We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the bestknown force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted.

The vertices of most graphs that appear in real applications are nonuniform. They can be circles, ellipses, rectangles, or other geometric elements of varying shapes and sizes. Unfortunately, current force directed methods for laying out graphs are suitable mostly for graphs whose vertices are zero-sized and dimensionless points. It turns out that naively extending these methods to handle nonuniform vertices results in serious deficiencies in terms of output quality and performance. In this paper we try to remedy this situation by identifying the special characteristics and problematics of such graphs and offering several algorithms for tackling them. The algorithms can be viewed as carefully constructed extensions of force-directed methods, and their output quality and performance are similar.

We discuss several criteria for clustering graphs, and identify two criteria which are not biased towards certain cluster sizes: the node-normalized cut (also called cut ratio) and the edge-normalized cut. We present two energy models whose minimum energy drawings reveal clusters with respect to these criteria.

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.

The Unified Modelling Language (UML) is a visual language for modelling software designs and is currently the most widely accepted standard for software diagrams in the software engineering field. Force Directed Layout algorithms are methods for arranging the elements of a graph in an aesthetically pleasing and clear manner.

Graphs are ubiquitous, finding applications in domains ranging from software engineering to computational biology. While graph theory and graph algorithms are some of the oldest, most studied fields in computer science, the problem of visualizing graphs is comparatively young. This problem, known as graph drawing, is that of transforming combinatorial graphs into geometric drawings for the purpose of visualization. Most published algorithms for drawing general graphs model the drawing problem with a physical analogy, representing a graph as a system of springs and other physical elements and then simulating the relaxation of this physical system. Solving the graph drawing problem involves both choosing a physical model and then using numerical optimization to simulate the physical system. In this

We introduce energy models for drawing clustered small-world graphs.

Abstract. When drawing graphs whose nodes contain text or graphics, the non-trivial node sizes must be taken into account, either as part of the initial layout or as a post-processing step. The core problem is to avoid overlaps while retaining the structural information inherent in a layout using little additional area. This paper presents a new node overlap removal algorithm that does well by these measures. 1

Abstract. Several hierarchical clustering techniques have been proposed to visualize large graphs, but fewer solutions suggest a focus based approach. We propose a multilevel clustering technique that produces in linear time a contextual clustered view depending on a user-focus. We get a tree of clusters where each cluster- called meta-silhouette- is itself hierarchically clustered into an inclusion tree of silhouettes. Resulting Multilevel Silhouette Tree (MuSi-Tree) has a specific structure called multilevel compound tree. This work builds upon previous work on a compound tree structure called MO-Tree. The work presented in this paper is a major improvement over previous work by (1) defining multilevel compound tree as a more generic structure, (2) proposing original space-filling visualization techniques to display it, (3) defining relevant interaction model based on both focus changes and graph filtering techniques and (4) reporting from case studies in various fields: co-citation graphs, relateddocument graphs and social graphs. 1

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