A hierarchical layout method for large-scale networks

Abstract

This paper discloses a hierarchical layout method for large-scale networks. This method uses the Poincare disk model in hyperbolic space to visualize the hierarchical information and clustering structure of the network. This method is implemented in three steps: enhancing the network's hierarchical properties through preprocessing; calculating radial coordinates based on node centrality; and iteratively calculating angular coordinates based on edge relationships. Compared to previous hierarchical structure visualization methods, this method is more versatile and better demonstrates the hierarchical information and clustering properties of the network.

Description

Technical Field

[0001] The present invention belongs to the field of information visualization network layout / embedding technology, which mainly involves the reasonable layout of large-scale networks so that they can reflect the hierarchical information contained in the global structure of the network while maintaining the local structure of the network. Background Art

[0002] Graph (network) layout algorithm is a type of visualization method for graph (network) structure. It usually displays abstract network data concretely according to some specific models and algorithms. It is widely used in social networks, knowledge graphs and other fields.

[0003] There are many graph layout algorithms available today, the most widely used being force-directed layout algorithms, such as the Fruchterman-Reingold method and the Yifan Hu layout algorithm. Force-directed layout algorithms utilize natural rule models (spring or charge forces) to layout nodes, presenting the overall structure of the graph in an easily understandable manner. However, for large-scale networks, force-directed layouts cannot effectively represent the hierarchical structure of the network.

[0004] There are many algorithms for hierarchical layout, such as the hyperbolic tree algorithm and the circular tree algorithm. CN103049580B proposes a method for visualizing large-scale hierarchical data using radial rings. However, these methods are only applicable to hierarchical data. In practice, most network structures are hierarchical, not tree-like, and therefore cannot be directly applied to these methods.

[0005] Recent research has shown that complex networks possess the geometric structure of hyperbolic space. Therefore, networks can be embedded in hyperbolic space. However, these methods require computing gradients in hyperbolic space, which results in high time complexity. Furthermore, these algorithms focus on solving downstream tasks and are not well suited for visualization layouts. Summary of the Invention

[0006] This paper presents a layout method for large-scale network data that highlights the global hierarchical structure while preserving local network information, thereby enabling visualization of large-scale networks. This method embeds the network into a Poincare disk in hyperbolic space. This method is implemented in three steps: network preprocessing, calculation of node radial coordinates, and calculation of node angular coordinates. Ultimately, the radial coordinates on the disk characterize the node hierarchy within the network, while the angular coordinates reflect the node's local edge relationships.

[0007] The present invention is applicable to a large network G=(V, E, W), where V represents the node set of the network G, E represents the edge set of the network, and W represents the weight matrix of the edges.

[0008] The Poincare disk is a form of two-dimensional hyperbolic space with the following properties: (1) The entire space lies on a unit disk with a radius of 1. When the disk radius varies linearly from 0 to 1, the corresponding hyperbolic distance and area grow exponentially. (2) Conformal property: when any two straight lines in Euclidean space are mapped onto the Poincare disk, the angle between them remains unchanged. Due to property (1), networks of any size can be embedded on the unit disk, and nodes closer to the center of the network are closer to the center of the disk. Due to property (2), the local structure of the network can be continuously mapped onto the disk. Therefore, the Poincare disk is very suitable for visualizing large-scale networks.

[0009] The result of the present invention is to embed the network G into the Poincare disk H, that is, each network node V i , the corresponding polar coordinates on the disk will be calculated (ρ i ,θ i ).

[0010] The technical solutions of the present invention are as follows:

[0011] A hierarchical layout method for large-scale networks involves embedding the network into a Poincare disk in hyperbolic space in three steps: network preprocessing, computing the radial coordinates of nodes, and computing the angular coordinates of nodes. Ultimately, the radial coordinates on the disk characterize the node hierarchy within the network, while the angular coordinates represent the local edge relationships of the nodes.

[0012] 1. Network preprocessing

[0013] Network preprocessing is to remove redundant or minor connections in the network, highlight the hierarchical structure, and improve the computational efficiency of subsequent steps.

[0014] For large-scale networks, network density is a key factor influencing embedding effectiveness. Greater network density indicates more clustering and less pronounced hierarchical structure. Networks with pronounced hierarchical structures often have a small number of central nodes (high-level nodes) and a large number of leaf nodes (low-level nodes), demonstrating scale-free properties. In fact, most real-world networks exhibit some degree of scale-free properties. For a few dense networks, one can examine the edges within the network, retaining those that are important to both connected nodes and ignoring those with small weights.

