Experimental Analysis of the Accessibility of Drawings with Few SegmentsThis work was partially funded by the German Research Foundation (grant SCHU 2458/4-1). W. Meulemans is funded by the Netherlands eScience Center (NLeSC, grant 027.015.G02).

Experimental Analysis of the Accessibility of Drawings with Few Segmentsthanks: This work was partially funded by the German Research Foundation (grant SCHU 2458/4-1). W. Meulemans is funded by the Netherlands eScience Center (NLeSC, grant 027.015.G02).

Philipp Kindermann David R. Cheriton School of Computer Science, University of Waterloo, philipp.kindermann@uwaterloo.ca Wouter Meulemans Dept. of Mathematics and Computer Science, TU Eindhoven, w.meulemans@tue.nl André Schulz LG Theoretische Informatik, FernUniversität in Hagen, andre.schulz@fernuni-hagen.de
Abstract

The visual complexity of a graph drawing is defined as the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges, e.g., one needs only one line segment to draw two collinear incident edges. We investigate whether drawings with few segments have a better aesthetic appeal and help the user to assess the underlying graph. We design a user study that investigates two different graph types (trees and sparse graphs), three different layout algorithms for trees, and two different layout algorithms for sparse graphs. We asked the participants to give an aesthetic ranking on the layouts and to perform a furthest-pair or shortest-path task on the drawings.

1 Introduction

Algorithms for drawing graphs try to optimize (or give a guarantee on) certain formal quality measures. Typical measures include area, grid size, angular resolution, number of crossings, and number of bends. While each of these criteria is well motivated, we have no guarantee that we get a clear and legible drawing by optimizing only one of the measures. This is caused by most measures competing with each other, implying the best score according to one metric may require sacrificing another. For example, it is known that certain planar graphs cannot be drawn with good angular resolution and polynomial area [8]. The question arises how we can select an appropriate algorithm for a given graph drawing task. Instead of relying on a combinatorial or geometric measure of the drawing, one could also evaluate the results of the algorithms by measuring the efficiency of tasks carried out by the observer. Another option would be to just ask observers which drawing they considers “nicer”. By conducting such experiments we also hope to learn something about the formal measures. The goal is to identify formal measures and algorithms that are particularly suitable for typical tasks performed by using a graph visualization.

A path consisting of several edges may be drawn as a single segment if the edges happen to align their direction. Although the path may contain many edges, this can be counted as only one segment in the drawing. The total number of such segments is known as the visual complexity of the drawing. Instead of straight-line segments, one could also use other geometric objects to draw paths. One option that has been introduced by Schulz [12] is using circular arcs. In this paper our focus lies with drawings using segments.

It is an open question whether a small number of segments is a good quality measure for graph drawings. We present a study that investigates how drawings with few segments are perceived by the observer in contrast to other drawing styles. In other words, we want to find out if this design criterion makes drawings more aesthetically appealing for the observer and/or if they are helpful for executing tasks. The main difficulty is that we cannot control the visual complexity of a drawing while keeping other quality measures fixed. One way to avoid this problem is to adapt existing algorithms in such a way that we can reduce the number of segments in the final drawing without changing the layout “style” of the existing drawing too much.

In our study, we focus on two graph classes. The first class is that of (rooted) trees111In the remainder, we use “trees” to refer to rooted trees., for which many drawing algorithms are known. It is not hard to see that every tree can be drawn with segments, where denotes the number of odd-degree nodes in the tree [3]. It is unknown however, if every tree can be drawn with segments using only a polynomial grid size. We use a heuristic based on the algorithm of Hültenschmidt et al. [6] that draws a tree with minimal visual complexity and quasi-polynomial area and compare its drawing in the user study against drawings of other algorithms. In particular, we use the algorithms of Walker II [14] and of Rusu et al. [11] as alternatives. The former mimics the standard style in which trees are typically drawn in the computer science literature. The latter aims to draw trees with good angular resolution on a small grid.

