Effects of Crossing Angles

Transcription

1 Effects of Crossing Angles Weidong Huang Seok-Hee Hong Peter Eades School of Information Technologies, University of Sydney, Australia ABSTRACT In visualizing graphs as node-link diagrams, it is commonly accepted and employed as a general rule that the number of link crossings should be minimized whenever possible. However, little attention has been paid to how to handle the remaining crossings in the visualization. The study presented in this paper examines the effects of crossing angles on performance of path tracing tasks. It was found that the effect varied with the size of crossing angles. In particular, task response time decreased as the crossing angle increased. However, the rate of the decrease tended to level off when the angle was close to 90 degrees. One of the implications of this study in graph visualization is that just minimizing the crossing number is not sufficient to reduce the negative impact to the minimum. The angles of remaining crossings should be maximized as well. Keywords: Graph visualization, graph drawing, aesthetic criteria, link crossing, crossing angle, evaluation. Index Terms: H.1.2 [Models and Principles]: User/Machine Systems Human Factors; H.5.0 [Information Interfaces and Presentation]: User Interfaces Evaluation/Methodology 1 INTRODUCTION Much of the information in the real world has a relational structure. This type of structures can be modeled as graphs and visualized as node-link diagrams, in which nodes represent entities in the original data and links represent the relationships between the entities. Such diagrams have broad applications in many fields. In particular, these diagrams are used in law enforcement (for example, to detect fraud and money laundering), in biotechnology (to understand metabolic pathways, for example) and web engineering (to monitor usage). The primary objective of graph visualization is to enable data information to be more available by helping the viewer to effectively carry out various tasks. Simply providing a diagram does not guarantee good performance; research has shown that layout significantly affects user preference and task performance [14, 15]. A given graph data set can be visualized in many different ways by changing the spatial positions of nodes. Which way is the best for the viewer s comprehension of the underlying data has long been a concern in graph visualization in general and automatic graph drawing in particular. Graph drawing research concerns the problem of constructing geometric representations of abstract graphs. That is, given a set of nodes and links, calculate the positions of the nodes to optimize predefined layout requirements. The layout requirements can be modeled in terms of three fundamental parameters of graph drawing methodologies: drawing conventions, aesthetics and constraints [4]. A drawing convention is a basic rule that a drawing whua5569@it.usyd.edu.au shhong@it.usyd.edu.au peter@it.usyd.edu.au IEEE Pacific Visualisation Symposium March, Kyoto, Japan /08/$ IEEE must meet. For example, drawing graphs without link crossings (planar drawing). An aesthetic is a layout property that is intended to make the resulting drawing visually pleasant and easy to read and understand. For example, display the symmetries of the graph. A constraint is a rule that only applies to subsets of the graph or parts of the drawing. For example, place a given node close to the center of the drawing. Graph drawing research aims to design algorithms that display graphs reasonably fast, and conform to predefined layout requirements at the same time. In the past two decades, computational efficiency of algorithms has been researched extensively. However, little is known about the relevance of those layout requirements to humans in graph understanding (effectiveness). Recently, the research community has seen a growing interest in evaluation which is to ensure effectiveness [3, 7]. To be more specific, empirical studies have been conducted to examine layout effects. Link crossings, one of most discussed graph drawing aesthetics, has been empirically validated and identified as the most important factor negatively affecting graph reading performance [14, 16]. Consequently, in designing automatical graph drawing algorithms, a large amount of research has been devoted to crossing minimization. However, not every graph is planar. Particularly when a drawing convention is applied, crossings are often not removable. How to handle the remaining crossings in the drawing remains unanswered. Prior research indicated that different crossing styles may have different degrees of impact. For example, links may cross each other either at nearly 90 degrees, or at very acute angles. This paper describes a user study investigating how effects of crossings change with the size of crossing angles. The remainder of this paper is organized as follows. In Section 2, link crossings and related studies on effects of crossings are introduced. Section 3 describes the details of the experiment and Section 4 presents the results and findings. The experimental findings and their implications for graph visualization are discussed in Section 5. Finally, Section 6 concludes the paper. 2 BACKGROUND 2.1 Link Crossings In visualizing graphs as node-link diagrams, it is commonly accepted and employed as a general rule that the number of link crossings should be minimized whenever possible. As Bertin [2, p. 271] mentioned, the simplest, most efficient construction is one which presents the fewest meaningless intersections, while preserving the groupings, oppositions, or potential orders contained in the component. Moreno [13, p. 141], who was the first to introduce node-link diagrams (or sociograms) into social network analysis [19, p. 9], also stated, the fewer the number of lines crossing, the better the sociogram. Crossings not only hide important information from the viewer, but also make the viewer reluctant to approach the diagram in the first place [4, 14]. Although minimizing the number of crossings has long and widely been applied in producing diagrams (e.g., [5]), it is Purchase and her colleagues who provided empirical evidence validating the aesthetic of link crossings (e.g., [14, 16]). Purchase et al. [16] conducted the fist study to examine the effects of three common aesthetics including symmetry, link crossings and bends on human understanding of graphs. In this study, one dense 41

