A network structure analysis method based on inter-node point mutual information

Abstract

The application relates to a network structure analysis method based on inter-node point mutual information, which comprises the following steps: calculating inter-node point mutual information between all node pairs of a graph to be analyzed, constructing an inter-node point mutual information matrix, and determining information distances between the node pairs; using a spectral clustering method to calculate each community division when the graph to be analyzed is divided into different community numbers, and constructing a hierarchical community structure as a set of each community division; determining mutual information density of a node based on information distances between the node and each other node, and defining a node with a density greater than that of all adjacent nodes as a core node; calculating dominant adjacent nodes of the nodes in the graph to be analyzed, and constructing a core propagation forest of the graph to be analyzed; and using the core propagation forest constructed based on the dominant adjacent relationship of the nodes, which is beneficial to stripping a topological skeleton of a complex network from bottom to top, stably and clearly, and accurately positioning core nodes and bridge nodes connecting communities.

Description

Technical Field

[0001] This invention relates to the fields of computer science and technology, and in particular to a network structure analysis method based on inter-node mutual information. Background Technology

[0002] Complex networks abstract various systems into topological structures, where entities within a system are mapped to nodes in the network, and the relationships or interactions between entities are mapped to edges. Research on complex networks helps reveal the underlying evolutionary patterns, dynamic characteristics, and intrinsic mechanisms of various complex systems, thereby predicting system behavior.

[0003] Network structure analysis aims to extract valuable information from network structure data using methods from mathematics, graph theory, and computer science. Existing network structure analysis methods cover multiple dimensions of research, mainly including: macroscopic statistical characteristic analysis of networks (such as degree distribution and calculation of average path length), mesoscopic topology of networks (such as community structure partitioning), and microscopic node attribute evaluation (such as node centrality evaluation). Through these analytical methods, key characteristics of networks such as connectivity, stability, and information propagation efficiency can be effectively characterized.

[0004] Current mainstream approaches to community detection algorithms aim to achieve hierarchical community structures by adjusting resolution parameters or using hierarchical clustering to obtain community partitions of varying granularities. One existing method constructs a randomized external uniform partitioning model and calculates edge probabilities based on this model to achieve bottom-up agglomerative hierarchical clustering. However, hierarchical clustering suffers from error accumulation; if a node is initially assigned to an incorrect community, subsequent incorrect assignments can affect a large number of its surrounding nodes and even the entire community. Furthermore, while clustering can reveal the hierarchical community structure, it fails to differentiate the importance and role of nodes within the network structure. Existing core node identification methods typically consider global topological path features or local structural features to calculate node importance or influence, and then infer a node's role within the network or community based on influence ranking. Another existing approach proposes a key node identification method based on a comprehensive weighted calculation of global and local paths. This method constructs an efficiency matrix between nodes and a quantity influence matrix for various paths, reducing the dimensionality of complex topological edges and synthesizing them into a single comprehensive influence score to assess node importance. However, such methods cannot reconstruct network topology information. High-influence / high-importance nodes often cannot correspond to the hierarchical structure of the network, and the most important nodes may even be in the same community. Summary of the Invention

[0005] This invention addresses the technical problems existing in the prior art by providing a network structure analysis method based on inter-node mutual information. The method analyzes network structure based on inter-node mutual information density, core propagation forest, etc., which can effectively solve the problem that network structure analysis is limited by resolution limits and resolution bias, and it is difficult to clearly show the nested small community structure within a large community at a single scale.

[0006] According to a first aspect of the present invention, a network structure analysis method based on inter-node mutual information is provided, comprising: Step 1: Calculate the mutual information between all pairs of nodes in the graph to be analyzed, construct a mutual information matrix between nodes based on the mutual information between each pair of nodes, and determine the information distance between each pair of nodes based on the mutual information matrix between nodes. Step 2: Based on the information distance, use the spectral clustering method to calculate the community divisions of the graph to be analyzed when it is divided into different numbers of communities, and construct a hierarchical community structure as the set of the community divisions; Step 3: Determine the mutual information density of a node based on the information distance between the node and each other node, and define the node whose density is greater than the density of all its neighboring nodes as a core node. Step 4: Based on the mutual information density of the nodes and the information distance, calculate the dominant neighboring nodes of the nodes in the graph to be analyzed, and construct the core propagation forest of the graph to be analyzed based on the core nodes and the dominant neighboring nodes.

