The analysis of large graphs plays a prominent role in various fields of research and is relevant in many important application areas. Effective visual analysis of graphs requires appropriate visual presentations in combination with respective user interaction facilities and algorithmic graph analysis methods. How to design appropriate graph analysis systems depends on many factors, including the type of graph describing the data, the analytical task at hand, and the applicability of graph analysis methods. The most recent surveys of graph visualization and navigation techniques were presented by Herman et al. [HMM00] and Diaz [DPS02]. The first work surveyed the main techniques for visualization of hierarchies and graphs in general that had been introduced until 2000. The second work concentrated on graph layouts introduced until 2002. Recently, new techniques have been developed covering a broader range of graph types, such as time-varying graphs. Also, in accordance with ever growing amounts of graph-structured data becoming available, the inclusion of algorithmic graph analysis and interaction techniques becomes increasingly important. In this State-of-the-Art Report, we survey available techniques for the visual analysis of large graphs. Our review firstly considers graph visualization techniques according to the type of graphs supported. The visualization techniques form the basis for the presentation of interaction approaches suitable for visual graph exploration. As an important component of visual graph analysis, we discuss various graph algorithmic aspects useful for the different stages of the visual graph analysis process.

The Sugiyama framework is the most commonly used concept for visualizing directed graphs. It draws them in a hierarchical way and operates in four phases: cycle removal, leveling, crossing reduction, and coordinate assignment. However, there are situations where cycles must be displayed as such, e. g., distinguished cycles in the biosciences and scheduling processes which repeat in a daily or weekly turn. This excludes the removal of cycles. In their seminal paper Sugiyama et al. introduced recurrent hierarchies as a concept to draw graphs with cycles. However, this concept has not received much attention in the following years. In this paper we supplement our cyclic Sugiyama framework and investigate the coordinate assignment phase. We provide an algorithm which runs in linear time and constructs drawings which have at most two bends per edge and use quadratic area.

Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving one-sided 2-level crossing minimization problems, which are repeated in up and down sweeps over all levels. Crossing minimization problems are generally N P-hard. We introduce a global crossing reduction, which at any particular time considers all crossings between all levels. Our approach is based on the sifting technique. It yields an improvement of 5 – 10 % in the number of crossings over the level-by-level one-sided 2-level crossing reduction heuristics. In addition, it avoids type 2 conflicts which are crossings between edges whose endpoints are dummy vertices. This helps straightening long edges spanning many levels. Finally, the global crossing reduction approach can directly be extended to cyclic, radial, and clustered level graphs achieving similar improvements. The running time is quadratic in the size of the input graph, whereas the common level-by-level approaches are faster but operate on larger graphs with many dummy vertices for long edges. Submitted:

Directed graphs are generally drawn as level drawings using the hierarchical approach. Such drawings are constructed by a framework of algorithms which operates in four phases: cycle removal, leveling, crossing reduction, and coordinate assignment. However, there are situations where cycles should be displayed as such, e. g., distinguished cycles in the biosciences and scheduling processes repeating in a daily or weekly turn. In their seminal paper on hierarchical drawings Sugiyama et al. [31] also introduced recurrent hierarchies. This concept supports the drawing of cycles and their unidirectional display. However, it had not been investigated. In this paper we complete the cyclic approach and investigate the coordinate assignment phase. The leveling and the crossing reduction for recurrent hierarchies have been studied in the companion papers [3, 4]. We provide an algorithm which runs in linear time and constructs an intermediate drawing with at most two bends per edge and aligned edge segments in an area of quadratic width times the preset number of levels height. This area bound is optimal for such drawings. Our approach needs new techniques for solving cyclic dependencies, such as skewing edges and cutting components. The drawings can be transformed into 2D drawings displaying all cycles counterclockwise around a center and into 3D drawings winding the cycles around a cylinder. Submitted:

... indeed one of the most important drawing methods for graphs. It places the vertices on parallel level lines and attempts to map the directions of the edges to a uniform geometric direction, e. g., from top to bottom. Then the resulting drawing visualizes a common direction of information flow stored by the structure of the input graph. In this thesis we generalize the traditional hierarchic drawing style in two ways: radial drawings with level lines forming concentric cycles and cyclic drawings with levels forming a star, i. e., recurrent hierarchies. Further, we allow edges between vertices within the same level, which often occur in practice. Our main results are a complete framework for both layout conventions and a major enhancement of the reduction of edge crossings using a new global optimization method. This approach also upgrades traditional horizontal level drawings. Applications of (drawings of) general level graphs include for example state or flow charts, schedules, UML class diagrams, entity relationship diagrams, and biochemical pathways. Radial drawings are especially useful to visualize social or policy networks, i. e., to map structural centrality like importance or role of the vertices/actors on geometric

Directed graphs are generally drawn as level drawings using the hierarchical approach. Such drawings are constructed by a framework of algorithms which operates in four phases: cycle removal, leveling, crossing reduction, and coordinate assignment. However, there are situations where cycles should be displayed as such, e. g., distinguished cycles in the biosciences and scheduling processes repeating in a daily or weekly turn. In their seminal paper on hierarchical drawings Sugiyama et al. [31] also introduced recurrent hierarchies. This concept supports the drawing of cycles and their unidirectional display. However, it had not been investigated. In this paper we complete the cyclic approach and investigate the coordinate assignment phase. The leveling and the crossing reduction for recurrent hierarchies have been studied in the companion papers [3, 4]. We provide an algorithm which runs in linear time and constructs an intermediate drawing with at most two bends per edge and aligned edge segments in an area of quadratic width times the preset number of levels height. This area bound is optimal for such drawings. Our approach needs new techniques for solving cyclic dependencies, such as skewing edges and cutting components. The drawings can be transformed into 2D drawings displaying all cycles counterclockwise around a center and into 3D drawings winding the cycles around a cylinder. Submitted: