Dynamic projection visual layout method based on nonlinear programming and multi-objective optimization

Abstract

A dynamic projection visual layout method based on nonlinear programming and multi-objective optimization optimizes a visual presentation effect of a projection result from two aspects of reducing visual occlusion of the projection result and enhancing sample stability in a dynamic iteration process. In this way, a user can be effectively helped to track a dynamically changing data mode, and continuous data exploration and analysis decision are supported. Besides, the method also adopts a mathematical optimization method of nonlinear programming and multi-objective optimization, aims of reducing visual overlapping or enhancing a data visual mode and the like are fulfilled, and the visual definition and the user interaction experience are improved.

Description

Technical Field

[0001] The present invention belongs to the field of projection layout methods, and relates to a dynamic projection visual layout method based on non - linear programming and multi - objective optimization. Background Art

[0002] In the era of the rapid development of artificial intelligence, there is a need to process and analyze a large amount of high - dimensional data. To improve the performance and adaptability of the model, model data usually needs to be processed in multiple iterative cycles, and each cycle may update data samples or introduce new samples, thus increasing the complexity of data management.

[0003] To better understand and display this high - dimensional data, dimensionality reduction techniques and clustering techniques have become important tools. By projecting high - dimensional data onto a two - dimensional plane and performing clustering, effective visual analysis can be carried out to help data analysts better observe the relationships and patterns between data. Projection usually involves using techniques such as principal component analysis (PCA) or t - Distributed Stochastic Neighbor Embedding (t - SNE), which, while reducing the data dimension, also retain as much important information in the data as possible. Commonly used clustering algorithms include K - means, DBSCAN, and hierarchical clustering, etc. Each algorithm has its unique advantages and applicability according to different application scenarios and data characteristics. By analyzing the similarity of data points in high - dimensional space, clustering can group data into multiple clusters with similar characteristics. This not only helps to simplify the data structure but also highlights the natural segmentation in the data, providing an intuitive group division for subsequent analysis.

[0004] Projection layout methods can optimize the layout of data points on the projection plane to achieve goals such as reducing visual overlap or enhancing data visual patterns. For example, by adjusting the relative positions of data points or changing their aggregation degree, the overall visual presentation can be improved. This optimization is particularly important for making the patterns and trends hidden in complex data structures more obvious. Appropriate layout adjustment can enhance the visual effect, making it more direct and obvious to identify groups, anomalies, or any important data structure relationships.

[0005] When performing visual representation of high - dimensional data sets based on dimensionality reduction and clustering techniques, the distribution information and pattern information of the data can be effectively displayed. However, the increase in the number of nodes leads to a large number of overlaps between data points, causing visual burden and confusion. In addition, during the data dynamic iteration process, data samples are updated or new samples are introduced, resulting in dramatic changes in the projection results, which poses challenges for users to adapt to and track the samples retained across iterations. Summary of the Invention

[0006] To overcome the deficiencies of the prior art, the present invention proposes a dynamic projection visual layout method based on non-linear programming and multi-objective optimization. From the two perspectives of reducing visual occlusion in the projection results and enhancing the stability of samples during the dynamic iteration process, the visual presentation effect of the projection results is optimized. This method helps users track dynamically changing data patterns and supports continuous data exploration and analysis decisions.

[0007] To solve the above technical problems, the present invention provides the following technical solutions:

[0008] A dynamic projection visual layout method based on non-linear programming and multi-objective optimization, the method comprising the following steps:

[0009] 1) Data description; in each iteration process, the set S t is used to represent the sample set of the current iteration, and this set consists of two subsets: and where, contains the samples retained from the previous iteration, while includes the newly added samples, where u represents a new sample and i represents an old sample. That is

[0010] 2) Maintain the layout stability of samples; when laying out the samples in S t , first maintain the distribution of the samples in , and arrange the newly added samples according to the positions of the samples;

[0011] 3) Eliminate occlusion between samples; after determining the coordinates of the points in S t , cluster the samples through DBSCAN clustering technology to form clearly recognizable clustering nodes, and these nodes represent sets of similar samples. The samples within the nodes are arranged according to the attribute values generated by the model, and the central coordinates of each node are determined according to the centroid of the samples within the node; to make the best use of the screen space, the shape of the nodes is designed to be close to a square. By introducing a dynamic adjustment mechanism, the system is allowed to dynamically adjust the node positions according to the results of real-time analysis to minimize the occlusion between clustering nodes. The node position fine-tuning strategy includes considering the local density and relative position of the nodes, calculating the spatial density around each node through an algorithm, and thus determining the optimal amplitude and direction of node movement.

