A method for generating high-order network models based on embedding multiple clique structures

Abstract

The present invention belongs to the field of network information analysis technology, and is specifically a method for generating a high-order network model based on embedding multiple small group structures. The present invention includes: given an initial matrix, obtaining an edge probability matrix through Kronecker inner product iteration, and generating an initial edge graph, obtaining the first-order degree k1 and generalized degree k of each node in the initial graph. m and maximum degree k1‑max; starting from the 2nd-order clique, select the degree-dependent function based on the characteristic attributes of the network, select nodes with smaller generalized degrees, and obtain a labeled edge graph containing labeled nodes; then generate the order m of the clique based on the first-order degree k1, and then obtain an m-order clique structure, and embed the clique structure to obtain the generated graph; gradually increase the order of the embedded clique structure, and repeat the above steps of node labeling and clique structure embedding until a k1‑max-order clique structure is embedded. This method can well simulate real networks with different clustering coefficients and characterize the high-order structural characteristics of real networks.

Description

Technical Field

[0001] The present invention belongs to the technical field of network information analysis, and in particular relates to a method for generating a high-order network model. Background Art

[0002] In recent years, with the advancement of science and in-depth research in big data analysis, the academic community has gradually discovered that with the increasing complexity of natural and social networks, the nonlinear interactions between the system components have made the previous studies that attempted to explain and predict various behavioral patterns from a single pair of interactions no longer reasonable and effective. In many complex systems in the real world, such as interpersonal networks, biological networks, language systems, and technological systems, complex interactions do not just stay between two individuals, but often occur among three or more individuals and evolve into a group interaction. [4] .

[0003] By viewing multiple nodes that work together as a high-order organization or high-order structure, it is found that network models based on high-order clustering connection patterns can exhibit rich high-order organizational structures. For example, in the study of brain functional networks, researchers found that although the neural network of the human brain appears sparse, the internal high-order structure is very dense, and the high-order structure in the cerebral cortex plays an important role in perspective and cognitive function. [5] In the field of biology, researchers have been trying to understand and explain the diversity and stability of biological systems. Recently, it has been found that the stability of high-order interactions allows diverse communities to persist and has a huge impact on closed ecological community models, thereby introducing new species into the community. This explains the persistent stability of large-scale ecological communities and the phenomenon of diversity in the absence of interactions between species. [6] .

[0004] The continuous development of mathematical theory has effectively promoted the research progress of complex networks. The introduction of algebraic topology tools can describe the high-order structure of cross-scale scientific and technological knowledge networks. The introduction of theories such as high-order Laplacian matrices and boundary operators [7] Taking cliques as basic units, a series of vector spaces on binary domains are constructed, and the structure of the vector space is described using high-order Laplace matrices. The connection between two vector spaces is established through boundary operators, and invariants such as characteristic numbers and Betti numbers that characterize the properties of high-order networks are given.

[0005] Employing higher-order systems can help us understand and predict the dynamic behavior of everything from human communication to chemical reactions to ecosystems. A better understanding of higher-order features can help us accurately describe given types of real-world networks. Clustering coefficients and higher-order clustering coefficients are important characteristics of real-world networks. The inventors have discovered that existing network models either fail to accurately characterize higher-order clustering features of networks or are too complex and cumbersome to accurately capture the key characteristics of real-world networks. Summary of the Invention

[0006] In view of the above situation, the purpose of the present invention is to provide a high-order network model generation method based on embedding of multiple small group structures with high accuracy and low computational complexity.

