Graph layout

We document an open-source toolbox for drawing large-scale undirected graphs. This toolbox is based on a previously implemented closed-source algorithm known as VxOrd. Our toolbox, which we call OpenOrd, extends the capabilities of VxOrd to large graph layout by incorporating edge-cutting, a multi-level approach, average-link clustering, and a parallel implementation. At each level, vertices are grouped using force-directed layout and average-link clustering. The clustered vertices are then re-drawn and the process is repeated. When a suitable drawing of the coarsened graph is obtained, the algorithm is reversed to obtain a drawing of the original graph. This approach results in layouts of large graphs which incorporate both local and global structure. A detailed description of the algorithm is provided in this paper. Examples using datasets with over 600K nodes are given. Code is available at www.cs.sandia.gov/∼smartin.

We present a discussion and initial empirical investigation of user interface designs for a set of three Web browsers. The target end user population we identified were experienced software engineers who maintained large web sites or portals. The user study demonstrated the strengths and weaknesses of two conventional 2D browsers for this target user, as well as that of XML3D, a novel browser that integrates an interactive 3D hyperbolic graph view with a more traditional 2D list view of the data. A standard collapse/expand tree browser and a Web-based hierarchical categorization similar to Yahoo!, were competitively evaluated against XML3D. No reliable difference between the two 2D browsers was observed. However, the results showed clear differences between XML3D and the 2D user interfaces combined. With XML3D, participants performed search tasks within existing categories reliably faster with no decline in the quality of their responses. It was informally observed that integrating the ability to view the overall structure of the information space with the ability to easily assess local and global relationships was key to successful search performance. XML3D was the only tool of the three that efficiently showed the overall structure within one visualization. The XML3D browser accomplished this by combining a 3D graph layout view as well as an accompanying 2D list view. Users did opt to use the 2D user interface components of XML3D during new category search tasks, and the XML3D performance advantage was no longer obtained in those conditions. In addition, there were no reliable differences in overall user satisfaction across the three user interface designs. Since we observed subjects using the XML3D features differently depending on the kind of search task, future studies should explore optimal ways of integrating the use of novel focus + context visualizations and 2D lists for effective information retrieval. The contribution of this paper is that it includes empirical data to demonstrate where novel focus + context views might benefit experienced users over and above more conventional user interface techniques, in addition to where design improvements are warranted. 3 Risden et al. IJHCS 5/18/00

We present an expansion of the popular open source Visualization Toolkit (VTK) to support the ingestion, processing, and display of informatics data. The result is a flexible, component-based pipeline framework for the integration and deployment of algorithms in the scientific and informatics fields. This project, code named "Titan", is one of the first efforts to address the unification of information and scientific visualization in a systematic fashion. The result includes a wide range of informatics-oriented functionality: database access, graph algorithms, graph layouts, views, charts, UI components and more. Further, the data distribution, parallel processing and client/server capabilities of VTK provide an excellent platform for scalable analysis.

Social Network Analysis (SNA) has evolved as a popular, standard method for modeling meaningful, often hidden structural relationships in communities. Existing SNA tools often involve extensive pre-processing or intensive programming skills that can challenge practitioners and students alike. NodeXL, an open-source template for Microsoft Excel, integrates a library of common network metrics and graph layout algorithms within the familiar spreadsheet format, offering a potentially low-barrierto-entry framework for teaching and learning SNA. We present the preliminary findings of 2 user studies of 21 graduate students who engaged in SNA using NodeXL. The majority of students, while information professionals, had little technical background or experience with SNA techniques. Six of the participants had more technical backgrounds and were chosen specifically for their experience with graph drawing and information visualization. Our primary objectives were (1) to evaluate NodeXL as an SNA tool for a broad base of users and (2) to explore methods for teaching SNA. Our complementary dual case-study format demonstrates the usability of NodeXL for a diverse set of users, and significantly, the power of a tightly integrated metrics/visualization tool to spark insight and facilitate sensemaking for students of SNA.