[0007] Based on the above technical solution, the present invention can also be improved as follows.

[0008] Optional, any node and nodes Inter-point information The calculation formula is: in, Indicates random transfer to node The probability, Indicates random transfer to node The probability, Indicates from node Transfer to node The probability of; The inter-node mutual information matrix The Line number The value of the column is: .

[0009] Optionally, the probability , and The calculation process includes: Define an infinite-order cumulative transition probability matrix : in, It is the first-order transition probability matrix between nodes: in, Let be the adjacency matrix of the graph to be analyzed; For degree matrix, The values ​​of the diagonal elements For nodes The degree of the element is 0, and the value of the other elements is 0. For the first-order transition probability matrix Normalization yields the matrix : The matrix Symmetricization yields the transition probability matrix for: Determine the joint probability The value is relative to the transition probability matrix. elements Marginal probability and The transition probability matrix is ​​respectively The row and number The sum of the elements in the column.

[0010] Optionally, any node in step 1 and nodes Information distance between The calculation formula is: in, .

[0011] Optionally, step 2 includes: calculating the community division of the graph to be analyzed into 2, 3...n-1 communities based on the spectral clustering method, respectively: n is the number of nodes in the graph to be analyzed; Constructing the hierarchical community structure .

[0012] Optionally, the graph to be analyzed is divided into: When there are a number of communities, The process of calculating community division using spectral clustering in step 2 includes: Step 201: Using the K-nearest neighbor algorithm, construct a weighted adjacency matrix based on the information distance. : in, Refers to nodes The recent Adjacent nodes, For any node and nodes Information distance between them The variable is a Gaussian distribution; Step 202, based on the adjacency matrix Calculate the regularized Laplacian matrix : in, It is an adjacency matrix degree matrix ; It is the Laplace matrix ; Step 203, for the matrix Perform eigenvalue decomposition and obtain the matrix. the smallest The eigenvectors corresponding to each eigenvalue; Step 204, will One of the aforementioned feature vectors is used as a column to generate a... OK, Column matrix ; Step 205, for the matrix of indivual k-means clustering of 3D row vectors The clustering results obtained are used to determine the community division of the graph to be analyzed. .

[0013] Optionally, any node in step 3 The information density The calculation formula is: in, The number of nodes in the graph to be analyzed. For any node and nodes The information distance between them.

[0014] Optionally, in step 4, the node The dominant neighbor node Defined as a node High density and the densest adjacent nodes .

[0015] Optionally, in step 4, the core propagation forest is constructed. The process includes: Step 401, construct the directed edge set : For the core node set, Let the set of nodes in the graph to be analyzed be... Mapping of dominant adjacent nodes ; Step 402, construct a directed graph The core propagation forest of the graph to be analyzed: .

[0016] Optionally, step 4 further includes: calculating the core propagation information distance of all nodes in the graph to be analyzed. ; node The core message of communication is far from satisfy: .

