2014 IEEE Symposium on Visual Languages and Human-Centric Computing (VL/HCC)
Properties of Euler Diagrams and Graphs in Combination Mithileysh Sathiyanarayanan, Gem Stapleton, Jim Burton and John Howse University of Brighton, UK Email:m.sathiyanarayanan, g.e.stapleton, j.burton,
[email protected]
Abstract—Euler diagrams and graphs are used as visualisations individually in a large variety of application areas such as network analysis, medicine and engineering. Existing methods which combine both Euler diagrams and graphs such as Bubble Sets and Euler View provide somewhat limited results with suboptimal layout. In particular, they do not produce diagrams that are known to be most effective for performing user-driven tasks. That said, our knowledge is rather limited about what constitutes an effective layout for Euler diagrams and graphs in combination. Our ultimate aim is to automatically visualise large networks in an effective manner. To produce effective layouts, we need to identify properties that may correlate with effective layouts of Euler diagrams combined with graphs. Such properties are considered in this paper. In future, empirical studies will be conducted to inform and validate the combined properties.
I. I NTRODUCTION Euler diagrams have frequently appeared at VL/HCC, with contributions ranging from theoretical investigations into logics based on them to automated layout techniques. This topical content will be of interest to the VL/HCC community because we combine two existing visual languages (Euler diagrams and graphs), so that grouped network data such as social networks, ontologies etc. can be visualized effectively. In order to draw Euler diagrams to best effect, their topological properties [5] (such as curve simplicity) can be controlled [2]; sometimes these are called well-formedness properties. An Euler diagram with desirable topological properties is likely to be more understandable than one without those properties [4]. Euler diagram generation techniques typically aim to produce layouts which exhibit all, or most, of the desirable properties. Geometric properties are also known to impact on comprehension [1], but they are not our focus due to space limitations. In Fig. 1 (a), we illustrate which desirable properties are not possessed. The lefthand diagram exhibits a non-simple curve (labelled P ) and there is a duplicated curve label (S). A triple point (3-point) exists between curves P , Q and R and there is concurrency between Q and R. The righthand diagram exhibits a brushing point where P and Q meet. There is a disconnected zone; a zone is a region that is inside some set of contours (a contour is the largest set of curves with some given label) and outside the rest of the contours. The zone inside the contours Q and S is disconnected. Lastly, this diagram exhibits disconnected basic regions – a new (sixth) topological property that we introduce here in this paper. A basic region is inside some set of curves and outside the rest of the curves. The zone inside Q and S comprises two connected
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basic regions whereas the disconnected zone inside just R is also a disconnected basic region. Likewise, the topological properties possessed by graphs are often controlled. In fact, the topological properties of Euler diagrams have analogies in graphs. For example, in Fig. 1 (b), there are two edges crossing each other: this is a 2-point and graph layout algorithms often attempt to avoid edge crossings. Likewise, non-simple edges tend to be avoided; here there is a non-simple edge between d and h. Vertices b and c brush each other. Edges could also brush each other, potentially making it hard to trace the paths followed by the brushing edges; in this example the edge between b and f brushes that between c and d. Also, the edge between vertices e and h runs concurrently, in part, with the edge between g and f . Topological properties of graphs are generally considered when aiming to produce a layout that possesses certain properties which help humans to understand the resulting diagrams [3]. We extend the topological properties to Euler diagrams and graphs drawn in combination. This combined notation has the potential to be a powerful technique for visualizing and interpreting data. Since social systems, for instance, have grown in complexity, data analysts or users could be aided in understanding sets (represented by Euler diagrams), data items (represented by graph vertices) and binary relationships between data items (represented by graph edges). In addition, many diagrammatic logics (like spider diagrams and Euler/Venn diagrams) augment Euler diagrams with graphs. This, in part, motivates our provision of new properties for Euler diagrams and graphs in combination, which will be presented as a poster. II. E ULER D IAGRAMS AND G RAPHS IN C OMBINATION All the topological properties of Euler diagrams and graphs apply to the combined notation and here we provide some new properties. Given that user comprehension is affected for the individual notations, the properties possessed by visualizations that combine Euler diagrams with graphs will almost certainly affect user comprehension too. So when drawing Euler diagrams with graphs, we want to produce a layout that possesses certain desirable properties. A first step towards this is to define topological properties for the combined notation, that reflect how Euler diagrams and graphs ‘interact’. The topological properties are inspired by those for the individual notations and are now presented, beginning with the most simple. Throughout what follows, we make reference to Fig. 2.