A new divide-and-conquer framework for VLSI graph layout is introduced. Universally close upper and lower bounds are obtained for important cost functions such as layout area and propagation delay. The framework is also effectively used to design regular and configurable layouts, to assemble large networks of processors using restructurable chips, and to configure networks around faulty processors. It is also shown how good graph partitioning heuristics may be used to develop a provably good layout strategy.

Frequently, large knowledge bases are represented by graphs. Many visualization tools allow users or other applications to interact with and adjust the layouts of these graphs. One layout adjustment problem is that of showing more detail without eliding parts of the graph. Approaches based on a fisheye lens paradigm seem well suited to this task. However, many of these techniques are non-trivial to implement and their distortion techniques often cannot be altered to suit different graph layouts. When distorting a graph layout, it is often ...

Many graph drawing and visualization algorithms, such as force-directed layout and line-dot rendering, work very well on relatively small and sparse graphs. However, they often produce extremely tangled results and exhibit impractical running times for highly non-planar graphs with large edge density. And very few graph layout algorithms support dynamic time-varying graphs; applying them independently to each frame produces distracting temporally incoherent visualizations. We have developed a new visualization technique based on a novel approach to hierarchically structuring dense graphs via stratification. Using this structure, we formulate a hierarchical force-directed layout algorithm that is both efficient and produces quality graph layouts. The stratification of the graph also allows us to present views of the data that abstract away many small details of its structure. Rather than displaying all edges and nodes at once, resulting in a convoluted rendering, we present an interactive tool that filters edges and nodes using the graph hierarchy and allows users to drill down into the graph for details. Our layout algorithm also accommodates time-varying graphs in a natural way, producing a temporally coherent animation that can be used to analyze and extract trends from dynamic graph data. For example, we demonstrate the use of our method to explore financial correlation data for the U.S. stock market in the period from 1990 to 2005. The user can easily analyze the time-varying correlation graph of the market, uncovering information such as market sector trends, representative stocks for portfolio construction, and the interrelationship of stocks over time.

In this paper, we present a novel approach for constructing bundled layouts of general graphs. As layout cues for bundles, we use medial axes, or skeletons, of edges which are similar in terms of position information. We combine edge clustering, distance fields, and 2D skeletonization to construct progressively bundled layouts for general graphs by iteratively attracting edges towards the centerlines of level sets of their distance fields. Apart from clustering, our entire pipeline is image-based with an efficient implementation in graphics hardware. Besides speed and implementation simplicity, our method allows explicit control of the emphasis on structure of the bundled layout, i.e. the creation of strongly branching (organic-like) or smooth bundles. We demonstrate our method on several large real-world graphs.

We present interactive visual aids to support the exploration and navigation of graph layouts. They include Fisheye Tree Views and Composite Lenses. These views provide, in an integrated manner, overview+detail and focus+context. Fisheye Tree Views are novel applications of the well known fisheye distortion technique. They facilitate the exploration of the hierarchy trees associated with clustered graphs. Composite Lenses are the result of the integration of several lens techniques. They facilitate the display of local graph information that may be otherwise difficult to grasp in large and dense graph layouts.

Using a set of geometric containers to speed up shortest path queries in a weighted graph has been proven a useful tool for dealing with large sparse graphs. Given a layout of a graph G = (V, E), we store, for each edge (u, v) ∈ E, the bounding box of all nodes t ∈ V for which a shortest u-t-path starts with (u, v). Shortest path queries can then be answered by Dijkstra's algorithm restricted to edges where the corresponding bounding box contains the target.

Background: We investigate the accuracy of different similarity approaches for clustering over two million biomedical documents. Clustering large sets of text documents is important for a variety of information needs and applications such as collection management and navigation, summary and analysis. The few comparisons of clustering results from different similarity approaches have focused on small literature sets and have given conflicting results. Our study was designed to seek a robust answer to the question of which similarity approach would generate the most coherent clusters of a biomedical literature set of over two million documents.