2 and one sparse graph were used. Each graph was drawn three times to show each aesthetic at three different levels ( few, some and many ); eighteen drawings were used in total. Three graph reading tasks were used including one shortest path task. Their results [16] indicated that both minimizing bends and minimizing crossings improve task performance, while the effects of symmetry still need further examination. Purchase [17] found in another study that minimizing crossings was overwhelmingly beneficial in understanding graph structure; link crossings was the most important aesthetic. A study conducted by Ware et al. [21] further indicated that for shortest path tasks, it is the number of edges that cross the shortest path itself that is important, rather than the total number of edge crossings in the drawing. 2.2 Crossing angles Given their negative effects, crossings have been a main concern in graph visualization. However, in practice, crossing minimization is a hard problem in designing algorithms for automatic graph drawing [6]. It is difficult to draw graphs with minimum crossings. On the other hand, little attention has been paid to handling the remaining crossings to reduce their impact to the minimum. Prior research has indicated that in some situations, crossings may not be as bad as we normally think. For example, drawing graphs without crossings can make some important structural features, such as symmetry, less apparent. Huang et al. [10] found in a study that in perceiving sociograms, crossings are important only for tasks that involve path tracing. Even when sociograms are drawn to convey information about groups, it is more desirable to cross links connecting the group members. Figure 1: According to Ware et al. [21], the crossing on the right is less confusing than that on the left. Prior research also indicated that how links cross each other may have different degrees of impact. For example, based on recent results from neurophysiology, Ware at al. [21] suggested that links that cross at nearly 90 degrees are less likely to be confusing than those crossing at acute angles (see Figure 1). Eye movement evidence provided by Huang et al. [8, 9] also supported this. From the eye movement data, it appeared that close-to-90-degree crossings were ignored by subjects, while acute-angle crossings caused very slow eye movements, with extra back-and-forths around crossing points. Despite the facts mentioned above, the impact of crossing angles on graph reading performance has not been empirically validated with quantitative evidence [21]. The controlled experiment presented in the next section was to fill this gap. 3 EXPERIMENT This experiment 1 was conducted to test the effects of crossing angles. According to Ware et al. [21] and the results of previous eye tracking experiments [8, 9], small-angle crossings are more confusing than large-angle crossings. 1 Ethical clearance for this study was granted by the University of Sydney, November Design To test the effects of crossing angles on path search tasks, we draw the same graph in different ways; the graph has only one path between two preselected nodes. In each drawing, links cross the path at the same angle, while in different drawings, the size of crossing angles varied from 10 to 90 degrees. The layout of the path remains the same across the drawings. The task was to follow the path from one end to the other and determine the length of the path (i.e. the number of links that form the path). Effects of crossing angles on task performance were measured in terms of response time. We expected that more time would be spent on drawings with small angles than on those with large angles. 3.2 Stimuli Examples of the stimuli can be seen from Table 1. We randomly generated 16 graphs. In these graphs, there was only one path connecting two specified nodes with the path length ranging from 4 to 7. This range was to avoid subjects counting the number of links without their eyes traveling on the path. There were also separated links to cross the path. There were seven crossing angles used; they were 10, 15, 20, 30, 50, 70, 90 degrees. The intervals between angles were larger when angles are larger. Based on the results of pilot studies, small angles appeared to have more dramatic effects than large angles. Therefore relatively more small angles were used with the aim of catching possible changes in effects. Each graph was drawn 7 times, with each drawing having one specific crossing angle. Across the 7 drawings, the layout of the target path remained unchanged, and links crossed at the same locations of the path. The length of links crossing the target path could be changed to make sure that a single link crossed the path only once. From graph to graph, the layout of the target path, the location of crossing points, and the number of crossing links can be different. For each graph, we also produced a path graph that only contained the target path without crossing links. The path graph was drawn with the same layout as that of the target path in other 7 drawings. The path graph was added as a control condition to compare possible differences in response time between its drawing (such as the top left drawing in Table 1) and corresponding large-crossingangle drawings (such as the 90-degree or the 70-degree drawing in Table 1). Therefore, 16 sets of drawings were obtained with each set containing seven drawings of the same graph and one drawing of the corresponding path graph; 16 (7+1)=128 drawings in total. 