Large language model enhanced complex network community visual discovery method

Abstract

The invention discloses a big language model enhanced complex network community visual discovery method, which combines node division and link division, applies a big language model to carry out semantic information deep analysis on node and link data in a complex network, realizes accurate capture of potential associations in the complex network, and realizes visual discovery of the complex network community. And a large language model is combined with a visualization technology, so that the relationship between the communities and the topological structure of the whole network are accurately analyzed, and a community discovery result is displayed by an innovative visualization method. According to the method, interaction analysis is carried out through the large language model, the operation steps are guided, the recognition and analysis capability of the complex network community is enhanced, and the interpretability of the community discovery result is further improved. The application scenarios of the method cover the fields of marine transportation networks, traffic situations, complex clusters and the like, and can provide in-depth visual analysis services in a complex network environment to assist in intelligent decision making.

Description

Technical Field

[0001] The present invention relates to the fields of visual discovery analysis of complex network communities and artificial intelligence, and in particular to complex network community analysis in the fields of shipping networks, traffic situations, complex clusters, etc. Background Art

[0002] With the acceleration of global networking, complex networks play an increasingly important role in various fields, including but not limited to shipping networks, traffic situations, complex clusters, etc. In fact, these complex networks can also be regarded as graphs in computer science, consisting of nodes and edges. By analyzing these networks, we can reveal the potential relationships between different nodes and communities, and identify the key factors affecting the operation of the system. In particular, in these complex networks, it becomes crucial to discover the group structure of nodes and clusters.

[0003] Complex networks are usually composed of multiple communities. Community discovery is an important means to understand the relationship between nodes in complex networks. Traditional community discovery methods mainly rely on the topological structure of the network and aim to divide communities based on the link relationship between nodes. In order to better identify communities, recent research has further combined visualization technology, which can intuitively display the links and mutual influences between communities through an interactive graphical interface, improving analytical capabilities and depth of understanding.

[0004] Although community discovery and visualization analysis techniques for complex networks have made significant progress in recent years, existing methods still have certain limitations in practical applications such as shipping networks, traffic trends, and complex clusters. Traditional complex network analysis models mainly rely on the topological structure of the network. Although this method can identify the connections between communities to a certain extent, it performs poorly in dealing with dynamically changing network data and capturing semantic information. At the same time, although existing visualization techniques can intuitively present community structures, they often lack sufficient flexibility and interaction capabilities when dealing with highly complex node attributes and link relationships, resulting in a lack of in-depth understanding and accurate capture of semantic information when analyzing complex network characteristics, making it difficult to fully reveal potential associations in complex networks. Therefore, an innovative method that combines large language models with visual discovery of complex network communities is urgently needed. By optimizing the semantic understanding ability of the model, the interpretability of complex network community discovery is enhanced, and the community structure is dynamically presented using advanced visualization techniques. Through this method, the visual discovery effect of communities in complex networks can be effectively improved, providing strong support for intelligent decision-making in practical applications. Summary of the invention

[0005] In order to solve the problems existing in the background technology, the present invention proposes a complex network community visual discovery method enhanced by a large language model. By combining node partitioning and link partitioning, a large language model is used to perform deep semantic analysis of node and link data in a complex network, accurately capture the potential correlation in the complex network, and design an interactive visualization layout to dynamically adjust the node distribution and the overlapping area of ​​the community to achieve an innovative visualization of the complex network structure. By adopting a visualization analysis method that combines a large language model with interactive operations, the visualization effect of complex network community discovery is improved, and a more interpretable complex network community discovery analysis is achieved.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A complex network community visual discovery method enhanced by a large language model includes the following steps:

[0008] S1. Preprocess the real-world complex network data to complete the data set integration, and construct the initial complex network through the topological structure and link characteristics between nodes;

[0009] S2. Initially partitioning the complex network by a node partitioning method to determine the basic community structure of the complex network;

[0010] S3. Combining the complex network with force-oriented layout technology to construct a dynamic visualization interface to enhance the display effect of the complex network structure;

[0011] S4. Use a large language model to deepen the analysis of the complex network structure and enhance the visualization and interactive analysis capabilities;

[0012] S5. Classify the association relationship between the nodes by link partitioning method to further optimize the community partitioning;

[0013] S6. Combine the large language model with visual interactive analysis to mine the complex network features and explain the segmentation results, while generating real-time feedback on the interactive analysis results.