UML diagramshave becomeincreasinglyimportant in the engineeringandreengineering processes for softwaresystems.Of particular interestare UML classdiagramswhosepurposeis to display classhierarchies(generalizations), associations, aggre gations, and compositionsin one picture. The combinationof hierarchical and non-hierarchical relationsposesa specialchallengeto a graphlayout tool. Existing layout tools treathierarchicalandnonhierarchical relationseitheralike or asseparate tasksin a two-phase processasin, e.g., ].Wesuggest anew approach for visualizingUML classdiagramsleadingto a balancedmixture of thefollowing aestheticcriteria: Crossingminimization,bendminimization,uniform directionwithin eachclasshierarch y, no nesting of oneclasshierarch y within another , orthogonallayout, merging of multiple inheritanceedges,and good edgelabelling. We have realizedour approachwithin the graphdrawing library GoVisual. Experimentsshow the superiorityto state-of-the-art andindustrial standardlayouts.

A standard approach to large network visualization is to provide an overview of the network and a detailed view of a small component of the graph centred around a focal node. The user explores the network by changing the focal node in the detailed view or by changing the level of detail of a node or cluster. For scalability, fast force-based layout algorithms are used for the overview and the detailed view. However, using the same layout algorithm in both views is problematic since layout for the detailed view has different requirements to that in the overview. Here we present a model in which constrained graph layout algorithms are used for layout in the detailed view. This means the detailed view has high-quality layout including sophisticated edge routing and is customisable by the user who can add placement constraints on the layout. Scalability is still ensured since the slower layout techniques are only applied to the small subgraph shown in the detailed view. The main technical innovations are techniques to ensure that the overview and detailed view remain synchronized, and modifying constrained graph layout algorithms to support smooth, stable layout. The key innovation supporting stability are new dynamic graph layout algorithms that preserve the topology or structure of the network when the user changes the focus node or the level of detail by in situ semantic zooming. We have built a prototype tool and demonstrate its use in two application domains, UML class diagrams and biological networks.

The cutwidth of a graph G is defined to be the smallest integer k such that the vertices of G can be arranged in a vertex ordering [v 1 , . . . , v n ] in a way that, for every i = 1, . . . , n − 1, there are at most k edges with one endpoint in {v 1 , . . . , v i } and the other in {v i+1 , . . . , v n }. We examine the problem of computing in polynomial time the cutwidth of a partial w-tree with bounded degree. In particular, we show how to construct an algorithm that, in n O(w 2 d) steps, computes the cutwidth of any partial w-tree with vertices of degree bounded by a fixed constant d. Our algorithm is constructive in the sense that it can be adapted to output a corresponding optimal vertex ordering. Also, it is the main subroutine of an algorithm computing the pathwidth of a bounded degree partial w-tree with the same time complexity.

This paper presents a new algorithm for force directed graph layout on the GPU. The algorithm, whose goal is to compute layouts accurately and quickly, has two contributions. The first contribution is proposing a general multi-level scheme, which is based on spectral partitioning. The second contribution is computing the layout on the GPU. Since the GPU requires a data parallel programming model, the challenge is devising a mapping of a naturally unstructured graph into a well-partitioned structured one. This is done by computing a balanced partitioning of a general graph. This algorithm provides a general multi-level scheme, which has the potential to be used not only for computation on the GPU, but also on emerging multi-core architectures. The algorithm manages to compute high quality layouts of large graphs in a fraction of the time required by existing algorithms of similar quality. An application for visualization of the topologies of ISP (Internet Service Provider) networks is presented.

When exploiting the power of node-link diagrams to represent real-world data such as web structures, airline routes, electrical, telecommunication and social networks, link congestion frequently arises. Such areas in the diagram-with dense, overlapping links-are not readable: connectivity, node shapes, labels, and contextual information are obscured. In response, graph-layout research has begun to consider the modification of link shapes with techniques such as link routing and bundling. In this paper, we delve into the interactive techniques afforded by variant use of link curvature, delineating a six-dimensional design space that is populated by four families of interactive techniques: bundling, fanning, magnets, and legends. Our taxonomy encompasses existing techniques and reveals several novel link interactions. We describe the implementation of these techniques and illustrate their potential for exploring dense graphs with multiple types of links.