3.3 Subjects Twenty-two subjects were recruited on a completely voluntary basis. They were either research or coursework postgraduate students from the School of Information Technologies at the University of Sydney. The subjects had normal or correct-to-normal vision and were familiar with node-link diagrams. Upon the completion of their assigned tasks, they were paid $10 each for their time and effort. 3.4 Online System The drawings were displayed by an online system. The system was designed to highlight the two nodes at the ends of the target path as red, and display the drawing in the center of the screen. When it was started, the system displayed one of the highlighted nodes first. The subject was asked to look at the node and press the space key on the keyboard to show the whole drawing. This was to make sure that the eye was at the same position across the subjects when the drawing was shown. The subject was also expected to start path searching from this node. Once the answer was determined, 42

3 Table 1: One set of stimuli examples (the number below the drawing indicates the size of the crossing angle in degrees. 0 means no crossings) 43

4 the space key was pressed; the picture disappeared and an answer screen appeared. On the right side of the answer screen, there were six boxes with each representing one possible answer. The subject was asked to click on one of the boxes. Once the answer box was clicked, a red node of the next drawing was displayed, and so on. The response time for each subject and each drawing was recorded in real time by the system. The time started when the drawing was completely shown and ended when the space key was pressed. 3.5 Procedure There were two blocks with each block containing 8 sets of the drawings. The order of the blocks was counterbalanced across the subjects. The order of drawings in each block was randomized with the constraint that two drawings from the same set should be separated by seven drawings from other sets. The experiment was conducted in a quiet laboratory room on an individual basis. Before the experiment, the subject was asked to read the tutorial material, practice, ask questions, and sign the consent form. The experiment was started by running the online system. During task performance, if a wrong answer was given, the corresponding drawing was placed at the end of the block until the correct answer was given. The subject was required to have a break between blocks and at the middle of each block. After the online task, a short questionnaire and a brief interview were conducted. The whole experiment took about 35 minutes on average. Each subject was required to view all the drawings. During the preparation time, the subject was reminded that there was one path only between red nodes. The subject was asked to follow the path from the starting node immediately after the picture was shown, without having an overall examination or looking at the other red node first. The subject was instructed to determine the answer as quickly and as accurately as possible. 3.6 Hypotheses Based on prior research presented in Section 2, we hypothesize that: Figure 2: Means of response time (sec.) (crossing angle=0 means no crossings) From Figure 2, it can be seen that in general, the subjects took more time to complete the task when the crossing angle was small. An analysis of variance with repeated measures was conducted on the seven crossing conditions. The results indicated that there was a significant effect of crossing angle on response time, F(6,126) = 28.11, p < 0.001,η 2 = 0.57 (η 2 represents partial eta squared, a measure of effect size). This confirms the first hypothesis. 1. The crossing angle significantly affects response time. 2. There is a negative linear correlation between response time and the crossing angle. That is, response time decreases with increases in the crossing angle. 3. Response time with large crossing angles is not significantly different from that with no crossings. In particular, the third hypothesis was formulated also based on the following considerations. 1. Crossings have negative impact on performance. 2. When the crossing angle is nearly 90 degrees, the impact can be very small. 3. Given the drawings used in this experiment, when there are no crossings on the path, the viewer may need more effort to concentrate. When links cross the path at large angles, the path appears outstanding and might be easier to follow. 4 RESULTS Only times of correct responses were analyzed. Each subject viewed 128 drawings; =2,816 time entries were obtained. The time analysis was based on medians. For each subject, the median time in a condition was calculated. These individual medians across the subjects were averaged as the mean time that the subjects spent in that condition. This method eliminates the effects of possible outliers, and was also used by Korner et al. [12]. The means of median times were summarized in Figure 2. Figure 3: Illustration of the trend relationship between time and angle To test the second hypothesis, the means of the crossing conditions were plotted in Figure 3. From Figure 3, it appears as expected that there was a general tendency for response time to decrease with increases in the crossing angle. We formulated a set of linear trend coefficients based on the experimental angles. Analysis of linear trend [11] indicated that this trend between time and angle was significant, F(1,21) = 36.47, p < 0.001,η 2 = Further, as can be seen From Figure 3, the negative correlation between time and angle turns to positive around the angle of 70 degrees. When the crossing angle is larger than 70 degrees, response time started to gradually increase with the angle. To test whether this trend change was systematic, we formulated a set of quadratic trend coefficients orthogonal to the linear trend coefficients. Anal- 44