The second class of graphs we consider consists of sparse but not necessarily planar graphs, as provided by the ROME library [15]. In this setting, it is even harder to control more than one formal measure. We therefore selected only two algorithms to compare. The first is the popular Fruchterman-Reingold spring embedder algorithm [4]. The second is an adaptation of it, aimed at reducing the visual complexity as measured by the number of segments: this is done by adding constraints that force certain edges to be collinear and hence form a straight-line segment. As argued also above, we introduce this new adaptation in order to generate drawings that have a “similar feel” but use fewer segments.

We selected two tasks for the participants to evaluate the drawings presented to them. The first task addresses the question which of the drawings are aesthetically more appealing to the participant. In the second task, we asked the participants to answer questions. For the trees, we asked to identify pair of nodes realizing the largest distance; for the sparse graphs we asked to select a shortest path between two designated vertices. The user study was implemented as a voluntary online questionnaire in order to reach a significant number of participants.

2 Algorithms

Trees.

For trees, we used three algorithms that produce grid drawings as illustrated in Fig. 2: Tidier, Quad, and FewSegments. All three algorithms take as input a rooted tree. Many more examples can be found online [7].

The algorithm Tidier was presented by Walker II [14] and builds upon the classic algorithm by Reingold and Tilford [10]. This algorithm produces a drawing on a size grid that satisfies three criteria: {enumerate*}

nodes at the same level of the tree should lie along a straight line, and the straight lines defining the levels should be parallel;

a parent should be centered over its offspring;

a subtree should be drawn the same way regardless of where it occurs in the tree.

The algorithm Quad was presented by Rusu et al. [11]. This algorithm allows the user to specify an angular coefficient and draws edges such that the angles are above the angular coefficient if possible and evenly spread out otherwise. It also allows the user to specify how many quadrants may be used to place the children of a vertex. We chose an angular coefficient of  and allowed the algorithm to use all four quadrants. Higher values for the angular coefficient would lead to poorer angular resolution in the subtrees; our chosen value gave a well balanced layout for the tree complexity used in this study. For this algorithm, no bound on the grid size was given by Rusu et al.

Finally, the algorithm FewSegments is based on the algorithm by Hülten-schmidt et al. [6] that draws trees on a quasi-polynomial grid with a minimum number of segments. On a high level, that algorithm uses a heavy path decomposition of a tree, which decomposes the tree in heavy edges and light edges. The paths formed by the heavy edges are drawn as a single segment. It recursively (guided by the heavy-path decomposition) embeds each subtree such that the heavy path of its root is drawn with a vector specified by the parent edge of its root and all subtrees lie in disjoint regions. The children around a vertex  that are not connected by a heavy path edge are evenly placed to the top-right of  with decreasing -coordinates from left to right and to the bottom-left of  with increasing -coordinates from right to left with common slopes; see Fig. 1 for an illustration. We use three heuristics to reduce the size of the drawing.

Figure 1: One step of the algorithm by Hültenschmidt et al. [6].

The first heuristic is applied during the layout of the tree. When the algorithm assigns a vector to a subtree, we allow it to increase the length of the vector slightly such that the new vector is an integer multiple of a smaller primitive vector. For example, if the algorithm would assign a vector , then this heuristic would change the vector to . This implies that the segments on the heavy path in this subtree do not have to use vectors that are integer multiples of , but only integer multiples of . Although this makes one segment a bit longer, the subtree might use less area by this change.

In particular, our algorithm takes as an additional parameter some constant . We place the subtrees in pairs around a vertex from inside to outside (note that Hültenschmidt et al. [6] placed them from outside to inside, but the order does not matter for their algorithm). Let  and  be the subtrees to be placed next around  with the same slope, let  be the vector assigned to the heavy path of  before applying the heuristic. Hence, the vector assigned to the heavy path of  is . Note that, if  has an odd number of light children, then  might not exist. Assume without loss of generality that . For all integers and with , the algorithm computes the width and the height for the drawing of  if the vector assigned to the heavy path of  is , and the width and the height for the drawing of  analogously. For every choice of  and , the algorithm assigns a cost function

Then, the algorithm chooses  and  such that

and assigns the vector to the heavy path of  and the vector to the heavy path of . By only allowing slopes that are not larger than , we make sure that the edge from  to the root of  does not intersect any already drawn edge; since we are placing the subtrees from inside to outside, all previously drawn ancestors of  that are placed to the top-right of  lie above  if drawn with vector , so drawing  with a vector of smaller slope cannot create any crossings. The symmetric argument applies for the edge from  to the root of . We chose for all tree drawings used in this study.