Background: We investigate the accuracy of different similarity approaches for clustering over two million biomedical documents. Clustering large sets of text documents is important for a variety of information needs and applications such as collection management and navigation, summary and analysis. The few comparisons of clustering results from different similarity approaches have focused on small literature sets and have given conflicting results. Our study was designed to seek a robust answer to the question of which similarity approach would generate the most coherent clusters of a biomedical literature set of over two million documents.

VISVIP allows web site developers and usability engineers to visualize the paths taken through the site by the subjects of usability experiments. They can dynamically customize and simplify the graphical layout of the web site, and select which subjects' paths to view. An animated representation of progress along the path through the web site is also available. The third dimension of the 3D display is used to represent the time spent on each page visit. The graph layout provided by VISVIP can be governed by either the web site topology or the intrinsic structure of the subject's path.

We present a semantics-driven contour tree visualization approach to support exploring and comparing cohort-level brain
white matter connectivities. A contour tree is a topological method that stores the nesting relationships of the contours in a scalar
field, here a brain water molecule movement measure of factional anisotropy (FA) values. Previous contour tree visualizations have
lacked the capability to effectively relate two-dimensional (2D) tree structures to the three-dimensional (3D) counterparts reflecting
the semantic structures. Our approach is semantics-driven in that contour trees are labeled with brain anatomical regions, thus are
stable in their structures for comparative studies. We further explore the contour tree approach as a tool for interactive exploration and
for comparative studies of patient and normal cohorts. Our approach is novel not only in summarizing the 3D topological structures
but also in showing spatial attributes associated with brain connectivity represented in fiber tracts, thus allowing brain scientists to
examine and compare critical differences between cohorts.

Many classical graph visualization algorithms have already been developed over the past decades. However, these algorithms face difficulties in practice, such as the overlapping node problem, large graph layout and dynamic graph layout. In order to solve these problems, this paper aims to systematically address algorithmic issues related to a novel framework that describes the process of graph visualization applications.

We show how to improve the Sugiyama scheme by using edge bundling. Our method modifies the layout produced by the Sugiyama scheme by bundling some of the edges together. The bundles are created by a new algorithm based on minimizing the total ink needed to draw the graph edges. We give several implementations that vary in quality of the resulting layout and execution time. To diminish the number of edge crossings inside of the bundles we apply a metroline crossing minimization technique. The method preserves the Sugiyama style of the layout and creates a more readable view of the graph.

This paper proposes an organized generalization of Newman and Girvan's modularity measure for graph clustering. Optimized via a deterministic annealing scheme, this measure produces topologically ordered graph clusterings that lead to faithful and readable graph representations based on clustering induced graphs. Topographic graph clustering provides an alternative to more classical solutions in which a standard graph clustering method is applied to build a simpler graph that is then represented with a graph layout algorithm. A comparative study on four real world graphs ranging from 34 to 1 133 vertices shows the interest of the proposed approach with respect to classical solutions and to self-organizing maps for graphs.

We present a new network diagram authoring tool, Dunnart, that provides continuous network layout. It continuously adjusts the layout in response to user interaction, while still maintaining the layout style and, where reasonable, the current layout topology. The diagram author uses placement constraints, such as alignment and distribution, to tailor the layout style and can guide the layout by repositioning diagram components or rerouting connectors. The key to the flexibility of our approach is the use of topology-preserving constrained graph layout.