5 ysis of quadratic trend [11] indicated that this trend was significant, F(1,21) = 46.65, p < 0.001,η 2 = The third hypothesis requires equivalence testing. Equivalence testing involves two steps: define equivalency and perform two simultaneous one-tailed t tests (see, e.g., [18] for more details). Defining equivalency is to determine a small tolerance value δ so that two means that differ by less than δ can be considered equivalent. For this hypothesis, δ was defined as 10% of the time mean of the control condition [18]. The mean time for the non-crossing condition is 3.09 seconds. Therefore, δ= %=0.31 second. We compared two angle conditions: the 70- and 90-degree conditions with the non-crossing condition, respectively. For each comparison, the null hypotheses for the two one-tail paired t tests were: the difference between time means is greater than or equal to 0.31 and the difference between time means is less than or equal to The mean difference between the 70-degree and non-crossing conditions was =0.10 second. The tests indicated that this difference was significantly less than 0.31, t(21) = 6.37, p < 0.001, and significantly greater than -0.31, t(21) = 3.23, p < Both null hypotheses were therefore rejected, and it was concluded that response time with the 70-degree angle was statistically equivalent to that with no crossings. The mean difference between the 90-degree and non-crossing conditions was =0.18 second. The tests indicated that this difference was significantly less than 0.31, t(21) = 6.25, p < 0.001, but not significantly greater than -0.31, t(21) = 1.56, p = The null hypotheses could not be rejected at the same time, and the alternative hypothesis that the 90-degree condition was equivalent to the non-crossing condition was not supported in this experiment. 5 DISCUSSION The experimental results showed that crossing angles had significant impact on response time in tracing a path. When crossing angles were large, the subjects performed significantly faster than when crossing angles were small. The results also indicated that as illustrated in Figure 3, there was a negative linear component and also a quadratic component in the relationship between response time and the crossing angle. That is, response time decreased significantly as the crossing angle increased. However, the rate of the decrease tended to level off when the angle was close to 90 degrees. It is important to note that our analysis of trend was only based on seven specific angles. However, given the nature of the experimental task, it is not expected that when a different set of angles are used, the overall trend will be very different from that illustrated in Figure 3. The equivalence testing showed that when crossing angles were, for example, around 70 degrees, the response time was sufficiently near to that in the non-crossing condition to be considered equivalent (within the predefined δ). In other words, drawing graphs with large-angle crossings can be equally effective as drawing graphs without crossings, in terms of response time for path tracing tasks. It is not clear why the subjects in this experiment took slightly more time with the 90-degree condition than with the 70-degree and 50-degree conditions. The subjects generally commented that it was easier to follow the path with close-to-90-degree angles than with acute angles. For example, When the crossing has a very acute angle, it affects the path following task most. In this case, the crossing lines are almost parallel to each other, which makes it hard to identify the true path. Crossings make it hard to follow the lines, especially when the angles are small. It is harder when there are lots of crossings almost parallel to the links I am tracing. Crossings would require more concentration for small angles than large angles. Some subjects also commented that the right angle seems to have no effects, sometimes even makes path tracing easier, while the acute angle is confusing. Crossings help when the link is perpendicular to the path; crossings give me doubt when they are very close and parallel to the path. It terms of the effects of crossings and crossing angles on eye movements, the comments from the subjects were largely in line with the observations of Huang [8]. For example, Eye movement was faster and smoother for crossings with wider angles than for crossings with narrower angles. More crossed lines made my eye movements slow. Crossed lines made it hard to count and remember the number of links. Sometimes I had to go back to count again. I needed to slow down eye movement to find where the link goes. Crossings blur the paths that I am interested in, hence making my eyes more tired and making me blink more. Certainly, it decreases my accuracy as there is more cognitive load and strain on my eyes. Pictures with less crossings and larger crossing angles are much easier to read. Crossings with acute angles may lead my eye to the wrong path. 5.1 Crossing Number Minimization and Crossing Angle Maximization In automatic graph drawing, the aesthetic of link crossings has been researched extensively in terms of the number of crossings [4]. However, the crossing