(a) Tidier Layout
(b) Quad Layout
(c) FewSegments Layout
Figure 2: A drawing of a tree with each of the three considered layouts.

The second and the third heuristic are applied in alternating order after a layout has been found. We apply both of them five times.

The second heuristic tries to compress vectors: given a edge  that is drawn as a vector  that is an integer multiple of a primitive vector , it redraws the tree such that  is drawn with the smallest integer multiple of  without destroying planarity. This heuristic is applied to every edge in a post-order traversal of the tree.

The third heuristic takes an edge  that is drawn with a long vector and tries to find a smaller vector to draw  and all edges drawn with the same segment as  such that the resulting drawing is still planar. This is a more drastic approach and can change the way a subtree is drawn completely. Let  be the width of the current drawing and let  be its height. Let  be a subtree with root  and parent  such that is not a heavy edge, it is drawn with vector , and . Depending on the number of children of , there might be another subtree  with parent  such that the edge  drawn with vector for some integer . For all integers with (in order of rising ), the algorithm computes a drawing of the tree where is drawn with vector , the vector is assigned to the heavy path of , the edge is drawn with vector , the vector is assigned to the heavy path of , and the drawing of all other edges does not change. If the resulting drawing is planar and its width and height are not higher than those of the original drawing, then the algorithm keeps the drawing; otherwise, it uses the next values of  and  until all of them are used. This heuristic is again applied to every edge in a post-order traversal of the tree, but only those that fulfill the required conditions.

Graphs.

For sparse graphs we used the algorithms ForceDir and FDFewSeg; example drawings are provided in Fig. 3 and online [7]. The former is an implementation of the spring embedder by Fruchterman and Reingold [4]. This algorithm computes a force between each pair of vertices. If there is an edge between two vertices, then there is an attractive force between them, where  is the distance between the vertices and  is their optimal distance defined as , where  is some constant,  is the maximum area of the drawing, and  is the number of vertices in the graph. If there is no edge between two vertices, then there is a repulsive force  between them. By an addition of these forces at every vertex , we obtain a movement  of the vertex described by a 2-dimensional vector. Fruchterman and Reingold use simulated annealing to control the movement of the vertices such that the adjustments become smaller over time and the algorithm terminates.

(a) ForceDir Layout
(b) FDFewSeg Layout
Figure 3: ROME graph 2282.20 using both force-directed layouts.

The FDFewSeg algorithm is an extension of the ForceDir spring embedder that we developed for this paper. It takes as an additional input a set of edge-disjoint paths . First, the movement  for every vertex is computed. Let . The algorithm places the internal vertices  evenly spaced onto the segment between the endvertices  and . To this end, for every vertex , the movement becomes

Note that this procedure does not necessarily draw all paths in  as segments: if a vertex  of a path is an internal vertex of another path that is processed later, then  will be moved away from its path segment. Hence, the user should input paths in an order that avoids this problem: Let . Every vertex can be the internal vertex of at most one path. If some vertex If  is an internal vertex of a path  and the endvertex of another path , then  has to be input after .

For the creation of the drawings for the user study, we picked the paths to be placed on segments manually; most of the time, we picked paths that are (somewhat) close to being segments in the drawing of the ForceDir algorithm.