Background: Biological networks are widely used to represent processes in biological systems and to capture interactions and dependencies between biological entities. Their size and complexity is steadily increasing due to the ongoing growth of knowledge in the life sciences. To aid understanding of biological networks several algorithms for laying out and graphically representing networks and network analysis results have been developed. However, current algorithms are specialized to particular layout styles and therefore different algorithms are required for each kind of network and/or style of layout. This increases implementation effort and means that new algorithms must be developed for new layout styles. Furthermore, additional effort is necessary to compose different layout conventions in the same diagram. Also the user cannot usually customize the placement of nodes to tailor the layout to their particular need or task and there is little support for interactive network exploration.

The Unified Modeling Language (UML) has become the software industry's standard notation for representing software architecture and design models. UML diagrams play an important role in the engineering and re-engineering processes of software systems. Of particular interest from the Graph Drawer's perspective are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combination of hierarchical and non-hierarchical relations poses a special challenge to a graph layout tool. We present an implementation of our technology within well-known modelling tools.

Social Network Analysis (SNA) has evolved as a popular, standard method for modeling meaningful, often hidden structural relationships in communities. Existing SNA tools often involve extensive pre-processing or intensive programming skills that can challenge practitioners and students alike. NodeXL, an open-source template for Microsoft Excel, integrates a library of common network metrics and graph layout algorithms within the familiar spreadsheet format, offering a potentially low-barrierto-entry framework for teaching and learning SNA. We present the preliminary findings of 2 user studies of 21 graduate students who engaged in SNA using NodeXL. The majority of students, while information professionals, had little technical background or experience with SNA techniques. Six of the participants had more technical backgrounds and were chosen specifically for their experience with graph drawing and information visualization. Our primary objectives were (1) to evaluate NodeXL as an SNA tool for a broad base of users and (2) to explore methods for teaching SNA. Our complementary dual case-study format demonstrates the usability of NodeXL for a diverse set of users, and significantly, the power of a tightly integrated metrics/visualization tool to spark insight and facilitate sensemaking for students of SNA.

Most Semantic Web data visualization tools structure the representation according to the concept definitions and interrelations that constitute the ontology's vocabulary. Instances are often treated as somewhat peripheral information, when considered at all. These instances, that populate ontologies, represent an essential part of any knowledge base, and are often orders of magnitude more numerous than the concept definitions that give them machine-processable meaning. We present a visualization technique designed to enable users to visualize large instance sets and the relations that connect them. This hybrid visualization uses both node-link and adjacency matrix representations of graphs to visualize different parts of the data depending on their semantic and local structural properties, exploiting ontological knowledge to drive the graph layout. The representation is embedded in an environment that features advanced interaction techniques for easy navigation, including support for smooth continuous zooming and coordinated views.

Social Network Analysis (SNA) has evolved as a popular, standard method for modeling meaningful, often hidden structural relationships in communities. Existing SNA tools often involve extensive pre-processing or intensive programming skills that can challenge practitioners and students alike. NodeXL, an open-source template for Microsoft Excel, integrates a library of common network metrics and graph layout algorithms within the familiar spreadsheet format, offering a potentially low-barrierto-entry framework for teaching and learning SNA. We present the preliminary findings of 2 user studies of 21 graduate students who engaged in SNA using NodeXL. The majority of students, while information professionals, had little technical background or experience with SNA techniques. Six of the participants had more technical backgrounds and were chosen specifically for their experience with graph drawing and information visualization. Our primary objectives were (1) to evaluate NodeXL as an SNA tool for a broad base of users and (2) to explore methods for teaching SNA. Our complementary dual case-study format demonstrates the usability of NodeXL for a diverse set of users, and significantly, the power of a tightly integrated metrics/visualization tool to spark insight and facilitate sensemaking for students of SNA.

Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods that help to analyze the validity of these models. Here, we focus on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; thus they are huge and high dimensional. Their most salient features, however, are encapsulated by their so-called barrier tree that reflects the local minima and their connecting saddle points. For each length of the growing RNA chain there exists a folding landscape. We visualize the sequence of folding landscapes by an animation of the corresponding barrier trees. To generate the animation, we adapt the foresight layout with tolerance algorithm for general dynamic graph layout problems. Since it is very general, we give a detailed description of each phase: constructing a supergraph for the trees, layout of that supergraph using a modified DOT algorithm, and presentation techniques for the final animation.