[0012] The process of the above 1) is:

[0013] (1-1) Coordinates of points; represent the coordinates of the points in S t in the layout algorithm, and represent the coordinates of the points in S tThe coordinates of sample i are represented by where and are both continuous variables;

[0014] (1 - 2) Dynamic projection visual layout; When at the midpoint position of layout S t , first ensure the stability of the search space layout to maintain the visual stability of the user during the iteration process; By maintaining the consistency of the sample positions between iterations, ensure that the user can easily track the changes in the data, thereby improving the user experience and the efficiency of data analysis; Secondly, through an effective layout strategy, eliminate the occlusion between samples to better utilize the screen space, making the relationship between data points clearer and facilitating observation and analysis. The layout algorithm includes two steps: The first step is to maintain the layout stability of the samples in S t . By calculating the relative position relationships between samples and maintaining the stability of these relationships, avoid large position changes due to iteration; The second step is to fine-tune the positions of the samples in S with the goal of minimizing the occlusion between samples. t

[0015] In (2) above, make the distribution of the samples that remain unchanged during the iteration process consistent to help the user with visual tracking. For this purpose, use the following model as the layout requirement:

[0016]

[0017]

[0018] minα 1 f 1 +α 2 f 2 +α 3 f 3 (3)

[0019] In this model, the meaning of f 1 in formula (1) is to retain the sum of the position offset distances of the samples in the t-th iteration and the (t - 1)-th iteration. The function of this formula is to reduce the coordinate displacement of the maintenance samples in two iterations to stabilize the coordinate distribution of these retained samples; In f 2 of formula (2), the feature distance between sample u and i is represented as d ui , and the coordinate distance between the old image and the new image is represented as The meaning of this formula is the sum of the differences in feature distance and coordinate distance between each retained sample and the newly added samples. Its role is to minimize the differences between the feature distance and coordinate distance between the retained samples and the newly added samples, so as to arrange the newly added samples while maintaining the feature relationships of the existing samples; f in formula (3) 3 represents the Kullback-Leibler divergence in t-SNE, and its role is to reduce the information loss when data is transformed from high-dimensional vectors to low-dimensional vectors;

[0020] Finally, f 1 、f 2 and f 3 constitute the objective function of the model, and this model is an unconstrained multi-objective optimization model. Among them, α 1 , α 2 , α 3 are positive weights that balance the three objective functions. After multiple tests, 1, 1, 0.1 are selected as their values. Because the f 3 objective function has a larger range than the f 1 and f 2 two objective functions; the momentum method is used to solve this model, and it is solved through the Gurobi solver. The momentum method helps to accelerate the convergence of the model, and at the same time the Gurobi solver provides powerful computing power to ensure the efficiency and accuracy of the model when dealing with large-scale data sets.

[0021] In the above-mentioned 3), the following model is adopted as the layout requirement:

[0022] (3-1) Optimize the objective function; the objective function of this model aims to eliminate the occlusion between nodes by moving the node positions and maintain the distribution of nodes based on the local density of the nodes. In this model, the initial coordinates of each node are represented by (x i1 , x i2 ), and the coordinates after movement are represented by (x′ i1 , x′ i2 ). This change in coordinates reflects the spatial adjustment of the nodes during the optimization process; in order to precisely control this movement, the weight ω i is introduced. This weight is assigned according to the number of neighboring nodes around the node, which means that the weight ω i of each node is a function of its local density. The more neighbors there are around the node, the greater its weight; this design ensures that the movement amplitude of the nodes in the area with high node density is small to avoid frequent overlaps and chaos, while in the area with low node density, the nodes can move within a larger range, thus minimizing the change in the overall distribution; in actual computational processing, in order to simplify the problem and improve computational efficiency, the upper left corner position of the node is used as its coordinate point, and the width and height of each node are represented by w iand h i It is shown that the objective function is expressed as:

[0023]

[0024] (3-2) Constraints: Constraints are designed to check whether the overlap between nodes is eliminated. When designing the constraints, it is noted that the moving direction of the nodes needs to be strictly controlled to prevent local layout chaos caused by arbitrary movement, thus maintaining the clarity and order of the overall layout. To achieve this goal, it is specified that the nodes can only move along the shortest path. This moving strategy not only simplifies the dynamic interaction between nodes but also effectively reduces node overlap and visual interference, making the overall layout more stable and coordinated. In addition, by making the nodes move along the shortest path;