[0007] The method for generating a high-order network model based on embedding multiple clique structures provided by the present invention includes: given an initial matrix, obtaining an edge probability matrix through Kronecker inner product iterations for several times, generating edges according to the connection probabilities between node pairs in the edge probability matrix, thereby obtaining an initial edge graph, and obtaining the first-order degree k1 and generalized degree k of each node in the initial graph. m and the maximum degree k1-max; starting from the 2nd-order clique, select the appropriate degree dependency function f(m) according to the characteristic attributes of the real network, tend to select nodes with smaller generalized degrees, and obtain a labeled edge graph containing labeled nodes; if the first-order degree k1 of the selected node is greater than or equal to the order m of the generated clique, then the selected labeled node randomly selects m nodes from its neighboring nodes for full connection, and obtains an m-order clique structure; the graph obtained after embedding the clique structure is called the generated graph; gradually increase the order of the embedded clique structure, repeat the node labeling and clique structure embedding in the above steps until the k1-max order clique structure is embedded. The specific steps are:

[0008] Step 1: Given an initial matrix, iterate the Kronecker inner product several times to obtain the edge probability matrix, generate edges based on the connection probability between node pairs in the edge probability matrix, obtain the initial edge graph, and obtain the first-order degree k1 and generalized degree k of each node in the initial graph. m and the maximum degree k1-max; here, the first-order degree k1 is the number of edges connected to the node, and the m-order generalized degree k m Indicates the number of m-order cliques in which the node is located, and the maximum degree k1-max is the largest first-order degree of all nodes;

[0009] Step 2: Starting from the 2nd-order clique, select an appropriate degree dependence function f(m) based on the characteristic attributes of the real network, tend to select nodes with smaller generalized degrees, and obtain a labeled edge graph containing labeled nodes;

[0010] Step 3: If the first-order degree k1 of the selected node is greater than or equal to the order m of the generated small group, then the selected marked node randomly selects m nodes from its neighboring nodes for full connection to obtain an m-order small group structure. The graph obtained after embedding the small group structure is called the generated graph;

[0011] Step 4: Gradually increase the order of the embedded clique structure. Repeat the node marking in step 2 and the clique structure embedding in step 3 when generating cliques of each order until the process of embedding the k1-max order clique structure is completed.

[0012] In the present invention, step 1 specifically includes:

[0013] Step 1-1: Use the simulated moment method to determine the parameters of the model's initial generator matrix. The initial generator matrix is determined by the degree distribution and shortest path of the real network.

[0014] Step 1-2: Iterate the Kronecker inner product of the initial generator matrix n times to obtain a matrix of size 2 n ×2 n The edge probability matrix of ;

[0015] Step 1-3: For each element of the edge probability matrix, randomly generate a random number between 0 and 1 and compare it with the element. If the random number is less than the corresponding element, an edge is generated between the two nodes, and the corresponding element of the initial adjacency matrix is set to 1.

[0016] Step 1-4: Traverse all elements of the edge probability matrix to obtain the initial edge graph and initial adjacency matrix;

[0017] Steps 1-5: Count the first-order degree k1 and generalized degree k of each node in the initial graph m And the maximum degree k1-max.

[0018] In the present invention, step 2 specifically includes:

[0019] Step 2-1: According to the relative difference between the second-order clustering coefficient and the third-order clustering coefficient of the real network, select an appropriate degree dependence function f(m): the selected degree dependence function is inversely proportional to the order m, because the probability of generating a small group with a larger order is smaller; at the same time, ensure that the degree dependence function is consistent with the generalized degree k m Inversely proportional, because the more small groups a node has, the smaller the probability of generating;

[0020] Step 2-2: Starting from the 2nd-order clique, calculate the degree dependence function value f(m) of each node, where m represents the clique order and ranges from 2 to k1-max. Calculate the degree dependence function value and compare it with a randomly generated random number between 0 and 1. If the random number is smaller than the degree dependence function value, select the node as a marked node.

[0021] Step 2-3: Repeat the marking process of step 2-2 several times; if m is equal to 2 and k1 is greater than 1, select k1-1 times with the probability of the dependent function value; if m is greater than 2 and k m-1 = 0, select once with probability based on the dependency function value; if m is greater than 2 and k m-1 Greater than 0, select k with probability based on the dependency function value m-1 Second-rate;

[0022] Step 2-4: Traverse all nodes to obtain the labeled edge graph.