We chose not to use an automated way to select the paths for the FDFewSeg algorithm. The aim of this paper is to compare the aesthetic criteria of the drawing styles, which could be negatively influenced by a “bad” path selection algorithm. Automated strategies for selecting paths only become relevant if the aesthetic criterion is worthwhile. We note that, since there is some hint on the criterion being worthwhile, finding a good automated strategy is something to be done in future work.

3 Hypotheses

We designed a user study to compare aesthetics and legibility of drawings produced by the above-described algorithms. For the study we posed the following four hypotheses:

  1. For trees, the aesthetics ranking is

    • Tidier FewSegments Quad for people with a mathematics or computer science background, and

    • FewSegments Tidier Quad for people from a different background.

  2. For trees, path finding is easiest with the FewSegments layout, followed by Tidier, and hardest with Quad.

  3. For sparse graphs, the ForceDir layout is more aesthetically pleasing than the FDFewSeg layout.

  4. For sparse graphs, path finding is easier with the FDFewSeg layout than with the ForceDir layout.

For Hypothesis H1, we expect that the uniformity of the Tidier and the FewSegments layout make them preferred over the Quad layout. For mathematicians or computer scientists, we expect that the Tidier layout is preferred, since it creates a drawing in the standard way that trees are drawn in the literature. For people with different background, we expect that the FewSegments layout is preferred, because it seems to be more schematic. Hence, this hypothesis is split into two parts, H1a and H1b.

The same idea underlies Hypotheses H2 and H4: placing paths onto few segments makes it easier for the user to follow a path between two nodes, since the eye only has to move along few directions and can traverse several nodes quickly along a segment. Evenly distribution the nodes along a path in the force-directed layout should help the reader to quickly determine the number of nodes on a segment and thus to judge the combinatorial length of such a path.

For Hypothesis H3, we think that the smooth curves in the ForceDir layout look nicer to a reader than the drawings of the FDFewSeg layouts because the latter ones can have sharp corners at the meeting point of two path segments; for example, Bar and Neta [1] argue that sharp corners have a negative effect on aesthetics as such bends are identified with threat. On the other hand, Vessel and Rubin [13] studied the objectiveness of taste—their conclusion is that there is typically agreement for natural images, abstract depictions are influenced more by individual taste. Though they cannot fully be eliminated, we believe that the uniformity of graphical presentation may mitigate personal preferences to allow for investigating an overall agreement in aesthetics.

4 Experimental design

Selecting tasks.

We used two tasks in the study: Aesthetics and Query. We created different graphs for each task. For the Aesthetics task, we showed the participant one drawing for each layout of the same graph next to each other. The order of the drawings was determined randomly. The participant was asked to determine a ranking on the aesthetics of the drawings by clicking on them in the desired order.

We used different Query tasks based on the graph class. We showed the participant one drawing at a time. Over the course of the experiment, every graph was presented to the participant once with each layout.

For the sparse graphs, we asked the participants to find the shortest path between two randomly marked vertices that have distance at least 3 (the pair of vertices was the same for each layout and each participant). The participant solved this task by clicking on the vertices (or edges) in the order that they appear on this path. To make sure that a participant does not get stuck on a question, we allowed them to submit their answer even if no path was found. We helped the participant with this task by marking (in a different color) the valid nodes and edges they can click on, which are those that are adjacent to the endpoint of one of the two paths starting in the two marked vertices.

For trees, shortest paths are uniquely defined which makes it unsuitable as a task. Hence, we asked the participant to find the furthest pair of vertices, that is, the pair of vertices such that the distance between them is maximized. Like finding shortest paths in general graphs, this also requires the participant to inspect several paths to determine the answer. The participant then had to click on the vertices that they determined as the furthest pair.

Generating stimuli.

All stimuli and their drawings have been made available online [7]. For trees, we have the following two variables for the stimuli:

  • Size. Two different sizes: (1) 20 nodes and (2) 40 nodes.

  • Depth. Three different tree depths as defined by the length of the longest root–leaf paths: (D) deep trees of depth 8 for size 1 and of depth 14 for size 2, (B) balanced trees of depth 5 for size 1 and of depth 9 for size 2, and (W) wide trees of depth 3 for size 1 and of depth 5 for size 2.