[0014] In S1, the real-world complex network data is preprocessed to complete the data set integration, and the initial complex network is constructed through the topological structure and link characteristics between nodes. The specific process is as follows:

[0015] S11. Preprocess the complex network data to ensure the integrity and accuracy of the data, and integrate them into a basic data set that meets the network structure requirements.

[0016] S12. An initialized complex network is constructed through the topological structure and link characteristics between the nodes, the node size is dynamically adjusted according to the node degree attribute, and the color, thickness and transparency of the link are adjusted according to the link strength attribute between the nodes to form an initial visual hierarchy.

[0017] Furthermore, the specific process of step S2 is as follows:

[0018] S21. The initial partitioning of the complex network is realized by the node partitioning method, and the community partitioning quality of the complex network nodes is evaluated based on the modularity function Q. The closer the Q value is to 1, the stronger the community structure is. The difference between the actual links in the network and the expected random links is compared (Formula 1). Where A i,j represents the link weight matrix between nodes i and j, m represents all link weights in the network, k i =∑ j A i,j represents the sum of all link weights of node i, C i With C j Indicates the community to which the node belongs. When nodes i and j are in the same community, d(C i ,C j )=1, otherwise it is 0.

[0019]

[0020] S22. Based on the modularity function Q, the network node division and visualization are iterated through two steps: modularity optimization and community aggregation. First, a community is assigned to each node, and the community affiliation of the node is intuitively displayed through visualization. Node i is moved to the adjacent community in turn and the modularity increment ΔQ (Formula 2) is calculated. The change process of node affiliation is displayed through dynamic visualization. If ΔQ is greater than 0, node i is divided into the community with the largest modularity increment, otherwise the node community remains unchanged.

[0021]

[0022] Furthermore, the specific process of step S3 is as follows:

[0023] S31. Use a force-directed layout algorithm (Formula 3) to automatically arrange the nodes and links in the complex network, so that the nodes in the network form a reasonable and intuitive layout in space, avoid excessive overlap, and enhance the visualization effect of the complex network structure.

[0024]

[0025] where F ij represents the resultant force between nodes i and j, k dis the adjustment coefficient of attraction and repulsion, which is used to balance the repulsion and attraction between nodes, d ij represents the actual distance between node i and node j, l ij is the ideal distance between nodes, adjusted according to the node relevance and community structure, x i and x j Represent the coordinate positions of node i and node j respectively.

[0026] S32. Combined with the force-guided layout, the results of node division are clustered into different communities for display, and the communities are color-coded. The links between nodes are displayed in the form of lines, and the thickness of the lines is dynamically adjusted according to the weight attributes of the links, showing the strength of the association within and between communities, and enhancing the visualization effect.

[0027] S33. A characteristic glyph visualization method is provided to display the structural characteristics of the community in the force-directed layout. Specifically, the number of nodes of the community is displayed through the central circle, the color of the outer circle sector indicates the associated community, and the size of the sector indicates the number of nodes linked to the associated community, so as to clearly display the internal structure of the community, the node scale and the links with other communities.

[0028] Furthermore, the specific process of step S4 is as follows:

[0029] S41. Use large language models to deeply analyze nodes and communities in complex networks. First, by analyzing the contextual information, link relationships, and attributes of nodes, the large language model can dig out the semantic associations hidden behind the nodes; secondly, using the large language model's powerful ability to understand semantic information, the importance of nodes is evaluated and key nodes that have a greater impact on the overall structure of the network are identified. At the same time, by analyzing the closeness of nodes within communities and the strength of links between communities, the large language model can clarify the associations between different communities, reveal the strength of associations between communities, and improve the interpretability of complex network analysis and the accuracy of community discovery.

[0030] S42. Use an analysis method that combines a large language model with interactive operations. First, by selecting a node or community, the large language model analyzes the attribute characteristics of the node or community in real time, and generates an explanation based on contextual information to clarify the role of the node or community in the entire complex network; second, after screening and adjusting the nodes or communities, the large language model can explain the structural changes of the complex network after screening based on the properties of the screened nodes and the network topology, and at the same time feedback the changes in the relationships inside and outside the community. Through the continuous feedback of the large language model, the understanding of the changes in the structure of the complex network is enhanced.