[0017] The present invention provides a network structure analysis method based on inter-node mutual information, the beneficial effects of which include: Discovering hierarchical community structures based on inter-node mutual information can intuitively present the multi-resolution nesting characteristics of networks, which is beneficial for systematically analyzing the deep modularity and hierarchical nesting characteristics within various real-world complex networks (such as social networks and biological networks). The core propagation forest constructed using the dominant adjacency relationships of nodes facilitates the bottom-up, stable, and clear extraction of the topological skeleton of complex networks, accurately locating core nodes and bridge nodes connecting communities. Attached Figure Description

[0018] Figure 1 A flowchart illustrating an embodiment of a network structure analysis method based on inter-node mutual information provided by the present invention; Figure 2(a) is a schematic diagram of the KarateClub network structure obtained by applying a network structure analysis method based on an embodiment of the present invention to the KarateClub network. Figure 2(b) is a schematic diagram of the mutual information density of each node and its core propagation distance obtained by applying a network structure analysis method based on an embodiment of the present invention to the KarateClub network; Figure 2(c) is a schematic diagram of the core propagation forest obtained by applying a network structure analysis method based on an embodiment of the present invention to the KarateClub network; Figure 3(a) is a schematic diagram of the mutual information density of each node and its core propagation distance obtained by applying a network structure analysis method based on an embodiment of the present invention to a Polblogs network; Figure 3(b) is a schematic diagram of the core propagation forest obtained by applying a network structure analysis method based on an embodiment of the present invention to a Polblogs network; Figure 4(a) is a schematic diagram of the original PMI matrix obtained by applying a network structure analysis method based on an embodiment of the present invention to a Dolphins network, with nodes numbered sequentially. Figure 4(b) is a schematic diagram of the PMI matrix obtained by applying a network structure analysis method based on an embodiment of the present invention to a Dolphins network after clustering and sorting. Figure 5(a) is a schematic diagram of the structure of the Dolphins network obtained by applying a network structure analysis method based on an embodiment of the present invention to the Dolphins network. Figure 5(b) is a schematic diagram of the mutual information density of each node and its core propagation distance obtained by applying a network structure analysis method based on an embodiment of the present invention to a Dolphins network. Figure 5(c) is a schematic diagram of the core propagation forest obtained by applying a network structure analysis method based on an embodiment of the present invention to a Dolphins network. Detailed Implementation

[0019] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0020] Figure 1 A flowchart of a network structure analysis method based on inter-node mutual information provided by the present invention is shown below. Figure 1 As shown, the analytical method includes: Step 1: Calculate the mutual information between all pairs of nodes in the graph to be analyzed. Construct a mutual information matrix between nodes based on the mutual information between each pair of nodes. Determine the information distance between each pair of nodes based on the mutual information matrix between nodes.

[0021] Step 2: Based on information distance, use spectral clustering to calculate the community divisions of the graph to be analyzed when it is divided into different numbers of communities, and construct a hierarchical community structure as a set of community divisions.

[0022] Step 3: Determine the mutual information density of a node based on the information distance between the node and each other node, and define the node whose density is greater than the density of all its neighboring nodes as the core node.

[0023] Step 4: Based on the mutual information density and information distance of the nodes, calculate the dominant neighboring nodes of the nodes in the graph to be analyzed, and construct the core propagation forest of the graph to be analyzed based on the core nodes and the dominant neighboring nodes.

[0024] This invention provides a network structure analysis method based on inter-node mutual information. It analyzes network structure using node mutual information density and core propagation forests, effectively addressing the limitations of resolution constraints and resolution biases in network structure analysis, which makes it difficult to clearly display the nested small-community structure within a large community at a single scale. It can intuitively present the multi-resolution nesting characteristics of networks, facilitating the systematic analysis of the deep modularity and hierarchical nesting characteristics within various real-world complex networks. The core propagation forest, constructed using the dominant adjacency relationships of nodes, helps to steadily and clearly extract the topological skeleton of complex networks from the bottom up, accurately locating core nodes and bridge nodes connecting communities.

[0025] Example 1 Embodiment 1 provided by this invention is an embodiment of a network structure analysis method based on inter-node mutual information provided by this invention, combined with... Figure 1 As can be seen, embodiments of this analytical method include: Step 1: Calculate the mutual information between all pairs of nodes in the graph to be analyzed. Construct a mutual information matrix between nodes based on the mutual information between each pair of nodes. Determine the information distance between each pair of nodes based on the mutual information matrix between nodes.