We use rejection sampling to construct random trees of given size and depth. We first create a uniformly distributed random Prüfer sequence [9] and the corresponding labeled unrooted tree. We always choose the vertex with label  as the root to create a rooted tree and then check whether it has the given depth. It is known that Prüfer sequences provide a bijection between the set of labeled trees on  vertices and the set of sequences of  integers between  and . Hence, this algorithm gives us uniformly distributed random trees of a given depth. For each size and depth, we created four different graphs for the Aesthetics task and two different graphs for the Query task. This gives us graphs for the Aesthetic tasks (4 repetitions) and graphs for the Query task (2 repetitions).

For the sparse graphs, we have the following two variables for the stimuli:

  • Size. Two different sizes: (1) 20 nodes and (2) 40 nodes.

  • Type. Two different types: (A) graphs from the ROME library and (B) random graphs.

For graphs of type A, we randomly picked graphs of the given size from the ROME library [15] that consists of 11,535 sparse, but not necessarily planar, graphs with 10 to 100 vertices. For graphs of type B, we created a random graph by creating a number of nodes specified by the size and picking 30 random edges for graphs of size 1 and 60 random edges for graphs of size 2; we used the resulting graph if and only if it is connected. Our choice leads to comparatively sparse but not necessarily planar graphs, to ensure legible layouts for the user study. For each size and type, we again created four different graphs for the Aesthetics task and two different graphs for the Query task. This gives us graphs for the Aesthetic tasks (4 repetitions) and graphs for the Query task (2 repetitions).

We stored the graphs as JSON files, which contained the coordinates of the vertices and the set of edges. During the study, the graphs were drawn using the JavaScript library D3.js [2] as SVG figures to allow arbitrary resizing. The nodes were drawn using blue circles. Links were drawn in black with a small halo to increase separability between crossing links. The selected vertices and links in both Query tasks were marked in green and the selectable vertices and links in the shortest path task were marked in light blue.

Further considerations.

For trees, we created  stimuli for the Aesthetics task and  stimuli for the Query task (one per graph and layout algorithm). For the sparse graphs, we created  stimuli for the Aesthetics task and  for the Query task. This gives us  stimuli in total. This is beyond what is reasonable for an online study, assuming 15 to 25 seconds per trial. Since the study has two different graph classes with different tasks, we used the graph class as a between-subjects measure. This still leaves  stimuli for the tree tasks. Since the size of a graph is very likely to be an overall factor by the larger difficulty of the Query task on a larger graph, we used the size as an additional between-subjects measure for trees. This way, we obtain three groups of stimuli: (1) stimuli for trees of size , (2)  stimuli for trees of size , and (3)  stimuli for sparse graphs. A pilot study showed a completion time of about  minutes for each group.

We first show the Aesthetics task and then the Query task. We did this such that the participant does not get a bias for a specific drawing style based on the difficulty of the Query task and instead of the most aesthetic one picks the one that they preferred in the Query section. Though explicitly asking for visual preference could bias performance in the following Query section, we expect this effect to be negligible, since only one drawing is shown at any given time; and in any case less strong than the potential bias if the sections were to be inverted.

In order to account for learning effects, the orders of the stimuli for each task were randomized for each participant. Before each stimulus, the participant was given a pause screen to reduce memory effects and at the same time allow them to pace themselves and reduce the possible impact of interruptions. The participants received one example question with an answer revealed after providing one, from which they could go back to task description, to ensure that the task was understood before starting the actual questions. We opted not to provide a longer series of training questions to keep time investment to a minimum.

Setup.