[0031] Furthermore, the specific process of step S5 is as follows:

[0032] S51. Assign community labels to links according to the result of node division, and divide link types into core links with the same community labels of nodes at both ends and edge links with different community labels of nodes at both ends. Then, further divide the links between nodes through link division, directly assign community labels of nodes to core links, and mark them with different colors and shapes in the visualization graph.

[0033] S52. The edge link needs to calculate the modularity gain (Formula 4) generated by the link joining different communities, and generate graphical feedback of the gain in real time.

[0034]

[0035] in represents the number of links between node v and community C, k C represents the total node degree of community C, k v represents the node degree of v, m represents the weight of all links in the network, and γ is a resolution parameter used to adjust the number of communities in the network. When the difference in modularity gain is significant, the link is attributed to the community that can bring greater modularity gain through a visual icon. When the modularity gain is equal, it means that the link acts as a bridge between communities, and a new community label will be given to it, which will be distinguished by different symbols and colors in the visual graphic display.

[0036] Furthermore, the specific process of step S6 is as follows:

[0037] S61. Design multi-level node styles and interactive visualization layouts to intuitively display the link division results in complex networks. Design node styles based on multi-dimensional visual elements. First, assign different color codes to nodes according to the communities they belong to. Then, dynamically adjust the node size according to the importance and node degree of the node in the community, so as to highlight the key nodes.

[0038] S62. In the visualization layout, the interaction between links and communities is presented in a dynamic visualization manner. First, the selected links and the communities to which the links belong are obtained, and the link colors and transparency are adjusted to reflect the link characteristics. Then, the nodes of the core links are gathered in the center, and the nodes of the edge links are dispersed around, so as to realize the dynamic adjustment of the node distribution in the community and complete the visual display of the community structure;

[0039] S63. The results of the division of links between nodes are then explained in detail through the large language model. First, the large language model extracts various attribute information of the link, including link strength, contextual semantics, and node attributes of the link, and analyzes the rationality of the community to which the link belongs based on this information. Then, the large language model judges the community characteristics of the link in combination with the link relationship of the node, and identifies the key links that serve as bridges between communities. In addition, the large language model will automatically prompt to pay attention to closely connected communities and guide the analysis of potential relationships therein. At the same time, during the interactive operation, the large language model will be dynamically updated according to the input questions, and further provide targeted visual explanations and operation suggestions to obtain deeper association analysis and structural explanations.

[0040] Compared with the prior art, the present invention has the following beneficial effects:

[0041] The present invention significantly improves the visual discovery and interpretation capabilities of complex network communities through the deep integration of large language models and visual interaction technology. Through the large language model, contextual information and attributes are accurately parsed, and combined with innovative visualization methods, it not only provides efficient and intuitive visual feedback, but also can generate further explanations and guidance based on interactive operations and large language models, helping to fully understand the multi-level structure of complex networks. Through this innovative technical integration, the present invention can greatly improve the more accurate and dynamic visualization of complex network communities, while providing more interpretable result analysis. Therefore, the present invention is widely applicable to shipping networks, traffic situations, complex clusters and other fields, helping to make complex network analysis intelligent and achieve a higher level of visual interaction and analysis depth. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 Flowchart of the complex network community visual discovery method enhanced for large language models;

[0043] Figure 2 Schematic diagram of complex network node partitioning and community interaction visual analysis enhanced by large language model;

[0044] Figure 3 Schematic diagram of complex network link partitioning and link interaction visual analysis enhanced by large language model. DETAILED DESCRIPTION

[0045] The technical solutions in the embodiments of the present invention are clearly and completely described below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work belong to the protection scope of the present invention. In the following description, many specific details are set forth to facilitate a full understanding of the present invention, but the present invention can also be implemented in other ways different from those described herein, and those skilled in the art can make similar generalizations without violating the connotation of the present invention, so the present invention is not limited by the specific embodiments disclosed below.