[0015] The specific processing steps are:

[0016] (1) The undirected network G u Convert it to a directed weighted network G = (V, E, W). If it is a directed network, no further processing is required.

[0017] (2) For each node V in the network i , get its first-order neighbor N i, and sort them in descending order of weight;

[0018] (3) For node j∈N i and threshold∈, if the condition is met Then from network G

[0019] Remove the edge e ij .

[0020] 2. Calculate radial coordinates using node centrality

[0021] The radial coordinates of a node on the disk reflect its hierarchy within the network, which can be characterized by the node's centrality metric. In a Poincare disk, while the distance from the center of the disk increases linearly, the actual hyperbolic distance increases exponentially. The degree distribution of large-scale networks also typically exhibits a power-exponential distribution, so there is a good correlation between node degree and hierarchy. Let the node with the highest degree be at the center of the disk, and the nodes with the lowest degree be at the periphery. Based on the exponential distribution, the radial coordinates of other nodes can be derived.

[0022] If radial coordinates are calculated directly based on degree, the vast majority of nodes will be located around the disk, with only a few nodes in the center. This results in a highly unbalanced information density from a visualization perspective. To ensure a balanced layout, we introduce kcore centrality to calculate radial coordinates and weight them with the radial coordinates calculated using degree centrality to obtain the final radial coordinates.

[0023] Using degree centrality highlights the hub nodes in the network. Since the distribution of kcore is flatter than the degree distribution, using kcore centrality emphasizes the secondary hub nodes in the network. At the same time, most ordinary nodes are located around the disk. Therefore, the weighted degree and kcore well characterize the network hierarchy while taking into account the layout balance.

[0024] The specific processing steps are:

[0025] Here are the steps:

[0026] (1) Calculate the degree and kcore centrality of each node in the network G, represented by D and K

[0027] (2) The node diameter coordinate with the minimum degree in G is defined as ρ max , the node with the largest degree has a radius coordinate of 0, and for other nodes V i , whose radial coordinates are determined by degree centrality Solve it by the following formula:

[0028]

[0029] (3) The node diameter coordinate with the minimum kcore in G is defined as ρ max, the node with the largest kcore diameter coordinate is 0, and for other nodes V i , whose radial coordinates are determined by degree centrality Solve it by the following formula:

[0030]

[0031] (4) Let 0<η<1, node V i The radial coordinates are calculated by the following formula:

[0032]

[0033] 3. Use network edge relationships to determine angular coordinates

[0034] The angular coordinates of nodes on the disk are primarily determined by the local structure of the network: nodes with edges or small network distances are close together on the disk, while nodes without edges or large network distances are far apart. Therefore, by leveraging edge relationships, we can replace the calculation of coordinates in hyperbolic space with the calculation of angular coordinates in Euclidean space. For each node, we sample its neighboring nodes (positive samples) and non-neighboring nodes (negative samples), and adjust the angular coordinates of the positive and negative samples so that the positive samples are close to the current node and the negative samples are far away. After multiple rounds of iteration, when the angular coordinates stabilize, we can consider the corresponding angular coordinates to have been obtained.

[0035] The specific processing steps are:

[0036] (1) For each node V i Assign angular coordinate θ i ;

[0037] (2) Calculate the node distance matrix Dist, V of each node i Positive sample is pos i ={j|Dist ij ≤2}, negative samples are neg i ={j|Dist ij >2};

[0038] (3) Let the learning rate be lr, and for sample j, let it be equal to the current node V i The angle difference is Δθ ij , if j is a positive sample, update the angular coordinates of j to If j is a negative sample, update the angular coordinate of j to θ j =θ j +lr*Δθ ij , and finally θ j Adjust to the interval (0,2π);

[0039] (4) Let θ i ′ is the angular coordinate value calculated in the previous round, ∈θ is the angle threshold. i |θ i -θ i ′|≤∈ θ

[0040] When , the iteration ends.

[0041] Beneficial effects

[0042] The present invention provides a hierarchical layout method for large-scale networks, which has the following advantages:

[0043] First, a step-by-step method of determining radial coordinates and angular coordinates is adopted, so that radial coordinates and angular coordinates have clear meanings: radial coordinates reflect the hierarchy of nodes in the network, and angular coordinates reflect the local relationship of the network.