We developed our user study with PHP and the JavaScript library D3.js. The study was hosted on a web server222http://tutte.fernuni-hagen.de/web/userstudy/fewarcs and the data was stored in a MySQL database. Since the questionnaire was conducted online, we had no control over many parts of the experimental environment, e.g., device, pointing device, operating system, browser, screen size, interruptions. We asked the participants to fill in the questionnaire using a desktop or laptop computer, not a tablet or phone, and to use the pointing device they are most comfortable with. To make sure that the browser is suitable to run the questionnaire, the participants first had to set a slider to the value depicted in an SVG figure. We could not control the screen size, resolution, or distance of the participant to their screen, so we let the participant control the scale of the web page by providing a Shrink and a Grow button. Further, we asked them to put their browser in full-screen mode to reduce distractions. We requested the participants to not engage in other activities during the questionnaire and to minimize interruptions. At the end of the study, we asked participants to specify if any interruptions occurred during the questionnaire.

We recruited the voluntary participants of the user study using a mix of mailing lists, social networks, and social media. Some background and preference information was asked upon completion, although this remained optional for what may considered sensitive information (age, gender, country of residence).

5 Results

The data set for the analysis as well as all stimuli have been made available online [7]. In total, 84 people volunteered and completed the online questionnaire, which was open for participation for two weeks. We inspected all comments left by participants. One participant had a longer break during one of the questions, rendering this particular question unsuitable for the analysis. As to maintain a balanced design to allow for stronger analysis methods, we excluded this participant from the analysis. This gave us 21 participants for both group 1 and group 2, and 41 participants for group 3. Of the 83 participants, 75 provided their age with an average of , a median of , and a standard deviation of . In terms of country of residence, a majority of the participants live in Europe (63), predominantly in Germany (42).

For Hypothesis H1, we expected different results for people from mathematics or computer science background (H1a) than for people from a different background (H1b). However, we asked for the background of the participants only after they completed the questionnaire, so we could not influence the distribution between the three different groups. Unfortunately, this resulted in only two participants without a mathematics of computer science background to be assigned to the groups for drawings of trees, so we could not claim a significant effect for H1b. Hence, we started a follow-up study that consisted only of the aesthetics task on tree drawings, but for trees of both sizes. We advertised this study on Reddit333https://www.reddit.com/r/SampleSize/ to specifically get people from more diverse backgrounds. This resulted in 24 additional participants, 14 of them did not have a mathematics or computer science background. Of these participants, 23 provided their age with an average of , a median of , and a standard deviation of . The countries of residence also turned out to be more diverse, with 8 participants from Europe, 7 from USA, 4 from Canada, 2 from Australia, 1 from Japan, and 2 unspecified. The results from the follow-up study helped us to evaluate Hypothesis H1 more precisely, compared to the conference version where that data was not available yet.

Hypothesis H1.

For the tree aesthetics task, we had 42 participants from groups 1 and 2 and each of them was shown 12 stimuli, and we had 24 participants from the follow-up study and each of them was shown 24 stimuli. This gave us a total of 1,080 rankings between the three layouts. We used loglinear Bradley-Terry (LLBT) modeling [5] of the 3,240 pairwise aesthetic preference comparisons to produce ranked worth scores for each of the three layouts. The worth score allows the consistency of preference to be assessed in forming an overall ranking of the three classes. Fig. 4 shows the ranking of the three layouts in terms of aesthetic preference, broken down by the balance of the graph and by the background of the participants.

Figure 4: Worth scores of the three tree layout methods: overall and partitioned by tree balance or participant background.

Consistently, the Quad layout was considered as the least aesthetic tree layout. Over all answers, the Tidier layout performed the best. There was some effect based on the balance of the graph. For each balance, we received 360 rankings. However, the ranking for each balance is the same as the overall ranking, they differ only in the effect size. For balanced trees, the effect is the largest with worth scores  (Tidier),  (FewSegments), and  (Quad). For deep trees, we have worth scores  (Tidier),  (FewSegments), and  (Quad). For wide trees, the effect is the smallest, but still significant, with worth scores  (Tidier),  (FewSegments), and  (Quad).