[0025] A large value M is introduced as a penalty term to linearize the model and eliminate the non-linear terms. Four binary decision variables are defined, namely l ij 、r ij 、u ij and d ij , where l ij +r ij +u ij +d ij =1, representing the four moving directions of the nodes. Through these decision variables, the movement of each node is restricted to a clear direction, thus avoiding aimless movement and possible layout disorders;; This clear direction specification also greatly simplifies the decision-making process of node movement, making the interaction between nodes more explicit and controllable. The representation of this constraint is as follows:

[0026]

[0027] Finally, to ensure that the movement of the nodes remains within the screen space, that is, the coordinates of each node must be between the minimum and maximum coordinate values of the screen space, with the width of the maximum coordinate value being W and the height being H, the following constraints are established:

[0028]

[0029] Finally, the model is constructed and solved by the Gurobi solver.

[0030] The technical concept of the present invention is as follows: By introducing a dynamic projection visual layout method based on nonlinear programming and multi-objective optimization, the visualization representation of high-dimensional data sets in a two-dimensional space is processed and optimized. This method particularly emphasizes visual coherence during the dynamic iteration process of data. By precisely controlling the relative positions and distances between data points, visual occlusion is effectively reduced and visual presentation is optimized. The technical solution includes dynamically updating and adjusting the coordinates of data points, constructing a non-linear model from the perspectives of sample layout stability and elimination of node occlusion, and solving it through the Gurobi interface.

[0031] The beneficial effects of the present invention are as follows: The dynamic projection visual layout method maintains the layout stability of samples during iteration through dynamic adjustment algorithms and multi-objective optimization. The present invention enables users to more easily track and analyze data points with cross-iteration changes, thereby reducing visual chaos and cognitive burden. In addition, while maintaining the relationship of data features, the present invention effectively reduces visual occlusion, making each data point and cluster clearer and more recognizable, and enhancing the interpretability of the relationship between data. Brief Description of the Drawings

[0032] Figure 1 It is a flow chart of the dynamic projection visual layout method of the present invention;

[0033] Figure 2 It is a schematic diagram before the node layout optimization in the t-th iteration of the present invention;

[0034] Figure 3 It is a schematic diagram after the node layout optimization in the t-th iteration of the present invention;

[0035] Figure 4 It is a schematic diagram after the node layout optimization in the (t + 1)-th iteration of the present invention. Specific Embodiment

[0036] The present invention will be further described below in conjunction with the accompanying drawings.

[0037] Refer to Figures 1 to 4 , A dynamic projection visual layout method based on nonlinear programming and multi-objective optimization, includes the following steps:

[0038] 1) As Figure 1 shown, the main content of the implementation process of the solution includes obtaining the data samples of the t-th iteration, and using the existing unsupervised model for feature extraction to obtain a dynamic data set that can be used for projection; then projecting the data onto a two-dimensional plane through the tsne dimensionality reduction technology, and optimizing the layout according to the layout stability; then clustering the data samples through the DBSCAN clustering technology, and finally eliminating occlusion by slightly adjusting the node coordinates to generate a projection layout with better visual effects;

[0039] The present invention utilizes the set S t To represent the sample set of the current iteration, this set consists of two subsets: and in, contains samples retained from the previous iteration, and Then it includes the newly added samples. Where u represents a new sample and i represents an old sample. That is

[0040] The process of 1) is:

[0041] (1-1) The coordinates and styles of the points; the present invention uses S in the layout algorithm t The coordinates of the midpoint are expressed as S t The coordinates of sample i in Indicates that and are all continuous variables, each sample is represented as a square with a side length of 2 pixels, and the eigenvalues ​​obtained by the model are used to color the samples. Figure 2 As shown, it is the initial projection layout in the tth iteration (without layout optimization);

[0042] (1-2) Dynamic projection visual layout; the present invention is in the layout S t When the position of the midpoint is determined, the stability of the search space layout is first ensured to maintain the visual stability of the user during the iteration process. By maintaining the consistency of the sample position between each iteration cycle, it is ensured that the user can easily track the changes in the data, thereby improving the user experience and the efficiency of data analysis; secondly, the present invention eliminates the occlusion between samples through an effective layout strategy to better utilize the screen space, making the relationship between data points clearer and easier to observe and analyze; the layout algorithm of the present invention includes two steps: the first step is to maintain S t In the sample The layout stability of the samples is achieved by calculating the relative position relationship between the samples and maintaining the stability of these relationships to avoid large position changes due to iterations. The second step is to fine-tune S t The position of samples in the graph, the goal is to minimize the occlusion between samples;