[0023] In the present invention, step 3 specifically includes:

[0024] Step 3-1: Determine whether the first degree k1 of the node is greater than or equal to the degree m of the currently generated clique. If so, continue with the following steps. If less than, the node cannot generate a clique.

[0025] Step 3-2: Randomly select m neighbors with equal probability from the neighbors of the marked node and perform full connection to generate an m-order clique;

[0026] Step 3-3: Remove duplicate edges between two nodes, that is, only one edge can be generated between two nodes;

[0027] Step 3-4: Traverse all labeled nodes and embed the clique structure to obtain the generated graph.

[0028] In the present invention, step 4 specifically includes:

[0029] The order of the embedded clique structure is gradually increased, and the node marking and clique structure embedding in the above steps are repeated until the k1-max order clique structure is embedded.

[0030] The innovation of the present invention lies in: the method for generating a high-order network model based on embedding multiple small group structures proposed in the present invention analyzes the existing data network and extracts feature information. It can construct a high-order network based on this feature information and accurately and reliably characterize the high-order structural characteristics of the real world. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 This is a flow chart of the method for generating a high-order network model based on embedding multiple clique structures according to the present invention.

[0032] Figure 2 Schematic diagram of the small clique structure in the high-order network model generation method based on embedding of multiple small clique structures of the present invention.

[0033] Figure 3 Schematic diagram of the wedge structure in the method for generating a high-order network model based on embedding of multiple small group structures according to the present invention.

[0034] Figure 4 Schematic diagram of an initial edge graph according to an embodiment of the present invention.

[0035] Figure 5 Schematic diagram of a marked edge diagram according to an embodiment of the present invention.

[0036] Figure 6 Schematic diagram of an embedded second-order clique edge graph according to an embodiment of the present invention.

[0037] Figure 7 Schematic diagram of an embedded third-order clique edge graph according to an embodiment of the present invention.

[0038] Figure 8 Schematic diagram of a generation graph according to an embodiment of the present invention.

[0039] Figure 9 Generate a system block diagram for the high-order network model of the present invention. DETAILED DESCRIPTION

[0040] In order to make the above-mentioned objectives and innovations of this aspect easier to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0041] Step 1: Given an initial matrix, obtain the edge probability matrix through several iterations of Kronecker inner product. Generate edges based on the connection probabilities between node pairs in the edge probability matrix to obtain the initial edge graph, and obtain the degree of each node in the initial graph and the maximum degree k1-max of the node.

[0042] Step 1-1: Determine the initial generating matrix of the model using the simulated moment method

[0043] Step 1-2: Iterate the Kronecker inner product of the initial matrix n times to obtain a matrix of size 2 n ×2 n The edge probability matrix P of , the calculation process of a Kronecker inner product is as follows:

[0044]

[0045] Step 1-3: For each element p of the edge probability matrix ij , randomly generate a random number between 0 and 1 to compare with it, if the random number is less than the corresponding element p ij , then node i and node j generate an edge, and the corresponding element of the initial adjacency matrix is set to 1;

[0046] Step 1-4: Traverse all elements of the edge probability matrix to obtain the initial edge graph and the initial adjacency matrix A0;

[0047] Steps 1-5: Count the first-order degree k1 and generalized degree k of each node in the initial graph m and maximum degree k1-max; the first-order degree is the number of edges connecting the node; the m-order generalized degree represents the number of m-order cliques in which the node is located; the maximum degree k1-max is the largest first-order degree among all nodes.

[0048] Since the general edge probability matrix and initial edge graph are too large to be presented in this paper, we use an initial edge graph containing seven nodes to illustrate the generation process of the high-order network model. The process of selecting nodes through the degree dependence function f(m) is simplified to manual selection.