The hypothesis was split into two parts, depending on the background of the participants. Let us first consider the participants with a mathematics or computer science background. There were 286 rankings by 19 people with a mathematics background, but not computer science; for those, the Tidier layout was clearly preferred with a worth score of  over . There were 648 rankings by 45 participants with a computer science background, but not mathematics; for those, the Tidier layout was also clearly preferred with a worth score of  over . There were 192 rankings by 13 participants with both mathematics and computer science background; those slightly preferred the Tidier layout with a worth score of  over . Overall, this suggests that the suspected preference of the Tidier layout over the FewSegment layout, and of both layouts over the Quad layout exists; hence, we accept Hypothesis H1a.

Hypothesis H1b is about participants from neither mathematics nor computer science background. There were 336 rankings by 15 people with neither a mathematics background, nor a computer science background. These participants indeed preferred the FewSegment layout over the Tidier layout with a worth score of  over . This therefore also confirms Hypothesis H1b, so we may accept Hypothesis H1.

Hypothesis H2.

For the tree query task, we had 42 participants from groups 1 and 2 and each of them was shown 18 stimuli. This gave us a total of 756 tasks between the three layouts with 252 tasks per layout. We analyzed the error rates for finding a furthest pair for the three tree layouts defined by the difference of the distance between the picked pair and the distance between a furthest pair in the graph, broken down by the balance and by the size of the trees. The maximum response time was 53 seconds, so we did not have to exclude any participants. Fig. 5 shows the error rates and the answer times.

Figure 5: Error rates and answer times for finding a furthest pair for the three tree layout methods: overall and partitioned by tree balance or size group. Error bars indicate 95% confidence intervals.

We used a two-way RM-ANOVA test to analyze the effects of the layouts, tree balance, tree size, and their interaction. We used the logarithm of the response times to normalize the distribution. For the error rate, there are no interaction effects between layout, balance, and size. The analysis showed a weak effect of the layout on the error rate (, ). A post-hoc Tukey HSD test with Bonferroni adjustment showed a significant difference between layout Quad and FewSegments in favor of FewSegments () and a significant difference between layout Quad and Tidier in favor of Tidier (), but no significant difference between layout Tidier and FewSegments. Further, a post-hoc test showed a weak difference between the tree sizes () in favor of smaller trees. For small trees, there is some evidence that FewSegments outperforms Tidier (); for large trees the error rate seems lower for Tidier, though no statistically significant effect was found (). We conclude that the layouts FewSegments and Tidier perform better than the layout Quad, while the participants performed better on small trees than on large trees.

For the response time, there is some interaction between tree size and tree balance (, ), so we split according to sizegroup for further analysis. For small trees, there are no interaction effects between layout and tree balance. The analysis showed a very weak effect of layout (, ) on response time. A post-hoc test showed a very weak difference between layout Tidier and FewSegments in favor of Tidier () and no significant difference between the other two layout pairs. We conclude that the participants performed slightly faster for the Tidier layout than for the FewSegments layout

For large trees, there are also no interaction effects between layout and tree balance. The analysis showed significant effect of layout (, ) on response time. A post-hoc test showed a weak difference between layout Quad and FewSegments in favor of Quad () and a significant difference between layout Tidier and FewSegments in favor of Tidier (). We conclude that the participants performed slightly faster for the Quad layout than for the FewSegments layout and significantly faster for the Tidier layout than for the FewSegments layout.

Since the error rate was smaller for the FewSegments and Tidier layouts than for the Quad layout, but the response time for FewSegments was worse than for the other two, we can only partially accept Hypothesis H2: the layouts FewSegments and Tidier both outperform the layout Quad, but the layout Tidier outperforms the layout FewSegments.

Though not initially hypothesized, we also found a significant effect of the tree balance on the error rate (, ). A post-hoc test showed a significant difference between tree balances balanced and deep in favor of balanced () and a significant difference between tree balances wide and deep in favor of wide (), but no significant difference between tree balances wide and balanced.

For small trees, we found a significant effect of balance (, ) on the response time. There is a significant difference between both balances wide and balanced and the balance deep in favor of the former ( for both).

