We extend the popular force-directed approach to network (or graph) layout to allow separation constraints, which enforce a minimum horizontal or vertical separation between selected pairs of nodes. This simple class of linear constraints is expressive enough to satisfy a wide variety of application-specific layout requirements, including: layout of directed graphs to better show flow; layout with non-overlapping node labels; and layout of graphs with grouped nodes (called clusters). In the stress majorization force-directed layout process, separation constraints can be treated as a quadratic programming problem. We give an incremental algorithm based on gradient projection for efficiently solving this problem. The algorithm is considerably faster than using generic constraint optimization techniques and is comparable in speed to unconstrained stress majorization. We demonstrate the utility of our technique with sample data from a number of practical applications including gene-activation networks, terrorist networks and visualization of high-dimensional 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.

Abstract. Constrained graph layout is a recent generalisation of forcedirected graph layout which allows constraints on node placement. We give a constrained graph layout algorithm that takes an initial feasible layout and improves it while preserving the topology of the initial layout. The algorithm supports poly-line connectors and clusters. During layout the connectors and cluster boundaries act like impervious rubber-bands which try to shrink in length. The intended application for our algorithm is dynamic graph layout, but it can also be used to improve layouts generated by other graph layout techniques. 1

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Constrained stress majorization is a promising new technique for integrating application specific layout constraints into forcedirected graph layout. We significantly improve the speed and convergence properties of the constrained stress-majorization technique for graph layout by employing a diagonal scaling of the stress function. Diagonal scaling requires the active-set quadratic programming solver used in the projection step to be extended to handle separation constraints with scaled variables, i.e. of the form siyi + gij ≤ sjyj. The changes, although relatively small, are quite subtle and explained in detail.

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Abstract – The challenge of finding documents relevant to a task is pervasive in today’s information rich society. Current methods for finding documents focus on keyword search using resources such as Internet search engines or digital libraries provided by professional organizations. Visual representations can augment the search for information, yet the vastness of sources must be dealt with in a way that makes these large bodies of documents comprehensible and individual documents identifiable. The current work describes a system that provides a means to organize documents through graph based clustering and allows the user to display and interact with an abstracted multilevel visual display of information. The clustering techniques used by the system to provide abstracted views exploit the organizational characteristics of document collections and web based sources to supply users a seamless representation useful for overview, document identification, and document retrieval.