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(a)
(b)
Fig. 1. Topological properties: (a) Euler diagrams; (b) graphs
Fig. 2. New topological properties
No curve and vertex duplicated labels A potential source of confusion is from labels used for curves and vertices. Using a label for both a set and an element in a set is, therefore, not necessarily sensible. Thus, our first new property is that the labels of the curves and labels of the vertices are distinct. In Fig.2, the label P is duplicated. No concurrency between curves and edges Avoiding concurrency between curves is desirable for Euler diagrams which suggests that concurrency between a curve and an edge should also be avoided. Thus, our second new property is that curves and edges do not run concurrently. Fig. 2 has concurrency between R and the edge between vertices e and f . No n-points between curves and edges In Euler diagrams, n-points are passed through n or more times by the curves. Similarly, for graphs n points relate to the edges. At n-points (n > 2 for Euler diagrams and n > 1 for graphs) it can be difficult to follow the route taken by each curve or edge. Setting n = 3 for Euler diagrams means that the reader only has to determine the paths followed by up to two curves at any particular point. When an Euler diagram and a graph interact, the problem with tracing curves and edges that pass though any given point is compounded. The third new property is that 3-points between curves and edges should be avoided. Fig. 2 exhibits a 3-point between curves P and Q and the edge between vertices c and d. No brushing points between diagrammatic elements Similarly, brushing points should be avoided, because of the potential for confusion about the routes followed by curves and edges. In Fig. 2, there is a brushing point between the curve P and the edge between a and e. No edges disconnecting zones It is possible that one or more of the edges in a graph cause zones in the underlying Euler diagram to become disconnected. Disconnected zones
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Fig. 3. Layout choices
are known to be one of the worst topological properties to break from a user comprehension perspective, as is multiple label use. Thus, our fifth new property is that zones are not disconnected by the graph. No edges disconnecting basic regions Our last property is that basic regions are not disconnected by the graph. In Fig. 2, the edges incident with d disconnect basic regions (and zones). III. C ONCLUSION When devising automated layout tools for Euler diagrams and graphs in combination, topological properties will impact on user comprehension. As with the individual notations, there are choices of layout that have different sets of properties. For example in Fig. 3, in the lefthand diagram, due to a ‘bad’ choice of curves, some of the desirable topological properties are broken: the edge between vertices a and b disconnects some zones and basic regions, but this is avoided in the righthand diagram. This avoidance is possible because the zones which contain the vertices are topologically adjacent in the righthand diagram (but not in the lefthand diagram). Ideally, layout tools will ensure that the desirable properties all hold. However, just as with the individual notations, sometimes the properties must be broken. It is an interesting challenge to develop layout methods that ensure desirable properties are exhibited by the resulting Euler diagrams and graphs. R EFERENCES [1] F. Benoy, P. Rodgers. Evaluating the comprehension of Euler diagrams. In 11th Int. Conf. on Information Visualization, pp. 771–778, IEEE, 2007. [2] A. Fish, B. Khazaei, C. Roast. User-comprehension of Euler diagrams. J. Visual Languages and Computing, pp. 340–354, 2011. [3] H. Purchase. Metrics for graph drawing aesthetics. J. Visual Languages and Computing, pp. 501–516, 2002. [4] G. Stapleton, P. Rodgers, J. Howse, J. Taylor. Properties of Euler diagrams. In Layout of Software Eng. Diagrams, pp. 2–16. EASST, 2007.