[0043] 2) Maintain the layout stability of the sample; t When the samples in the present invention are laid out, the present invention first Keep the distribution of samples in The location of the samples is arranged as follows: This layout strategy not only takes into account the spatial relationship between new and old samples, but also minimizes the interference of new samples on the existing layout. The purpose of this is to make the samples that remain unchanged during the iteration The distribution remains consistent to assist users in visual tracking; as Figure 4 shown, this is a schematic diagram of the layout optimization of the present invention in the (t + 1)-th iteration based on the retained samples in the t-th iteration. Therefore, the following model is used as the layout requirement:

[0044]

[0045]

[0046] minα 1 f 1 +α 2 f 2 +α 3 f 3 (9)

[0047] In this model, the f in Equation (1) 1 means the sum of the position offset distances of the retained samples in the t-th iteration and the (t - 1)-th iteration. The function of this equation is to reduce the coordinate displacement of the maintenance samples in two iterations to stabilize the coordinate distribution of these retained samples; in the f of Equation (2) 2 , the feature distance between sample u and i is expressed as d ui , and the coordinate distance between the old image and the new image is expressed as The meaning of this equation is the sum of the differences between the feature distance and the coordinate distance between each retained sample and the newly added samples. Its function is to minimize the difference between the feature distance and the coordinate distance between the retained samples and the newly added samples to arrange the newly added samples while maintaining the feature relationship of the existing samples; the f in Equation (3) 3 represents the Kullback-Leibler divergence in t-SNE, and its function is to reduce the information loss when data is transformed from a high-dimensional vector to a low-dimensional vector;

[0048] Finally, f 1 、f 2 and f 3 constitute the objective function of the model. This model is an unconstrained multi-objective optimization model, where α 1 , α 2 , α 3 is the positive weight for balancing the three objective functions. After multiple tests, 1, 1, and 0.1 are selected as their values because the f 3 objective function has a larger range than the f 1 and f 2 objective functions. The present invention uses the momentum method to solve this model and solves it through the Gurobi solver. The momentum method helps to accelerate the convergence of the model, and at the same time, the Gurobi solver provides powerful computing power to ensure the efficiency and accuracy of the model when dealing with large-scale data sets;

[0049] 2) Eliminate occlusion between samples; after determining the coordinates of the S t midpoint, the samples are clustered by the DBSCAN clustering technique in the present invention to form clearly recognizable clustering nodes, and these nodes represent a set of similar samples. The samples within a node are arranged according to the attribute values generated by the model; the central coordinate of each node is determined according to the centroid of the samples within the node. To maximize the use of the screen space, the shape of the node is designed to be close to a square. However, as the number of samples increases, especially in large-scale datasets, occlusion may occur between nodes, which will seriously affect the user's visual analysis experience; the present invention introduces a dynamic adjustment mechanism that allows the system to dynamically adjust the node positions according to the results of real-time analysis to minimize the occlusion between clustering nodes. The node position fine-tuning strategy includes considering the local density and relative position of the nodes, calculating the spatial density around each node through an algorithm, and thus determining the optimal amplitude and direction of node movement; this density-based fine-tuning method not only helps reduce occlusion but also maintains the relative position relationship between nodes to a certain extent and maintains the structural integrity of the clustering. The present invention adopts the following model as the layout requirement:

[0050] The process of the said 3) is as follows:

[0051] (3-1) Optimize the objective function; the objective function of this model aims to eliminate the occlusion between nodes by moving the node positions and maintain the distribution of nodes based on the local density of the nodes. In this model, the initial coordinates of each node are represented by (x i1 , x i2 ), and the coordinates after movement are represented by (x' i1 , x' i2 ). This change in coordinates reflects the spatial adjustment of the nodes during the optimization process. To precisely control this movement, a weight ω i is introduced, and this weight is assigned according to the number of neighboring nodes around the node, which means that the weight ω i of each node is a function of its local density. The more neighbors a node has, the greater its weight; this design ensures that the movement amplitude of nodes in areas with high node density is small to avoid frequent overlaps and chaos, while in areas with low node density, the nodes can move over a larger range to minimize the change in the overall distribution. In actual computational processing, to simplify the problem and improve computational efficiency, the upper left corner position of the node is used as its coordinate point, and the width and height of each node are represented by w i and h i respectively. As shown in Figure 3 , it is a schematic diagram of the node after removing occlusion in the t-th iteration of the present invention, and its objective function is expressed as:

