measures the similarity of the overall trajectory morphology of a pair of nodes in a low-dimensional phase space; The similarity metric takes into account the trajectory's geometry, dynamic evolution pattern, and the characteristics of the attractor to which it belongs. The trajectory similarity matrix is ​​used as community structure feature data; The trajectory similarity matrix is ​​a symmetric positive semi-definite matrix, and the magnitude of the element values ​​is used to reflect the similarity of the corresponding node pairs in dynamic behavior.

6. The network community discovery system based on matrix analysis as described in claim 5, characterized in that: The process of identifying community boundaries in phase space through density clustering and constructing hierarchical community organizations through multi-scale analysis includes: Using the community structure feature data as a measure of the proximity of nodes in the feature space, a clustering algorithm is executed; The clustering algorithm forms a preliminary community division by identifying high-density node regions separated by low-density regions. Nodes located at the core of high-density regions are classified into the same community, while nodes located at the edges of different high-density regions or in low-density regions are identified as community boundaries or noise points. By dynamically adjusting the core parameters used to control the density of community formation in the density clustering algorithm, community partitioning results at different granularity scales are generated. The community partitioning results at different granularity scales are organized into a hierarchical tree structure, which includes coarse-grained and fine-grained scales. The coarse-grained community serves as the parent node, which contains and decomposes into multiple sub-communities at the fine-grained scale, thus demonstrating the hierarchical organizational relationship of the network community structure.

7. A method for discovering network communities through matrix analysis, characterized in that... Includes a network community discovery system based on matrix analysis as described in any one of claims 1-6; wherein: The static network topology is transformed into an information propagation model. By defining the information processing rules of nodes and the multi-round propagation mechanism, a network node state time-series dynamic model is established, and node state time-series data is output. The node state time series data is received, and the high-dimensional time series state sequence data is mapped to a low-dimensional phase space through a nonlinear dimensionality reduction method to form node trajectory distribution data. The node trajectory distribution data is received, and the dynamic characteristics of the node trajectories are analyzed, including trajectory convergence, oscillation mode and attractor features. Community structure feature data is generated through trajectory similarity. The system receives the community structure feature data, identifies community boundaries in phase space through density clustering, and constructs hierarchical community organizations through multi-scale analysis.

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