[0049] Step 2: Starting from the 2nd-order clique, select an appropriate degree dependence function f(m) based on the characteristic attributes of the real network, tend to select nodes with smaller generalized degrees, and obtain a labeled edge graph containing labeled nodes;

[0050] Step 2-1: According to the relative difference between the second-order clustering coefficient and the third-order clustering coefficient of the real network, select an appropriate degree dependence function f(m): the selected degree dependence function is inversely proportional to the order m, because the probability of generating a small group with a larger order is smaller; at the same time, ensure that the degree dependence function is consistent with the generalized degree k m Inversely proportional, because the more small groups a node has, the smaller the probability of generating one. The commonly used degree dependence function is as follows:

[0051]

[0052] Among them, f u (m) represents the probability of node u being selected when generating an m-order clique, m represents the order of the high-order structure, k i represents the number of i-order cliques in which node u is located, k max is the maximum degree of the network, i.e. k1-max; F is an adjustable parameter between 0 and 1; the degree dependence function f(m) has different generalized degrees k mThe value decreases approximately exponentially; since high-order cliques are generated on the basis of lower-order cliques, the probability of generating high-order cliques will be affected by the probability of low-order cliques; when generating 2nd-order cliques, there are no cliques in the entire network, and the degree dependence function is only related to the 1st-order degree, and decreases as the 1st-order degree increases; when generating high-order networks above 2nd order, the degree dependence function is not only related to the 1st-order degree but also to the m-1th-order degree, so the degree dependence function adds a coefficient term related to the number of m-1th-order cliques; in the coefficient term, m-1 different nodes are randomly selected around node u as a group, and the denominator of the subtracted term represents the number of combinations, which is also the number that node u can generate The number of the largest m-1 order groups formed; the numerator of the subtracted term is the number of m-1 order groups generated in the network; the subtracted term represents the proportion of the m-1 order groups of node u; if the subtracted term is 1, it means that an m-1 order small group has been formed between node u and any of its m-1 neighboring nodes, indicating that this node and its neighboring nodes are fully connected structures and have formed a small group of the highest order. No subsequent group embedding is required, so the coefficient term is 0 and the degree dependence function is 0; if the subtracted term is 0, it means that the node and its surrounding neighbors have not formed an m-1 order small group, the coefficient term is 1, the degree dependence function is restored to the case when m=2, and the selection probability is calculated according to the size of the first-order degree;

[0053] Step 2-2: Start embedding from the 2nd-order clique structure and calculate the degree dependence function value f(m) of each node, where m represents the clique order; compare the calculated degree dependence function value with a randomly generated random number between 0 and 1. If the random number is smaller than the degree dependence function value, the node is selected as a marked node;

[0054] Step 2-3: Repeat the marking process of step 2-2 several times; if m is equal to 2 and k1 is greater than 1, select k1-1 times with the probability of the dependent function value; if m is greater than 2 and k m-1 = 0, select once with probability based on the dependency function value; if m is greater than 2 and k m-1 Greater than 0, select k with probability based on the dependency function value m-1 Second-rate;

[0055] Step 2-4: Traverse all nodes to get the labeled edge graph, for example Figure 4 In the example, let node 2 be marked twice when embedded in a 2nd-order clique, once when embedded in a 3rd-order clique, and node 3 be marked once when embedded in a 3rd-order clique; marked nodes are represented by red nodes, and unmarked nodes are represented by blue nodes, so we get Figure 5 ; The marking process does not affect the connections between nodes in the initial graph.

[0056] Step 3: If the degree of the selected node is greater than or equal to the number of generated small clusters, the selected marked node randomly selects m nodes from its neighboring nodes for full connection to obtain an m-order small cluster structure. The graph obtained after embedding the small cluster structure is called the generated graph;

[0057] Step 3-1: Whether embedding a 2nd-order clique structure or a 3rd-order clique structure, if the number of neighbors of node 2 meets the requirements for generating a clique, you can proceed to the next step; the same applies to node 4.