[0052]

[0053] (3-2) Constraints; Constraints are designed to check whether the overlap between nodes has been eliminated. When designing the constraints, it is noted that the moving direction of the nodes needs to be strictly controlled. This is to prevent local layout chaos caused by arbitrary movement, thus maintaining the clarity and orderliness of the overall layout. To achieve this goal, it is specified that the nodes can only move along the shortest path. This movement strategy not only simplifies the dynamic interaction between nodes but also effectively reduces node overlap and visual interference, making the overall layout more stable and coordinated. In addition, by making the nodes move along the shortest path, we can minimize the time and computing resources required for node adjustment and improve the running efficiency of the algorithm.

[0054] A large value M is introduced as a penalty term to linearize the model and eliminate the non-linear terms. Four binary decision variables are defined, namely l ij , r ij , u ij and d ij (where l ij + r ij + u ij + d ij = 1), representing the four moving directions of the nodes. Through these decision variables, the movement of each node is restricted to a specific direction, thus avoiding aimless movement and possible layout disorders. This specific direction specification also greatly simplifies the decision-making process of node movement, making the interaction between nodes more clear and controllable. The representation of this constraint is as follows:

[0055]

[0056] Finally, to ensure that the movement of the nodes remains within the screen space, that is, the coordinates of each node must be between the minimum and maximum coordinate values of the screen space, with the width of the maximum coordinate value being W and the height being H, the following constraints are established:

[0057]

[0058] Finally, the present invention constructs and solves this model through the Gurobi solver.

[0059] In this embodiment, a dynamic projection visual layout method based on nonlinear programming and multi-objective optimization is introduced to process and optimize the visual representation of high-dimensional data sets in a two-dimensional space. This method particularly emphasizes visual coherence during the data dynamic iteration process. By finely controlling the relative positions and distances between data points, it effectively reduces visual occlusion and optimizes the visual presentation. The technical solution includes dynamically updating and adjusting the coordinates of data points, constructing a non-linear model from the perspectives of sample layout stability and eliminating node occlusion, and solving it through the Gurobi interface.

[0060] The content described in the embodiments of this specification is only an enumeration of the implementation forms of the inventive concept and is only for illustrative purposes. The protection scope of the present invention should not be regarded as limited to the specific forms stated in this embodiment. The protection scope of the present invention also extends to equivalent technical means that can be conceived by those of ordinary skill in the art based on the inventive concept of the present invention.

Claims

1. A dynamic projection visual layout method based on nonlinear programming and multi-objective optimization, characterized in that: The method comprises the following steps: 1) Data description: In each iteration, the set S is used t To represent the sample set of the current iteration, this set consists of two subsets: and in, contains samples retained from the previous iteration, and It includes newly added samples, where u represents a new sample and i represents an old sample, that is, 2) Maintain the layout stability of the sample; t When laying out the samples in Keep the distribution of samples in The location of the samples is arranged as follows: sample; 3) Eliminate occlusion between samples; when determining S t After obtaining the coordinates of the midpoint, the samples are clustered using the DBSCAN clustering technology to form clearly identifiable cluster nodes. These nodes represent a collection of similar samples. The samples within the nodes are arranged according to the attribute values ​​generated by the model, and the central coordinates of each node are determined based on the centroid of the sample within the node. In order to maximize the use of screen space, the shape of the nodes is designed to be close to a square. By introducing a dynamic adjustment mechanism, the system is allowed to dynamically adjust the node position based on the results of real-time analysis and minimize occlusion between cluster nodes. The node position fine-tuning strategy includes considering the local density and relative position of the nodes, and calculating the spatial density around each node through an algorithm to determine the optimal amplitude and direction of node movement.

2. The dynamic projection visual layout method based on nonlinear programming and multi-objective optimization according to claim 1, characterized in that: The process of 1) is: (1-1) The coordinates of point S; t The coordinates of the midpoint are expressed as S t The coordinates of sample i in Indicates that and All are continuous variables; (1-2) Dynamic projection visual layout; in layout S t When locating the midpoint of the search space, we first ensure the stability of the search space layout to maintain the visual stability of the user during the iteration process; by maintaining the consistency of the sample position between each iteration cycle, we ensure that the user can easily track the changes in the data, thereby improving the user experience and the efficiency of data analysis; secondly, through an effective layout strategy, we eliminate the occlusion between samples to better utilize the screen space, making the relationship between data points clearer and easier to observe and analyze. The layout algorithm contains two steps: the first step is to maintain S t In the sample The layout stability of the samples is achieved by calculating the relative position relationships between the samples and maintaining the stability of these relationships to avoid large position changes due to iterations; The second step is to fine-tune S t The goal is to minimize the occlusion between samples.