A standard approach to large network visualization is to provide an overview of the network and a detailed view of a small component of the graph centred around a focal node. The user explores the network by changing the focal node in the detailed view or by changing the level of detail of a node or cluster. For scalability, fast force-based layout algorithms are used for the overview and the detailed view. However, using the same layout algorithm in both views is problematic since layout for the detailed view has different requirements to that in the overview. Here we present a model in which constrained graph layout algorithms are used for layout in the detailed view. This means the detailed view has high-quality layout including sophisticated edge routing and is customisable by the user who can add placement constraints on the layout. Scalability is still ensured since the slower layout techniques are only applied to the small subgraph shown in the detailed view. The main technical innovations are techniques to ensure that the overview and detailed view remain synchronized, and modifying constrained graph layout algorithms to support smooth, stable layout. The key innovation supporting stability are new dynamic graph layout algorithms that preserve the topology or structure of the network when the user changes the focus node or the level of detail by in situ semantic zooming. We have built a prototype tool and demonstrate its use in two application domains, UML class diagrams and biological networks.

The typical use of force-directed layout is to create organiclooking, straight-edge drawings of large graphs while combinatorial techniques are generally preferred for high-quality layout of small to medium sized graphs. In this paper we integrate edge-routing techniques into a force-directed layout method based on constrained stress majorisation. Our basic procedure takes an initial layout for the graph, including polyline paths for the edges, and improves this layout by moving the nodes to reduce stress and moving edge bend points to straighten the edges and reduce their overall length. Separation constraints between nodes and edge bend points are used to ensure that nodes do not overlap edges or other nodes and that no additional edge crossings are introduced.

Food webs describe who eats whom among species within a habitat. These networks depict interlinked food chains and have long been a central ecological paradigm for studying biocomplexity. As food-web data encompasses more species, the network structure of the feeding links among those species results in a highly connected network that is difficult to visualize in 2D. Such simple representations often obscure important details critical for understanding the structure and function of ecosystems. Graph layout is thus critically important for enhanced understanding of complex food webs. In this paper, we describe 3D visualizations of food webs that use intuitive node placement, minimal edge-crossing and link length, hierarchical node aggregation, and analysis tool integration. We also describe our deployment of "Webs on the Web" (WoW) visualization and analysis tools on the WWW. These tools will facilitate food-web research, collaboration, and education. Given the variety of 3D visualization technologies on the WWW, our WoW visualization pipeline supports diverse formats and adapts to users' preferences by employing XML and a flexible architecture. Currently, WoW uses XML to semantically markup food-web data (FoodWebML), extracts visual information into X3D format, and then uses the information to create a direct display on VRML/X3D browsers or feed a Shockwave 3D visualization.

Packages that support the creation of pathway diagrams are limited by their inability to be readily extended to new classes of pathway-related data. Results: VitaPad is a cross-platform application that enables users to create and modify biological pathway diagrams and incorporate microarray data with them. It improves on existing software in the following areas: (i) It can create diagrams dynamically through graph layout algorithms; (ii) It is open-source and uses an open XML format to store data, allowing for easy extension or integration with other tools; and (iii) It features a cuttingedge user interface with intuitive controls, high-resolution graphics and fully customizable appearance.

We present MoleView, a novel technique for interactive exploration of multivariate relational data. Given a spatial embedding of the data, in terms of a scatter plot or graph layout, we propose a semantic lens which selects a specific spatial and attribute-related data range. The lens keeps the selected data in focus unchanged and continuously deforms the data out of the selection range in order to maintain the context around the focus. Specific deformations include distance-based repulsion of scatter plot points, deforming straight-line node-link graph drawings, and as varying the simplification degree of bundled edge graph layouts. Using a brushing-based technique, we further show the applicability of our semantic lens for scenarios requiring a complex selection of the zones of interest. Our technique is simple to implement and provides real-time performance on large datasets. We demonstrate our technique with actual data from air and road traffic control, medical imaging, and software comprehension applications.