[0058] Step 3-2: When embedding the 2nd-order clique structure, randomly select 2 neighbors with equal probability from the neighbors of node 2 for full connection, and repeat the selection twice. The first time, randomly select nodes 1 and 3 from the 3 neighbor nodes and node 2 to generate a 2nd-order clique. The second time, randomly select nodes 3 and 4 from the 3 neighbor nodes and node 2 to generate a 2nd-order clique. Figure 6 It represents the generated graph after the initial graph is embedded in the 2nd-order clique. When embedding the 3rd-order clique structure, since node 2 has only 3 neighbor nodes, the neighbor nodes are fully connected, and nodes 5, 6, and 7 are selected from the neighbors of node 4 to generate a 3rd-order clique. Figure 7 It represents the generated graph after the initial graph is embedded in the 3rd-order clique;

[0059] Step 3-3: When generating the 2nd-order clique and the 3rd-order clique, node 2 is repeatedly selected, forming clique structures of different orders on node 2. The 2nd-order clique structure is covered by the 3rd-order clique model. Since there are no multiple edges between nodes, the 2nd-order cliques generated by nodes 1, 2, 3 and nodes 2, 3, 4 are all covered by the clique generated by nodes 1, 2, 3, 4.

[0060] Step 3-4: Traverse all labeled nodes and embed the clique structure to obtain the generated graph.

[0061] Step 4: Gradually increase the order of the embedded clique structure, repeat the node marking and clique structure embedding in the above steps until the k1-max order clique structure is embedded. The final result is as follows: Figure 8 shown.

[0062] The public dataset was used as the experimental object, and the algorithm of the present invention was used to fit the real network. The present invention uses real networks of different sources, categories, and scales to evaluate the fitting. The C.elegans dataset is derived from the complete nervous system of Caenorhabditis elegans. [8] The network formed by the nervous system of Caenorhabditis elegans is an undirected graph. Each neuron is regarded as a node. If there is interaction between neurons, there is an edge between the corresponding two nodes in the network. The email dataset comes from the email communication list of members of Rovira-Virgili University.[9] , taking each email address as a node, if there is email communication between two nodes, there is an edge between the nodes, and an email network can be established; the Japanese dataset represents the word connection network in Japanese text

[10] , each node represents a word in the text. In the same sentence of the text, if a word appears after another word, it is considered that there is an edge between the two words. The basic properties of the real network are shown in Table 1:

[0063] Table 1

[0064] Dataset Number of nodes Number of connected edges Edge density Second-order clustering coefficient Third-order clustering coefficient C.elegans 297 2148 0.0046 0.3079 0.1366 Email 1133 5451 0.0085 0.2540 0.1390 Japanese 2704 7798 0.0022 0.3033 0.1301

[0065] The second-order clustering coefficient and the third-order clustering coefficient are used to measure the quality of the fitting results. The evaluation standard used is accuracy, and the calculation formula is as follows:

[0066]

[0067] Among them, RC2, SC2, RC3, and SC3 represent the second-order clustering coefficient of the real network, the second-order clustering coefficient of the fitted network, the third-order clustering coefficient of the real network, and the third-order clustering coefficient of the fitted network, respectively. The experiment was repeated 20 times for each data set, and the average value was taken as the final fitting result. The final results are shown in Table 2:

[0068] Table 2

[0069] Dataset RC2 SC2 RC3 SC3 ACCURACY C.elegans 0.3079 0.3026 0.1366 0.1367 0.9910 Email 0.2540 0.2575 0.1390 0.1386 0.9916 Japanese 0.3033 0.3003 0.1301 0.1450 0.9377