[0026] Point-wise Mutual Information (PMI) is a component of mutual information, measuring the correlation between specific events in two random variables. For discrete random variables... and A pair of values Their mutual information Defined as: in , and These are the marginal probability and joint probability of a specific event in a discrete random variable, respectively.

[0027] In one possible implementation, the random variable and Defined as random transfer to the network Any node can be defined by selecting one of the nodes. and nodes Inter-point information The calculation formula is: in, Indicates random transfer to node The probability, Indicates random transfer to node The probability, Indicates from node Transfer to node The probability of.

[0028] Constructing the mutual information matrix between nodes (PMI matrix for short). The Line number The value of the column is a node and nodes Point-to-point mutual information, that is: .

[0029] In one possible implementation, probability , and The calculation process includes: Define an infinite-order cumulative transition probability matrix This matrix is ​​calculated using an exponential decay model: in, It is the first-order transition probability matrix between nodes: in, Let be the adjacency matrix of the graph to be analyzed; For degree matrix, The values ​​of the diagonal elements For nodes The degree of the element is 0, and the value of the other elements is 0.

[0030] Considering the matrix at this time Since the sum of all elements is not 1, it is necessary to adjust the first-order transition probability matrix. Normalization yields the matrix : For undirected graphs Since the transition probabilities between any two nodes are equal, it is also necessary to adjust the matrix. Symmetricization yields the transition probability matrix for: Determine the joint probability The value is relative to the transition probability matrix. elements Marginal probability and These are the transition probability matrices. The row and number The sum of the elements in the column.

[0031] Combining point mutual information between nodes The formula can be used to calculate the PMI matrix. The values ​​of each element: in, and Represent matrices respectively The row and number The sum of the columns.

[0032] In one possible embodiment, for The matrix is ​​normalized to transform it into an effective similarity matrix. : Then any node in step 1 and nodes Information distance between The calculation formula is: Given an undirected network graph An information distance matrix can be constructed based on the information distance between each pair of nodes. The matrix of the first Line number The value of the column is a node and nodes Information distance between .

[0033] Step 2: Based on information distance, use spectral clustering to calculate the community divisions of the graph to be analyzed when it is divided into different numbers of communities, and construct a hierarchical community structure as a set of community divisions.

[0034] In one possible embodiment, step 2 includes: calculating the community divisions of the graph to be analyzed into 2, 3...n-1 communities based on the spectral clustering method, respectively: n is the number of nodes in the graph to be analyzed.

[0035] Building a hierarchical community structure ,in These correspond to different levels of community structure.

[0036] In one possible embodiment, the graph to be analyzed is divided into: When there are a number of communities, Step 2, which uses spectral clustering to calculate community partitioning, includes: Step 201: Use the K-Nearest Neighbors (KNN) algorithm to construct a weighted adjacency matrix based on information distance. : in, Refers to nodes The recent Adjacent nodes, For any node and nodes The information distance between nodes is represented by a distance matrix. measure, The variable is a Gaussian distribution. In one embodiment provided by this invention, .

[0037] Step 202, based on the adjacency matrix Calculate the regularized Laplacian matrix : in, It is an adjacency matrix The degree matrix, that is, the values ​​of the diagonal elements are the matrix. The sum of each row, with all other elements being 0, i.e. ; It is the Laplace matrix .

[0038] Step 203, based on the given number of communities For the matrix Perform eigenvalue decomposition and obtain the matrix. the smallest The eigenvectors corresponding to each eigenvalue.

[0039] Step 204, will One of the feature vectors is used as a column to generate a... OK, Column matrix ;matrix The The row corresponds to the first row in the network graph. Each node.

[0040] Step 205, for the matrix of indivual k-means clustering of 3D row vectors The clustering results obtained are used to determine the community divisions of the graph to be analyzed. .

[0041] The pseudocode for this spectral clustering algorithm based on mutual information between nodes is as follows: Situation of being divided into a community And the case of dividing each node into a separate community It can be obtained directly without calculation. For a given undirected network graph... It can output a hierarchical community structure of the network. .

