There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A compound graph is a frequently encountered type of data set. Relations are given between items, and a hierarchy is defined on the items as well. We present a new method for visualizing such compound graphs. Our approach is based on visually bundling the adjacency edges, i.e., non-hierarchical edges, together. We realize this as follows. We assume that the hierarchy is shown via a standard tree visualization method. Next, we bend each adjacency edge, modeled as a B-spline curve, toward the polyline defined by the path via the inclusion edges from one node to another. This hierarchical bundling reduces visual clutter and also visualizes implicit adjacency edges between parent nodes that are the result of explicit adjacency edges between their respective child nodes. Furthermore, hierarchical edge bundling is a generic method which can be used in conjunction with existing tree visualization techniques. We illustrate our technique by providing example visualizations and discuss the results based on an informal evaluation provided by potential users of such visualizations

Hierarchical Edge Bundles: Visualization of Adjacency Relations in Hierarchical Data

Graphs depicted as node-link diagrams are widely used to show relationships between entities. However, nodelink diagrams comprised of a large number of nodes and edges often suffer from visual clutter. The use of edge bundling remedies this and reveals high-level edge patterns. Previous methods require the graph to contain a hierarchy for this, or they construct a control mesh to guide the edge bundling process, which often results in bundles that show considerable variation in curvature along the overall bundle direction. We present a new edge bundling method that uses a self-organizing approach to bundling in which edges are modeled as flexible springs that can attract each other. In contrast to previous methods, no hierarchy is used and no control mesh. The resulting bundled graphs show significant clutter reduction and clearly visible high-level edge patterns. Curvature variation is furthermore minimized, resulting in smooth bundles that are easy to follow. Finally, we present a rendering technique that can be used to emphasize the bundling.

Force-directed edge bundling for graph visualization

Graphs have been widely used to model relationships among data. For large graphs, excessive edge crossings make the display visually cluttered and thus difficult to explore. In this paper, we propose a novel geometry-based edge-clustering framework that can group edges into bundles to reduce the overall edge crossings. Our method uses a control mesh to guide the edge-clustering process; edge bundles can be formed by forcing all edges to pass through some control points on the mesh. The control mesh can be generated at different levels of detail either manually or automatically based on underlying graph patterns. Users can further interact with the edge-clustering results through several advanced visualization techniques such as color and opacity enhancement. Compared with other edge-clustering methods, our approach is intuitive, flexible, and efficient. The experiments on some large graphs demonstrate the effectiveness of our method.

/pdf/geometry-based-edge-clustering-for-graph-visualization-3o468sfme6.pdf

Geometry-Based Edge Clustering for Graph Visualization

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.

/pdf/skeleton-based-edge-bundling-for-graph-visualization-kx9v9iqoc9.pdf

Skeleton-Based Edge Bundling for Graph Visualization

We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw, in a crossing-free manner, graphs—such as software interaction diagrams—that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as “tracks” (similar to train tracks). Producing such confluent diagrams automatically from a graph with many crossings is quite challenging, however, so we offer two heuristic algorithms to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.

/pdf/confluent-drawings-visualizing-non-planar-diagrams-in-a-17of57sbcs.pdf

Confluent drawings: visualizing non-planar diagrams in a planar way.

We combine the idea of confluent drawings with Sugiyama-style drawings in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed-style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge-coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.

/pdf/confluent-layered-drawings-3c2mxrw1e9.pdf

Confluent Layered Drawings

We generalize the tree-confluent graphs to a broader class of graphs called Δ-confluent graphs. This class of graphs and distance-hereditary graphs, a well-known class of graphs, coincide. Some results about the visualization of Δ-confluent graphs are also given.

/pdf/delta-confluent-drawings-2i12zk72st.pdf

Delta-confluent drawings

We introduce a new approach for drawing diagrams. Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way. This approach allows us to draw, in a crossing-free manner, graphs-such as software interaction diagrams-that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as tracks (similar to train tracks). Producing such confluent diagrams automatically from a graph with many crossings is quite challenging, however, so we offer two heuristic algorithms to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.

Jeremy Yu Meng

Papers

Confluent drawings: visualizing non-planar diagrams in a planar way.

Confluent Layered Drawings

Delta-confluent drawings

Confluent drawings: Visualizing non-planar diagrams in a planar way

Delta-confluent drawings