[0044] Second, using degree centrality and kcore centrality to jointly determine the radial coordinates of the nodes not only reflects the hierarchical structure of the network but also balances the layout effect.

[0045] 3. Using edge relationships and weights to iteratively calculate the angular coordinates of nodes can reflect the clustering structure of the network and has high computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is a flow chart of the method of the present invention

[0047] Figure 2 This is the network preprocessing flowchart

[0048] Figure 3 This is the flow chart for calculating node radius coordinates

[0049] Figure 4 This is the flow chart for calculating node angular coordinates

[0050] Figure 5 This is an example diagram of a concept network.

[0051] Figure 6 is the result of concept network preprocessing

[0052] Figure 7 is the result of the concept network layout DETAILED DESCRIPTION

[0053] To further illustrate the technical solution of the present invention, it will now be described in detail with reference to examples.

[0054] This section uses a concept network to illustrate the implementation method of the present invention. Concept networks often have a hierarchical structure, but not a strict tree structure. We selected a small-scale concept network ( Figure 5) is used to illustrate the implementation method of the present invention. In actual use, the advantages and effects of the present invention can be better reflected for large-scale networks.

[0055] 1. Network preprocessing

[0056] (1) To convert an undirected network into a directed network, just Figure 5 The edges in the figure can be replaced with bidirectional edges. Take the threshold as 0.1, traverse each node and its edges, and the edges that need to be deleted are as follows. The network after preprocessing is shown in Figure 6 .

[0057]

[0058]

[0059] 2. Calculate the radial coordinates

[0060] (1) Calculate the degree of each node in the network as follows:

[0061] D = {Complex Systems: 5, Machine Learning: 4, Quantum Computing: 3, Neural Networks: 3, Statistical Physics: 3, Multi-Agent Simulation: 4, Network Science: 4, Scale-Free Networks: 3, Random Networks: 1, Epidemic Transmission: 3, Cellular Automata: 1, Ising Model: 1, Optimization Algorithm: 4, Gradient Descent; 1, Genetic Algorithm: 2, Reproducible Number: 1, Preference Attachment: 1, Power-Law Distribution: 2}.

[0062] The kcore of each node in the network is calculated as follows:

[0063] K = {'Cellular Automata': 1, 'Reproducible Number': 1, 'Ising Model': 1, 'Preferred Attachment': 1, 'Gradient Descent': 1, 'Random Networks': 1, 'Statistical Physics': 2, 'Power-Law Distribution': 2, 'Optimization Algorithms': 2, 'Quantum Computing': 2, 'Scale-Free Networks': 2, 'Complex Systems': 2, 'Machine Learning': 2, 'Epidemic Spread': 2, 'Multi-Agent Simulation': 2, 'Network Science': 2, 'Genetic Algorithms': 2, 'Neural Networks': 2}.

[0064] Take η=0.75,ρ max =0.99 Taking multi-agent simulation as an example:

[0065] Depend on The solution is, The same goes for

[0066] In summary, ρ = 0.484*0.75+0.25*0 = 0.339.

[0067] The radial coordinates of other nodes can be calculated in the same way:

[0068] {'Complex Systems': 0.0, 'Multi-Agent Simulation': 0.339, 'Network Science': 0.339, 'Machine Learning': 0.339, 'Optimization Algorithms': 0.339, 'Quantum Computing': 0.55, 'Statistical Physics': 0.55, 'Neural Networks': 0.55, 'Epidemic Spread': 0.55, 'Scale-Free Networks': 0.55, 'Power-Law Distribution': 0.644, 'Genetic Algorithms': 0.644, 'Ising Model': 0.94, 'Cellular Automata': 0.94, 'Random Networks': 0.94, 'Preferential Attachment': 0.94, 'Reproducible Number': 0.94, 'Gradient Descent': 0.94}.

[0069] 3. Calculate angular coordinates

[0070] (1) Assign initial values ​​of angular coordinates to each node;

[0071] (2) Calculate the distance matrix and positive and negative samples of the nodes;

[0072] (3) Adjust the angular coordinates of the node and set the learning rate to lr = 0.1.

[0073] Taking node network science as an example, its positive samples are [random networks, scale-free networks, complex systems, epidemic spread, power-law distribution, statistical physics, quantum computing, reproductive numbers, preferential attachment, multi-agent simulation].