[0042] Step 3: Determine the mutual information density of a node based on the information distance between the node and each other node, and define the node whose density is greater than the density of all its neighboring nodes as the core node.

[0043] In one possible embodiment, any node in step 3... Information density The calculation formula is: Distance matrix The mean of the elements in the middle.

[0044] in, The number of nodes in the graph to be analyzed. For any node and nodes The information distance between them.

[0045] Given an undirected network graph ,picture The core node set is defined as the set of all nodes whose density is less than that of their neighboring nodes, i.e.: Step 4: Based on the mutual information density and information distance of the nodes, calculate the dominant neighboring nodes of the nodes in the graph to be analyzed, and construct the core propagation forest of the graph to be analyzed based on the core nodes and the dominant neighboring nodes.

[0046] In one possible embodiment, in step 4, the node dominant neighbor node Defined as a node High density and the densest adjacent nodes .

[0047] Right now: For any undirected network graph, all nodes except the core node have at least one dominant neighbor node.

[0048] Based on this, a dominant adjacency node mapping can be constructed. This ensures that for each non-core node, there is one and only one dominant neighbor node. Obviously, according to the definition of node mutual information density, it is possible for a node to have two or more neighbor nodes with the highest density. In this case, the following principle is used for filtering: If there are two or more dominant neighboring nodes, select the node with the largest PMI of the current node.

[0049] If its PMI is the same as the current node, then the node with the smallest number is selected.

[0050] In one possible implementation, the core propagation forest is constructed in step 4. The process includes: Step 401, construct the directed edge set : For the core node set, Let the set of nodes in the graph to be analyzed be... Mapping of dominant adjacent nodes .

[0051] Step 402, using node set and directed edge set Construct a directed graph The core propagation forest of the graph to be analyzed: .

[0052] Obviously in the diagram middle: Any core node Its out-degree is strictly 0, and any non-core node The out-degree is strictly 1.

[0053] The graph will not contain loops. have Each core node, Will contain A tree has a core node, which is the root node.

[0054] by Any non-core node in Starting from this point, we continuously traverse the nodes pointed to by its directed edges (dominant adjacent nodes), eventually forming a directed path leading to the core node. We call them nodes. The core propagation path can be formally represented as: The path includes Nodes, Nodes It is the end point of the sequence and a core node.

[0055] In one possible embodiment, step 4 further includes: calculating the core propagation information distance of all nodes in the graph to be analyzed. Given an undirected network graph ,picture Corresponding information distance matrix ,picture Mutual information density of each node and dominant adjacent node mapping ,node The core message of communication is far from Define as a node The sum of information distances between nodes along the core propagation path.

[0056] Specifically, nodes The core message of communication is far from satisfy: .

[0057] To address the limitations of existing network structure analysis methods, this invention proposes a method for analyzing complex network structures based on inter-node mutual information, aiming to: Given an undirected network graph Output the hierarchical community structure of the network. ,in This corresponds to different levels of community division. Output the core node set of this network. The core propagation forest that constructs this network. (or the core propagation tree, depending on the number of core nodes), and output any node. Core propagation path connecting to the core node and its core dissemination information distance .

[0058] Specifically, the following steps can be used to traverse the computation graph. The core propagation distance of all nodes: Step (1), for each node traverse its adjacent nodes If none of its neighboring nodes have a density greater than its own, then the node... Insert core node set Otherwise, according to Definition 8, the node with the highest density among adjacent nodes is mapped to a node. The dominant adjacent node.

[0059] Step (2), create the set to be searched , core node set Insert all nodes Create a searched collection .

[0060] Step (3), for the set to be searched Each node in The core propagation information distance is calculated using the core propagation information distance formula above. and remove the node from Move to In the middle, at the same time, all nodes not in the node Inserting adjacent nodes .