The problem of graph layout and drawing is fundamental to many aproaches to the visualization of relational information structures. As the data set grows, the visualization problem is compounded by the need to reconcile the user's need for orientation cues with the danger of information overload. Put simply: How can we limit the number of visual elements on the screen so as not to overwhelm the user yet retain enough information that the user is able to navigate and explore the data set confidently? How can we provide orientational cues so that a user can understand the location of the current viewpoint in a large data set? These are problems inherent not only to graph drawing but information visualization in general. We propose a method which extracts the significant features of a directed acyclic graph as the basis for navigation 1 .

This paper introduces an automatic procedure to assist on the interpretation of a large dataset when a similarity metric is available. We propose a visualization approach based on a graph layout methodology that uses a Quadratic Assignment Problem (QAP) formulation. The methodology is presented using as testbed a time series dataset of the Standard & Poor's 100, one the leading stock market indicators in the United States. A weighted graph is created with the stocks represented by the nodes and the edges' weights are related to the correlation between the stocks' time series. A heuristic for clustering is then proposed; it is based on the graph partition into disconnected subgraphs allowing the identification of clusters of highly-correlated stocks. The final layout corresponds well with the perceived market notion of the different industrial sectors. We compare the output of this procedure with a traditional dendogram approach of hierarchical clustering.

The increasing complexity of models for prediction of the native spatial structure of RNA molecules requires visualization methods that help to analyze and understand the models and their predictions. This paper improves the visualization method for sequences of barrier trees previously published by the authors. The barrier trees of these sequences are rough topological simplifications of changing folding landscapes -energy landscapes in which kinetic folding takes place. The folding landscapes themselves are generated for RNA molecules where the number of nucleotides increases. Successive landscapes are thus correlated and so are the corresponding barrier trees. The landscape sequence is visualized by an animation of a barrier tree that changes with time.

Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods that help to analyze the validity of these models. Here, we focus on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; thus they are huge and high dimensional. Their most salient features, however, are encapsulated by their so-called barrier tree that reflects the local minima and their connecting saddle points. For each length of the growing RNA chain there exists a folding landscape. We visualize the sequence of folding landscapes by an animation of the corresponding barrier trees. To generate the animation, we adapt the foresight layout with tolerance algorithm for general dynamic graph layout problems. Since it is very general, we give a detailed description of each phase: constructing a supergraph for the trees, layout of that supergraph using a modified DOT algorithm, and presentation techniques for the final animation.

Many graph drawing and visualization algorithms, such as force-directed layout and line-dot rendering, work very well on relatively small and sparse graphs. However, they often produce extremely tangled results and exhibit impractical running times for highly non-planar graphs with large edge density. And very few graph layout algorithms support dynamic time-varying graphs; applying them independently to each frame produces distracting temporally incoherent visualizations. We have developed a new visualization technique based on a novel approach to hierarchically structuring dense graphs via stratification. Using this structure, we formulate a hierarchical force-directed layout algorithm that is both efficient and produces quality graph layouts. The stratification of the graph also allows us to present views of the data that abstract away many small details of its structure. Rather than displaying all edges and nodes at once, resulting in a convoluted rendering, we present an interactive tool that filters edges and nodes using the graph hierarchy and allows users to drill down into the graph for details. Our layout algorithm also accommodates time-varying graphs in a natural way, producing a temporally coherent animation that can be used to analyze and extract trends from dynamic graph data. For example, we demonstrate the use of our method to explore financial correlation data for the U.S. stock market in the period from 1990 to 2005. The user can easily analyze the time-varying correlation graph of the market, uncovering information such as market sector trends, representative stocks for portfolio construction, and the interrelationship of stocks over time.