[0046] As a preferred embodiment of the present invention, the method proposed by the present invention comprises the following steps:

[0047] S1. Analyze and preprocess the real-world complex network data to complete the data set integration, and construct the initial complex network through the topological structure and link characteristics between nodes. The specific process is as follows:

[0048] S11. Preprocess the complex network data to ensure the integrity and accuracy of the data, and integrate them into a basic data set that meets the network structure requirements.

[0049] S12. Initialize the complex network through the topological structure between nodes and their link characteristics, dynamically adjust the node size according to the node degree attribute, and adjust the color, thickness and transparency of the link according to the link strength attribute between nodes to form an initial visual hierarchy.

[0050] S2. The initial division of the complex network is achieved through the node division method, and the basic community structure of the complex network is determined. The specific process is as follows:

[0051] S21. The initial partitioning of the complex network is achieved through the node partitioning method, and the quality of the community partitioning of the network nodes is evaluated based on the modularity function of formula 1. The closer the Q value is to 1, the stronger the community structure is.

[0052]

[0053] S22. Based on the modularity function, the network node division is iteratively implemented through two steps of modularity optimization and community aggregation in Formula 2, and the changing process of node affiliation is displayed using dynamic visualization.

[0054]

[0055] S3. Combine complex networks with force-oriented layout technology to build a dynamic visualization interface and improve the display effect of network structure. The specific process is as follows:

[0056] S31. The force-directed layout algorithm (Formula 3) is used to automatically arrange the nodes and links in the complex network, so that the nodes in the network form a reasonable and intuitive layout in space, avoid excessive overlap, and enhance the visualization effect of the network structure.

[0057]

[0058] S32. Cluster the node division results into different communities for display in combination with the force-directed layout.

[0059] S33. The structural characteristics of the community in the force-directed layout are displayed through a characteristic glyph visualization method, highlighting the internal structure of the community, the node scale, and the connection with other communities.

[0060] S4. Use the large language model to deepen the analysis of complex network structures and enhance the ability of visual interactive analysis. The specific process is as follows:

[0061] S41. Use large language models to deeply analyze nodes and communities in complex networks. First, by analyzing the contextual information, link relationships, and attributes of nodes, the large language model can dig out the semantic associations hidden behind the nodes; secondly, using the large language model's powerful ability to understand semantic information, the importance of nodes is evaluated and key nodes that have a greater impact on the overall structure of the network are identified. At the same time, by analyzing the closeness of nodes within communities and the strength of links between communities, the large language model can clarify the associations between different communities, reveal the strength of associations between communities, and improve the interpretability of complex network analysis and the accuracy of community discovery.

[0062] S42. Use an analysis method that combines a large language model with interactive operations. First, by selecting a node or community, the large language model can analyze the attribute characteristics of the node or community in real time, and generate explanations based on contextual information to clarify its role in the entire complex network; second, after screening and adjusting the nodes or communities, the large language model can explain the changes in the screened structure based on the screened node attributes and network topology, and at the same time feedback the changes in the relationships inside and outside the community. Through the continuous feedback of the large language model, the understanding of the changes in the complex network structure is enhanced.

[0063] S5. Classify the associations between nodes through the link partitioning method to further optimize the community partitioning. The specific process is as follows:

[0064] S51. Assign community labels to links based on the node division results, and divide link types into core links with the same community labels of nodes at both ends and edge links with different community labels of nodes at both ends. Then, further divide the links between nodes through link division. Core links can be directly assigned community labels of nodes.

[0065] S52. The edge link needs to calculate the modularity gain (Formula 4) generated by the link joining different communities, and generate graphical feedback of the gain in real time.

[0066]

[0067] S6. Combine the large language model with visual interactive analysis to mine complex network features and explain the segmentation results. At the same time, real-time feedback can be generated for the interactive analysis results. The specific process is as follows:

[0068] S61. Through multi-level node styles and interactive visualization layout, the link division results in complex networks are intuitively displayed: first, different color codes are assigned to the nodes according to the communities they belong to, and then the node sizes are dynamically adjusted according to the importance and node degree of the nodes in the community, so as to highlight the key nodes.