[0061] Step (4), repeat step (3) until The value is empty. At this point, the core propagation information distances of all nodes have been calculated.

[0062] The network structure analysis in this invention mainly focuses on the following two aspects: First, there's the discovery of hierarchical community structures. Complex networks typically exhibit significant modularity, meaning they can be divided into multiple subgraphs with tightly connected internal structures and sparse external connections; these subgraphs are called communities. Furthermore, real-world complex networks often display multi-resolution nested hierarchical relationships, where large communities contain smaller communities. Achieving accurate discovery of hierarchical community structures is crucial for understanding the organizational structure of complex systems and revealing the functional module divisions within networks of different granularities.

[0063] Second, the analysis of core nodes and core propagation paths. The importance of nodes in complex networks varies significantly. From a macroscopic perspective, nodes that play a crucial role in the network structure and function are the core nodes. From a microscopic perspective, ordinary nodes typically connect with other nodes through a dominant, directly adjacent node. Based on this inherent topological characteristic, there objectively exists a core propagation path from the core node to any other node in the network, consisting of a series of dominant adjacent nodes. Accurately identifying core nodes and analyzing the core propagation path of the entire network can effectively reveal the network's backbone architecture, and has significant application value in practical business applications such as information tracing, network vulnerability assessment, and resource link optimization.

[0064] This invention provides a network structure analysis method based on inter-node mutual information (PMI). It uses a PMI matrix (Point Information Matrix) based on information theory to measure distance and density between nodes. Compared to methods using hop count or the number of adjacent nodes to calculate distance or density, this method overcomes the shortcomings of traditional local metrics that are highly sensitive to network random noise and isolated edges. By introducing information-theoretic measures, the PMI matrix fully integrates global network structure information, accurately identifying node pairs that are physically far apart but have strong information connections. This measurement method, based on the essence of information transmission, makes the constructed core propagation path more consistent with the inherent dynamics of real complex systems.

[0065] Based on the mutual information density of nodes and dominant neighboring nodes, this method identifies core nodes in the network and analyzes the core propagation paths of each node. This approach deeply aligns with the objective laws governing information flow within complex systems. By calculating the information distance and density of nodes based on inter-node mutual information, it overcomes the problem of ranking node importance solely based on local topological connections, which fails to reveal the true hierarchical dependencies and propagation links between nodes. Within the information theory framework, the connection between a node and its dominant neighboring nodes essentially represents the optimal (or least resistant) channel for information transmission in the network. By tracing the direction of this local information flow step by step, this invention can accurately identify core nodes in the network and extract the core skeleton of information propagation, i.e., the core propagation forest.

[0066] Example 2 Embodiment 2 provided by the present invention is a specific application embodiment of a network structure analysis method based on inter-node mutual information provided by the present invention.

[0067] Figure 2(a) shows a schematic diagram of the KarateClub network obtained by applying a network structure analysis method provided in an embodiment of the present invention to the KarateClub network. In the embodiment shown in Figure 2(a), the network has two core nodes, represented by a pentagram shape. Figure 2(b) shows a schematic diagram of the mutual information density of each node and its core propagation distance obtained by applying a network structure analysis method provided in an embodiment of the present invention to the KarateClub network. In Figure 2(b), the horizontal axis represents the mutual information density, and the vertical axis represents the core propagation distance. Figure 2(c) shows a schematic diagram of the core propagation forest obtained by applying a network structure analysis method provided in an embodiment of the present invention to the KarateClub network.

[0068] As can be seen from Figures 2(a)-2(c), in the KarateClub social network, the network structure analysis method based on inter-node mutual information provided in this embodiment of the invention accurately locates the two core members (i.e., two core nodes) that caused the club members to split.

[0069] Figure 3(a) shows a schematic diagram of the mutual information density of each node and its core propagation distance obtained by applying a network structure analysis method provided in an embodiment of the present invention to a Polblogs network. In Figure 3(a), the horizontal axis represents the mutual information density, and the vertical axis represents the core propagation distance. Figure 3(b) shows a schematic diagram of the core propagation forest obtained by applying a network structure analysis method provided in an embodiment of the present invention to a Polblogs network. It can be seen that the Polblogs network contains two main communities.