angle has not been considered as one of the aesthetic criteria. This study indicated that only minimizing the number is not sufficient. To reduce the negative impact of crossings to the minimum, the angles of remaining crossings should be maximized as well. Finkel et al. [5] propose a force-directed curvilinear drawing method, which considers three aesthetics including node angular resolution, link separation and crossing number. This method may produce drawings with enlarged crossing angles, but it is not guaranteed. The finding that effects of crossings differ with the size of the crossing angle has two main implications for graph drawing: 1. It is well known that minimizing the number of crossings is computationally difficult and time consuming. Alternatively, to avoid this difficulty, we may draw graphs with a few more large-angle crossings. But the resulting drawings can be equally as effective as those with minimum crossings. 2. There are situations where crossings cannot be removed (for example, graphs are nonplanar, or a drawing convention is applied), or where effort needed to remove all crossings cannot be justified. In these cases, the crossing angle should be maximized. In summary, taking into account crossing number minimization and crossing angle maximization at the same time, and making tradeoffs between them when necessary, helps to fully reduce the negative impact of crossings. 5.2 Equivalence Testing and Superiority Testing In visualization evaluation, superiority testing is the most common statistical approach used. This approach asks whether or not the conditions under consideration differ. However, this approach has some limitations. For example: 1. The chance to detect a significant difference may be increased by increasing sample size or test power [11]. 2. Statistically significant differences may not be meaningful in practice. For example, the difference between two conditions can be too small to be considered as important; although the new method results in better visualizations, it may take relatively more time and effort to construct them. 45

6 In evaluating graph visualizations, it may be more desirable to take into consideration not just the visualization itself, but also the cost involved in producing and maintaining it [1, 20]. In some cases, the new visualization method may produce visualizations that can be considered as being equally effective as those by the current best method, but at a lower cost and greater convenience. In these cases, equivalence testing as we did in this experiment may be more suitable to use. This approach asks whether there is a more efficient way to achieve the same result [18]. 5.3 Confounding Factors In performing controlled experiments, one of the major concerns is the effect of confounding factors. This is particularly important for graphs given the interconnecting relationships between nodes and links; manipulating one factor may inevitably affect the other. Ware et al. [21] present an ideal way to work around this issue, by using real graph drawings and considering all possible factors at the same time. Since our aim in this experiment was only to examine how crossing angle affects response time when tracing a path, we used simple purpose-constructed graphs, such as those in Table 1, to control the number of possible confounding factors and their effects. In constructing the stimuli drawings, although it is impossible to exclude all confounding factors, every effort had been made. For example, in each set of drawings, layout of the target path remains the same, as well as the locations of crossing points. These were to make sure that the only difference between drawings was the crossing angle. One exception was the distance between the parallel links. The distance reduced while the crossing angle decreased from drawing to drawing (see Table 1). This was the result of keeping the crossing locations unchanged while varying the size of the crossing angle. The reduced distance made the drawings look denser. To minimize the effect, the subjects had been reminded that there was only one target path, and asked to trace the path without having a overall look. Despite this, we may still roughly consider that the denser look was the result of smaller crossing angles, therefore being able to attribute the difference in response time to the difference in crossing angles. 6 CONCLUSION We have conducted a controlled experiment to validate the effects of crossing angles. It was found that the the effects of crossings differ with the size of crossing angles. To be more specific, the performance of path tracing tasks improves with increases in the crossing angle. However, the rate of improvement slows down when the angle size is close to 90 degrees. These indicate that in drawing graphs, to fully reduce the impact of crossings, not just the number of crossings needs to be minimized, but also the size of crossing angles should be increased. This study also found that drawings with around-70-degree crossings are equally as effective as those with no crossings. Although minimum-crossing drawings are more desirable, the finding indicates that we may achieve the similar level of effectiveness by drawing graphs with a few more large-angle crossings. [5] Finkel, B. and Tamassia, R. (2004) Curvilinear Graph Drawing Using the Force-Directed Method. In Proc. 12th International Symposium on Graph Drawing (GD 04), LNCS 3383, [6] Garey, M.R. and Johnson, D.S. (1983) Crossing Number is NP- Complete. SIAM Journal on Algebraic and Discrete Methods, 4: [7] Ghoniem, M., Fekete, J.-D. and Castagliola, P. (2004) A Comparison of the Readability of Graphs Using Node-Link and Matrix-Based Representations. 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