[0074] Negative samples are [cellular automata, Ising model, machine learning, genetic algorithm, neural network, optimization algorithm, gradient descent].

[0075] Assume the initial value is:

[0076] θ 网络科学 =π / 6,θ 随机网络 =π,θ 伊辛模型 =π / 2.

[0077] In the first iteration, the random network is a positive sample of network science, and its angular coordinates are adjusted to: θ 随机网络 =π-0.1*3 / 3*(π-π / 6)=11 / 12π,

[0078] After adjustment, the random network is 1 / 12π closer to network science.

[0079] The Ising model is a negative sample in network science, and its angular coordinates are adjusted to

[0080] θ 伊辛模型 =π / 2+0.1*(π / 2-π / 6)=8 / 15,

[0081] After adjustment, the random network moves away from network science by 1 / 30π.

[0082] Take ∈ θ =0.1After 30 rounds of iterative adjustments, ∑ i |θ i -θ i ′|≤∈ θ , the iteration ends.

[0083] The final angular coordinates are Figure 7 .

[0084] observe Figure 7 It can be seen that the angles of the nodes connected by edges are close to each other, forming several ribbon clusters, such as {network science, scale-free network, power-law distribution, random network, preferential attachment}, {optimization algorithm, neural network, gradient descent}, etc., which are clustering results, and also show a good hierarchical relationship.

Claims

1. A hierarchical layout method for network information, characterized in that: This layout method uses the Poincare disk model in hyperbolic space to display hierarchical information and operates in three modules: Module 1 optimizes the network structure through preprocessing to highlight hierarchical information. The specific steps are as follows: 1-1) The undirected network G u Convert to a directed weighted network G = (V, E, W); 1-2) For each node V in the network i , get its neighbors N i , and sort by weight in descending order List; 1-3) For node j∈N i and threshold∈, if the condition is met Then remove edge e from the network G ij ; Module 2: Calculate the radial coordinates of nodes on the disk according to the centrality of network nodes; Module three: Calculate the angular coordinates of the nodes on the disk according to the network's edge relationships; Arrange the nodes on the disk according to the calculated radial and angular coordinates to complete the layout.

2. The hierarchical layout method of network information according to claim 1, characterized in that: The specific methods and steps of the module 2 are as follows: 2-1) Calculate the radial coordinates of the node according to degree centrality; 2-2) Calculate the radial coordinates of the nodes according to kcore centrality; 2-3) Calculate the radial coordinates of the nodes by weighting the degree centrality and kcore centrality; The specific steps are as follows: Step 1: Calculate the degree and kcore centrality of each node in the network G, denoted by D and K; Step 2: Define the node diameter coordinate with the minimum degree in G as ρ max , the node with the largest degree has a radius coordinate of 0, and for other nodes V i , whose radial coordinate ρ is determined by degree centrality di , solved by the following formula: Step 3: Define the node diameter coordinate with the smallest kcore in G as ρ max , the node with the largest kcore diameter coordinate is 0, and for other nodes V i , whose radial coordinates are determined by degree centrality Solve it by the following formula: Step 4: Let 0<η<1, node V i The radial coordinates are calculated by the following formula:

3. The hierarchical layout method of network information according to claim 1, characterized in that: The module three specifically includes the following methods and steps: 3-1) Initialization of angular coordinates; 3-2) Obtain positive and negative neighbor samples: For each node V i , take V i The first and second order neighbor nodes of are regarded as positive samples, and the other nodes are regarded as negative samples; 3-3) Adjust the angular coordinates of positive and negative samples according to their relative positions; 3-4) When the sum of the differences between two consecutive rounds of angular coordinates is less than the threshold, the iteration ends; Here are the steps: Step 1: For each node V i Assign angular coordinate θ i ; Step 2: Calculate the node distance matrix Dist, V of each node i Positive sample is pos i ={j|Dist ij ≤2}, negative samples are neg i ={j|Dist ij >2}; Step 3: Let the learning rate be lr, and for sample j, let it be equal to the current node V i The angle difference is Δθ ij , if j is a positive sample, update the angular coordinates of j to If j is a negative sample, update the angular coordinate of j to θ j =θ j +lr*Δθ ij , and finally θ j Adjust to the interval (0,2π); Step 4: Let θ′ i is the angular coordinate value calculated in the previous round, ∈ θ is the angle threshold, when ∑ i |θ i -θ′ i |≤∈ θ When , the iteration ends.

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