[0070] By applying the network structure analysis method provided by this invention to the Polblogs blog reference network, it can be seen that the embodiments of this invention successfully extracted the core nodes representing the two major camps; at the same time, the core propagation forest graph and the core propagation information distance-mutual information density graph clearly show the layer-by-layer transmission links of information within these two major camps.

[0071] Experimental results show that this method can effectively locate core nodes with global dominance and extract the backbone of network information propagation.

[0072] Figure 4(a) shows a schematic diagram of the original PMI matrix obtained by applying a network structure analysis method provided by an embodiment of the present invention to a Dolphins network, numbered sequentially by nodes. Figure 4(b) shows a schematic diagram of the clustered and sorted PMI matrix obtained by applying a network structure analysis method provided by an embodiment of the present invention to a Dolphins network. In the figure, the darker colored pixels represent nodes with higher PMI values.

[0073] Figures 4(a)-4(b) show the generated PMI matrix of the Lusseau dolphin social network (Dolphins dataset). After reordering the nodes using the clustering results, the PMI heatmap can intuitively show the hierarchical community structure of the network: that is, the entire network is mainly divided into two large communities, and one of the large communities can be further nested into three smaller communities at a finer resolution.

[0074] Figure 5(a) shows a schematic diagram of the Dolphins network obtained by applying a network structure analysis method based on an embodiment of the present invention to the Dolphins network. In the embodiment shown in Figure 5(a), the network has only one core node, represented by a pentagram shape. Figure 5(b) shows a schematic diagram of the mutual information density of each node and its core propagation distance obtained by applying a network structure analysis method based on an embodiment of the present invention to the Dolphins network. In Figure 5(b), the horizontal axis represents the mutual information density, and the vertical axis represents the core propagation distance. Figure 5(c) shows a schematic diagram of the core propagation forest obtained by applying a network structure analysis method based on an embodiment of the present invention to the Dolphins network.

[0075] As can be seen from Figures 5(a)-5(c), the core node (numbered sn100) becomes the key node connecting the four independent secondary dolphin families. This is consistent with the general consensus in the biological community regarding this dolphin population. The core propagation forest constructed by this invention using the dominant adjacency relationship of nodes is beneficial for bottom-up, stable, and clear extraction of the topological skeleton of complex networks, and for accurately locating the core node and the bridge nodes connecting the communities.

[0076] The present invention provides a network structure analysis method based on inter-node mutual information, the beneficial effects of which include: Discovering core nodes in a network and analyzing their dominant information propagation paths is beneficial for in-depth analysis of information flow paths and the evolution of online public opinion in complex networks, as well as for accurately locating key hub nodes that maintain network connectivity. By calculating the mutual information density of nodes, finding core nodes and dominant adjacent nodes can accurately identify key nodes globally or locally, and extract the core propagation path from core nodes to any ordinary node.

[0077] This invention proposes two complementary analytical methods. Firstly, the hierarchical community structure discovery method based on inter-node mutual information can intuitively present the multi-resolution nested characteristics of networks, facilitating systematic analysis of the deep modularity and hierarchical nesting features within various real-world complex networks (such as social networks and biological networks). Secondly, this invention utilizes a core propagation forest constructed from the dominant adjacency relationships of nodes, which helps to steadily and clearly extract the topological skeleton of complex networks from the bottom up, accurately locating core nodes and bridge nodes connecting communities. This effectively reveals the multi-resolution hierarchical nested community structure of complex networks.