3. The dynamic projection visual layout method based on nonlinear programming and multi-objective optimization according to claim 1 or 2, characterized in that: In 2), the samples that remain unchanged during the iteration process Keep the distribution consistent to help users track visually. To this end, use the following model as layout requirements: minα1f1+α2f2+α3f3 (3) In this model, f1 in formula (1) means the sum of the position offset distances of the retained samples in the tth iteration and the t-1th iteration. The function of this formula is to reduce the coordinate displacement of the maintenance samples in the two iterations to stabilize the coordinate distribution of these retained samples. In f2 of formula (2), the characteristic distance between samples u and i is expressed as d ui , the coordinate distance between the old image and the new image is expressed as The meaning of this formula is the sum of the differences in feature distance and coordinate distance between each retained sample and the newly added sample. Its function is to minimize the difference in feature distance and coordinate distance between the retained sample and the newly added sample, so as to arrange the newly added samples while maintaining the feature relationship of the existing samples. The f3 in formula (3) represents the Kullback-Leibler divergence in t-SNE, which is used to reduce the information loss when converting data from high-dimensional vectors to low-dimensional vectors. Finally, f1, f2 and f3 constitute the objective function of the model, which is an unconstrained multi-objective optimization model, where α1, α2, α3 are positive weights for balancing the three objective functions. After multiple tests, 1, 1, and 0.1 are selected as their values ​​because the f3 objective function has a larger range than f1 and f2. The model is solved using the momentum method and the Gurobi solver. The momentum method helps to accelerate the convergence of the model. At the same time, the Gurobi solver provides powerful computing power, ensuring the efficiency and accuracy of the model when processing large-scale data sets.

4. The dynamic projection visual layout method based on nonlinear programming and multi-objective optimization according to claim 1 or 2, characterized in that: In 3), the following model is used as the layout requirement: (3-1) Optimize the objective function; The objective function of this model aims to eliminate the occlusion between nodes by moving the node positions and maintain the distribution of nodes based on the local density of the nodes. In this model, the initial coordinates of each node are represented by (x i1 ,x i2 ), and the coordinates after the shift are represented by (x′ i1 ,x′ i2 ) indicates that the change of this coordinate reflects the spatial adjustment of the node during the optimization process; in order to accurately control this movement, the weight ω is introduced i , the weight is assigned according to the number of neighboring nodes around the node, which means that the weight of each node ω i is a function of its local density. The more neighbors a node has, the greater its weight. This design ensures that nodes move less in areas with high node density to avoid frequent overlap and confusion, while in areas with low node density, nodes can move over a larger range to minimize changes in the overall distribution. In actual calculations, in order to simplify the problem and improve calculation efficiency, the upper left corner of the node is used as its coordinate point, and the width and height of each node are represented by w. i and h i The objective function is expressed as: (3-2) Constraints: Constraints are designed to verify whether the overlap between nodes is eliminated. When designing constraints, it is noted that the movement direction of the nodes needs to be strictly controlled. This is to prevent arbitrary movement from causing chaos in the local layout, thereby maintaining the clarity and order of the overall layout. To achieve this goal, the specified nodes can only move along the shortest path. This movement strategy not only simplifies the dynamic interaction between nodes, but also effectively reduces node overlap and visual interference, making the overall layout more stable and coordinated. In addition, by making the nodes move along the shortest path; A large value M is introduced as a penalty term to linearize the model and eliminate nonlinear terms. Four binary decision variables are defined, namely, l ij 、r ij 、u ij and d ij , where l ij +r ij +u ij +d ij = 1, representing the four movement directions of the node. Through these decision variables, the movement of each node is limited to a clear direction, thus avoiding aimless movement and possible layout disorder; this clear direction specification also greatly simplifies the decision-making process of node movement, making the interaction between nodes more clear and controllable. The constraint is expressed as follows: Finally, to ensure that the movement of the node remains within the screen space, that is, the coordinates of each node must be between the minimum and maximum coordinate values ​​of the screen space, with the maximum coordinate value having a width of W and a height of H, the following constraints are established: Finally, the model is constructed and solved using the Gurobi solver.

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