[0069] S62. In the visualization layout, a dynamic visualization method is used to display the interaction between links and communities: first, the selected links and the communities to which the links belong are obtained, and the link colors and transparency are adjusted to reflect the link characteristics. Then, the nodes of the core links are gathered in the center, and the nodes of the edge links are dispersed around, so as to realize the dynamic adjustment of the node distribution in the community and complete the visual display of the community structure.

[0070] S63. The large language model is then used to explain in detail the division results of the links between nodes. First, the large language model extracts various attribute information of the link, including link strength, contextual semantics, and node attributes of the link, and analyzes the rationality of the community to which the link belongs based on this information; then, the large language model combines the link relationship of the node to judge the community characteristics of the link and identify the key links that serve as bridges between communities. In addition, the large language model will automatically prompt attention to closely connected communities and guide the analysis of potential relationships therein. At the same time, during the interactive operation, the large language model will dynamically update according to the input questions, and further provide targeted visual explanations and operation suggestions to obtain deeper correlation analysis and structural explanations.

[0071] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A complex network community visual discovery method enhanced by a large language model, comprising the following steps: S1. Preprocess the real-world complex network data to complete the data set integration, and construct the initial complex network through the topological structure and link characteristics between nodes; S2. Initially partitioning the complex network by a node partitioning method to determine the basic community structure of the complex network; S3. Combining the complex network with force-oriented layout technology to construct a dynamic visualization interface to enhance the display effect of the complex network structure; S4. Use a large language model to deepen the analysis of the complex network structure and enhance the visualization and interactive analysis capabilities; S5. Classify the association relationship between the nodes by link partitioning method to further optimize the community partitioning; S6. Combine the large language model with visual interactive analysis to mine the complex network features and explain the segmentation results, while generating real-time feedback on the interactive analysis results.

2. According to claim 1, a large language model enhanced complex network community visual discovery method is characterized in that: The specific process of step S1 is as follows: S11. Preprocess the complex network data to ensure the integrity and accuracy of the data and integrate them into a basic data set that meets the network structure requirements; S12. An initialized complex network is constructed through the topological structure and link characteristics between the nodes, the node size is dynamically adjusted according to the node degree attribute, and the color, thickness and transparency of the link are adjusted according to the link strength attribute between the nodes to form an initial visual hierarchy.

3. The method for visually discovering complex network communities enhanced by a large language model according to claim 1, characterized in that: The specific process of step S2 is as follows: S21. The initial partitioning of the complex network is achieved by the node partitioning method, and the quality of the community partitioning of the complex network nodes is evaluated based on the modularity function Q. The closer the Q value is to 1, the stronger the community structure is. By comparing the difference between the actual links in the network and the expected random links (Formula 1), A i,j represents the link weight matrix between nodes i and j, m represents all link weights in the network, k i =∑ j A i,j represents the sum of all link weights of node i, C i With C j Indicates the community to which the node belongs. When nodes i and j are in the same community, d(C i ,C j )=1, otherwise 0; S22. Based on the modularity function Q, the network node division and visualization are iterated through two steps: modularity optimization and community aggregation. First, a community is assigned to each node. The community affiliation of the node is intuitively displayed through visualization. Node i is moved to the adjacent community in turn and the modularity increment ΔQ (Formula 2) is calculated. The change process of node affiliation is displayed through dynamic visualization. If ΔQ is greater than 0, node i is divided into the community with the largest modularity increment, otherwise the node community remains unchanged.

4. The method for visually discovering complex network communities enhanced by a large language model according to claim 1, characterized in that: The specific process of step S3 is as follows: S31. Use a force-directed layout algorithm (Formula 3) to automatically arrange the nodes and links in the complex network, so that the nodes in the network form a reasonable and intuitive layout in space, avoid excessive overlap, and enhance the visualization effect of the complex network structure: where F ij represents the resultant force between nodes i and j, k d is the adjustment coefficient of attraction and repulsion, which is used to balance the repulsion and attraction between nodes, d ij represents the actual distance between node i and node j, l ij is the ideal distance between nodes, adjusted according to the node relevance and community structure, x i and x j Respectively represent the coordinate positions of node i and node j; S32. Combined with the force-oriented layout, the results of node division are clustered into different communities for display, and the communities are color-coded. The links between nodes are displayed in the form of lines, and the thickness of the lines is dynamically adjusted according to the weight attributes of the links, showing the strength of the association within and between communities, and enhancing the visualization effect; S33. A characteristic glyph visualization method is provided to display the structural characteristics of the community in the force-directed layout. Specifically, the number of nodes of the community is displayed through the central circle, the color of the outer sector represents the associated community, and the size of the sector represents the number of nodes linked to the associated community, thereby clearly displaying the internal structure of the community, the size of the nodes, and the links with other communities.