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Claims

1. A method for generating a high-order network model based on embedding multiple clique structures, wherein the high-order network includes a network formed by the C. elegans nervous system, a network formed by emails, and a network formed by word connections in Japanese text; the network formed by the C. elegans nervous system is an undirected graph, in which each neuron is regarded as a node, and an edge exists between two nodes, indicating that there is interaction between the two neurons; the network formed by emails, in which each email address is regarded as a node, and an edge exists between two nodes, indicating that there is email communication between the two nodes; the network formed by word connections in Japanese text, in which each node represents a word in the text; an edge exists between two words, indicating that one word appears after another word in the same sentence in the text; characterized in that The specific steps are as follows: Step 1: Given an initial matrix, iterate the Kronecker inner product several times to obtain the edge probability matrix, generate edges based on the connection probability between node pairs in the edge probability matrix, obtain the initial edge graph, and obtain the first-order degree k1 and generalized degree k of each node in the initial graph. m and the maximum degree k1-max; here, the first-order degree k1 is the number of edges connected to the node, and the m-order generalized degree k m Indicates the number of m-order cliques in which the node is located. The maximum degree k1-max is the largest first-order degree among all nodes. Specifically: Step 1-1: Use the simulated moment method to determine the parameters of the model's initial generator matrix. The initial generator matrix is determined by the degree distribution and shortest path of the real network. Step 1-2: Iterate the Kronecker inner product of the initial generator matrix n times to obtain a matrix of size 2 n ×2 n The edge probability matrix of ; Step 1-3: For each element of the edge probability matrix, randomly generate a random number between 0 and 1 and compare it with the element. If the random number is less than the corresponding element, an edge is generated between the two nodes, and the corresponding element of the initial adjacency matrix is set to 1. Step 1-4: Traverse all elements of the edge probability matrix to obtain the initial edge graph and initial adjacency matrix; Steps 1-5: Count the first-order degree k1 and generalized degree k of each node in the initial graph m and maximum degree k1-max; Step 2: Starting from the 2nd-order clique, select an appropriate degree dependence function f(m) based on the characteristic attributes of the real network, tend to select nodes with smaller generalized degrees, and obtain a labeled edge graph containing labeled nodes; specifically: Step 2-1: According to the relative difference between the second-order clustering coefficient and the third-order clustering coefficient of the real network, select an appropriate degree dependence function f(m): the selected degree dependence function is inversely proportional to the order m, because the probability of generating a small group with a larger order is smaller; at the same time, ensure that the degree dependence function is consistent with the generalized degree k m Inversely proportional, because the more small groups a node has, the smaller the probability of generating; Step 2-2: Starting from the 2nd-order clique, calculate the degree dependence function value f(m) of each node, where m represents the clique order and ranges from 2 to k1-max. Calculate the degree dependence function value and compare it with a randomly generated random number between 0 and 1. If the random number is smaller than the degree dependence function value, select the node as a marked node. Step 2-3: Repeat the marking process of step 2-2 several times; if m is equal to 2 and k1 is greater than 1, select k1-1 times with the probability of the dependent function value; if m is greater than 2 and k m-1 = 0, select once with probability based on the dependency function value; if m is greater than 2 and k m-1 Greater than 0, select k with probability based on the dependency function value m-1 Second-rate; Step 2-4: Traverse all nodes to obtain the labeled edge graph; Step 3: If the first-order degree k1 of the selected node is greater than or equal to the order m of the generated clique, then randomly select m nodes from its neighboring nodes and fully connect them to obtain an m-order clique structure. The graph obtained after embedding the clique structure is called the generated graph; specifically: Step 3-1: Determine whether the first degree k1 of the node is greater than or equal to the degree m of the currently generated clique. If so, continue with the following steps. If less than, the node cannot generate a clique. Step 3-2: Randomly select m neighbors with equal probability from the neighbors of the marked node and perform full connection to generate an m-order clique; Step 3-3: Remove duplicate edges between two nodes, that is, only one edge can be generated between two nodes; Step 3-4: Traverse all marked nodes and embed the clique structure to obtain the generated graph; Step 4: Gradually increase the order of the embedded clique structure. Repeat the node marking in step 2 and the clique structure embedding in step 3 when generating cliques of each order until the process of embedding the k1-max order clique structure is completed.

2. The high-order network model generation method according to claim 1, characterized in that The degree-dependent function f(m) is: Among them, f u (m) represents the probability of node u being selected when generating an m-order clique, m represents the order of the high-order structure, k i represents the number of i-order cliques in which node u is located, k max is the maximum degree of the network, i.e. k1-max; F is an adjustable parameter between 0 and 1.

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