The Visual Display of Quantitative Information

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 cover techniques that had been introduced until 2000 or concentrate only on graph layouts published until 2002. Recently, new techniques have been developed covering a broader range of graph types, such as timevarying 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 first 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. We also present main open research challenges in this field.

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Visual analysis of large graphs : state-of-the-art and future research challenges

Background
Recent advances with high-throughput methods in life-science research have increased the need for automatized data analysis and visual exploration techniques. Sophisticated bioinformatics tools are essential to deduct biologically meaningful interpretations from the large amount of experimental data, and help to understand biological processes.

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VANTED: A system for advanced data analysis and visualization in the context of biological networks

Networks are typically visualized with force-based or spectral layouts. These algorithms lack reproducibility and perceptual uniformity because they do not use a node coordinate system. The layouts can be difficult to interpret and are unsuitable for assessing differences in networks. To address these issues, we introduce hive plots (http://www.hiveplot.com) for generating informative, quantitative and comparable network layouts. Hive plots depict network structure transparently, are simple to understand and can be easily tuned to identify patterns of interest. The method is computationally straightforward, scales well and is amenable to a plugin for existing tools.

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Hive plots—rational approach to visualizing networks

Summary: MAVisto is a tool for the exploration of motifs in biological networks. It provides a flexible motif search algorithm and different views for the analysis and visualization of network motifs. These views help to explore interesting motifs: the frequency of motif occurrences can be compared with randomized networks, a list of motifs along with information about structure and number of occurrences depending on the reuse of network elements shows potentially interesting motifs, a motif fingerprint reveals the overall distribution of motifs of a given size and the distribution of a particular motif in the network can be visualized using an advanced layout algorithm.

Availability: MAVisto is platform independent and available free of charge as a Java webstart application at http://mavisto.ipk-gatersleben.de/

Contact: schwoebb@ipk-gatersleben.de

Supplementary information: Can be found at http://mavisto.ipk-gatersleben.de/

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MAVisto: a tool for the exploration of network motifs

A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is $${NP}$$NP-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph $$G$$G is o1p, then the algorithm computes an embedding and can augment $$G$$G to a maximal o1p graph. Otherwise, $$G$$G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size $$n$$n has at most $$\frac{5}{2} n - 4$$52n-4 edges and there are maximal o1p graphs with $$\frac{11}{5} n - \frac{18}{5}$$115n-185 edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in $$\fancyscript{O}(n^2)$$O(n2) area with all vertices in the outer face, a planar visibility representation in $$\fancyscript{O}(n \log n)$$O(nlogn) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time.

Outer 1-Planar Graphs

Every planar graph has a concentric representation based on a breadth first search, see [21]. The vertices are placed on concentric circles and the edges are routed as curves without crossings. Here we take the opposite view. A graph with a given partitioning of its vertices onto k concentric circles is k-radial planar, if the edges can be routed monotonic between the circles without crossings. Radial planarity is a generalisation of level planarity, where the vertices are placed on k horizontal lines. We extend the technique for level planarity testing of [18,17,15,16,12,13] and show that radial planarity is decidable in linear time, and that a radial planar embedding can be computed in linear time.

Radial Level Planarity Testing and Embedding in Linear Time

NIC-Planar Graphs

A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. 
We characterize embeddings of maximal NIC-planar graphs in terms of generalized planar dual graphs. The characterization is used to derive tight bounds on the density of maximal NIC-planar graphs which ranges between 3.2(n-2) and 3.6(n-2). Further, we prove that optimal NIC-planar graphs with 3.6(n-2) edges have a unique embedding and can be recognized in linear time, whereas the general recognition problem of NIC-planar graphs is NP-complete. In addition, we show that there are NIC-planar graphs that do not admit right angle crossing drawings, which distinguishes NIC-planar from IC-planar graphs.

We present linear-time algorithms for drawing phylogenetic trees in radial and circular representations. In radial drawings given edge lengths (representing evolutionary distances) are preserved, but labels (names of taxons represented in the leaves) need to be adjusted, whereas in circular drawings labels are perfectly spread out, but edge lengths adjusted. Our algorithms produce drawings that are unique solutions to reasonable criteria and assign to each subtree a wedge of its own. The linear running time is particularly interesting in the circular case, because our approach is a special case of Tutte's barycentric layout algorithm involving the solution of a system of linear equations.

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Christian Bachmaier

Papers

Outer 1-Planar Graphs

Radial Level Planarity Testing and Embedding in Linear Time

NIC-Planar Graphs

NIC-Planar Graphs

Drawing phylogenetic trees