5. The method for visually discovering complex network communities enhanced by a large language model according to claim 1, characterized in that: The specific process of step S4 is as follows: S41. Through the large language model, the nodes and communities in the complex network are deeply analyzed. First, by analyzing the context information, link relationship and attributes of the nodes, the large language model can dig out the semantic associations hidden behind the nodes. Secondly, by using the large language model's powerful understanding of semantic information, the importance of the nodes is evaluated and the key nodes that have a greater impact on the overall structure of the network are identified. At the same time, by analyzing the closeness of the nodes within the community and the strength of the links between the communities, the large language model can clarify the relationship between different communities, reveal the strength of the association between communities, and improve the interpretability of complex network analysis and the accuracy of community discovery. S42. An analysis method combining a large language model with interactive operations is adopted. First, by selecting a node or community, the large language model parses the attribute characteristics of the node or community in real time, and generates an explanation based on the context information to clarify the role of the node or community in the entire complex network. Secondly, after screening and adjusting the nodes or communities, the large language model can explain the structural changes of the complex network after screening based on the properties of the screened nodes and the network topology, and at the same time feedback the changes in the relationships inside and outside the community. Through the continuous feedback of the large language model, the understanding of the structural changes of the complex network is enhanced.

6. The method for visually discovering complex network communities enhanced by a large language model according to claim 1, characterized in that: The specific process of step S5 is as follows: S51. assign community labels to links according to the result of node division, and divide link types into core links with the same community labels of nodes at both ends and edge links with different community labels of nodes at both ends, and then further divide the links between nodes through link division, directly assign community labels of nodes to core links, and mark them with different colors and shapes in the visualization graph; S52. For edge links, it is necessary to calculate the modularity gain (Formula 4) generated by the link joining different communities, and generate graphical feedback of the gain in real time: in represents the number of links between node v and community C, k C represents the total node degree of community C, k v represents the node degree of v, m represents the weight of all links in the network, γ is the resolution parameter used to adjust the number of communities in the network, when the modularity gain difference is significant, the link is attributed to the community that can bring greater modularity gain through the visualization icon, when the modularity gain is equal, it means that the link acts as a bridge between communities, and a new community label will be given to it, which will be distinguished by different symbols and colors in the visualization graphic display.

7. The method for visually discovering complex network communities enhanced by a large language model according to claim 1, characterized in that: The specific process of step S6 is as follows: S61. Design multi-level node styles and interactive visualization layouts to intuitively display the link division results in complex networks. Design node styles based on multi-dimensional visual elements. First, assign different color codes to nodes according to the communities they belong to. Then, dynamically adjust the node size according to the importance and node degree of the node in the community, so as to highlight the key nodes. S62. In the visualization layout, the interaction between links and communities is displayed in a dynamic visualization manner. First, the selected links and the communities to which the links belong are obtained, and the link colors and transparency are adjusted to reflect the link characteristics. Then, the nodes of the core links are gathered in the center, and the nodes of the edge links are dispersed around, so as to realize the dynamic adjustment of the node distribution in the community and complete the visual display of the community structure. S63. The large language model is then used to explain in detail the division results of the links between nodes. First, the large language model extracts various attribute information of the links, including link strength, contextual semantics, and node attributes of the links. Based on this information, the rationality of the community to which the links belong is analyzed. Then, the large language model combines the link relationship of the nodes to judge the community characteristics of the links and identify the key links that serve as bridges between communities. In addition, the large language model will automatically prompt attention to closely connected communities and guide the analysis of potential relationships therein. At the same time, during the interactive operation, the large language model will be dynamically updated according to the input questions, and further provide targeted visual explanations and operation suggestions to obtain deeper association analysis and structural explanations.

Images