[0078] It should be noted that the descriptions of each embodiment in the above embodiments have different focuses. For parts that are not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0079] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0080] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0081] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0082] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0083] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0084] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A network structure analysis method based on inter-node mutual information, characterized in that, The analytical method includes: Step 1: Calculate the mutual information between all pairs of nodes in the graph to be analyzed, construct a mutual information matrix between nodes based on the mutual information between each pair of nodes, and determine the information distance between each pair of nodes based on the mutual information matrix between nodes. Step 2: Based on the information distance, use the spectral clustering method to calculate the community divisions of the graph to be analyzed when it is divided into different numbers of communities, and construct a hierarchical community structure as the set of the community divisions; Step 3: Determine the mutual information density of a node based on the information distance between the node and each other node, and define the node whose density is greater than the density of all its neighboring nodes as a core node. Step 4: Based on the mutual information density of the nodes and the information distance, calculate the dominant neighboring nodes of the nodes in the graph to be analyzed, and construct the core propagation forest of the graph to be analyzed based on the core nodes and the dominant neighboring nodes.

2. The analytical method according to claim 1, characterized in that, any node and nodes Inter-point information The calculation formula is: in, Indicates random transfer to node The probability, Indicates random transfer to node The probability, Indicates from node Transfer to node The probability of; The inter-node mutual information matrix The Line number The value of the column is: 。 3. The analytical method according to claim 2, characterized in that, The probability , and The calculation process includes: Define an infinite-order cumulative transition probability matrix : in, It is the first-order transition probability matrix between nodes: in, Let be the adjacency matrix of the graph to be analyzed; For degree matrix, The values ​​of the diagonal elements For nodes The degree of the element is 0, and the value of the other elements is 0. For the first-order transition probability matrix Normalization yields the matrix : The matrix Symmetricization yields the transition probability matrix for: Determine the joint probability The value is relative to the transition probability matrix. elements Marginal probability and The transition probability matrix is ​​respectively The row and number The sum of the elements in the column.

4. The analytical method according to claim 2, characterized in that, Any node in step 1 and nodes Information distance between The calculation formula is: in, .

5. The analytical method according to claim 1, characterized in that, Step 2 includes: calculating the community division of the graph to be analyzed into 2, 3...n-1 communities based on the spectral clustering method, respectively: n is the number of nodes in the graph to be analyzed; Constructing the hierarchical community structure .

6. The analytical method according to claim 5, characterized in that, The graph to be analyzed is divided into: When there are a number of communities, ; The process of calculating community division using the spectral clustering method in step 2 includes: Step 201: Using the K-nearest neighbor algorithm, construct a weighted adjacency matrix based on the information distance. : in, Refers to nodes The recent Adjacent nodes, For any node and nodes Information distance between them The variable is a Gaussian distribution; Step 202, based on the adjacency matrix Calculate the regularized Laplacian matrix : in, It is an adjacency matrix degree matrix ; It is the Laplace matrix ; Step 203, for the matrix Perform eigenvalue decomposition and obtain the matrix. the smallest The eigenvectors corresponding to each eigenvalue; Step 204, will One of the aforementioned feature vectors is used as a column to generate a... OK, Column matrix ; Step 205, for the matrix of indivual k-means clustering of 3D row vectors The clustering results obtained are used to determine the community division of the graph to be analyzed. .

7. The analytical method according to claim 1, characterized in that, Any node in step 3 The information density The calculation formula is: in, The number of nodes in the graph to be analyzed. For any node and nodes The information distance between them.

8. The analytical method according to claim 1, characterized in that, In step 4, the node The dominant neighbor node Defined as a node High density and the densest adjacent nodes .

9. The analytical method according to claim 1, characterized in that, In step 4, the core propagation forest is constructed. The process includes: Step 401, construct the directed edge set : For the core node set, Let the set of nodes in the graph to be analyzed be... Mapping of dominant adjacent nodes ; Step 402, construct a directed graph The core propagation forest of the graph to be analyzed: .

10. The analytical method according to claim 1, characterized in that, Step 4 further includes: calculating the core propagation information distance of all nodes in the graph to be analyzed. ; node The core message of communication is far from satisfy: 。

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