For large trees, the analysis showed a significant effect of balance (, ) on the response time. A post-hoc test showed a significant difference between balance wide and balanced in favor of balanced (), a significant difference between balance wide and deep in favor of deep (), and a weak difference between balance deep and balanced in favor of the balanced ().

This analysis suggests that it is easier to find a furthest pair in balanced and wide trees. We believe that this effect comes from a correlation between the distance of a furthest pair and the depth of the tree and that finding a furthest pair seems to be easier the shorter the distance between them is. However, since we have no hypothesis on this behavior, we cannot claim the statistical effect as a strong evidence of an effect.

Hypothesis H3.

For the sparse graph aesthetics task, we had 41 participants from group 3 and each of them was shown 16 stimuli. This gave us a total of 656 rankings between the two layouts. We again used LLBT modeling of the 656 pairwise aesthetic preference comparisons to produce ranked worth scores for both layouts. Fig. 6 shows the ranking of the both layouts in terms of aesthetic preference, broken down by the graph class and the size of the graph.

Figure 6: Worth scores of the two graph layout methods: overall and partitioned by type.

Over all 656 rankings, the ForceDir layout was preferred with a worth score of  over the FDFewSeg layout with a worth score of . For graphs from the ROME library, the FDFewSeg layout was slightly preferred by the participants. For small ROME graphs, the layouts were perceived similar, with worth scores of  (FDFewSeg) and  (ForceDir), but for large ROME graphs, there is a clearer (although still small) preference for FDFewSeg (worth score ) over ForceDir (worth score ). On the other hand, for randomly generated graphs, there is a clear preference for the ForceDir layout over the FDFewSeg layout. The effect is the largest for small random graphs with a worth score of  over ; for large random graphs, the worth scores are  (ForceDir) and  (FDFewSeg). Hence, we accept Hypothesis H3, although we remark that for the real-world graphs from the ROME library the layouts performed similary with a slight preference for FDFewSeg.

Hypothesis H4.

For the sparse graph query task, we had 41 participants from group 3 and each of them was shown 16 stimuli. This gave us a total of 656 tasks between the two layouts with 328 tasks per layout. We analyzed the error rates for finding a shortest path for the two layouts defined by the difference between the length of the selected path and the length of a shortest path, broken down by the four graph types (ROME small, ROME large, Random small, Random large). Fig. 7 shows the error rates and answer times by the participants.

Figure 7: Error rates and response times for finding a shortest path for the two layout methods: overall and partitioned by graph type. Error bars indicate 95% confidence intervals.

We used the same analysis as for the tree query task. For the error rate, there is some interaction between layout and graph type (, ), so we split according to graph type. For Random small graphs, we found a significant difference between the layouts in favor of ForceDir (, ); for the other graph types, there is no significant effect of the layouts.

For the response time, there is a strong interaction between layout and graph type (, ), so we split according to graph type. For ROME small (, ) and Random small graphs (, ), there is a significant effect of the layouts on the response time in favor of ForceDir. For ROME large graphs, there is a very weak effect of the layouts on the response time in favor of ForceDir (, ). For Random large graphs, there is a significant effect of the layouts on the response time in favor of FDFewSeg (, ).

Since the ForceDir layout outperformed the FDFewSeg on three of the four graph layouts, we have to reject Hypothesis H4 in general. However, the FDFewSeg performed better on large random graphs, so there is some evidence that this layout can give better results if the input graph has many vertices. The reason for this may be that the ROME graphs tend to have many degree-2 vertices. Similar effects can be observed for the small random graphs. Consequently, paths become easily traceable for these instances even if drawn with many segments.

6 Conclusion

We compared various graph layout algorithms to assess the effect of low visual complexity on aesthetics and performance. We have confirmed Hypothesis H1: people with a math or computer science background tend to prefer the classical top-down layout for trees, and people with no such background prefer the layouts produced by the algorithm assuring low visual complexity. We have also partially confirmed Hypothesis H2, by discovering evidence that finding a furthest pair is the easiest with the classical tree layout. We confirmed Hypothesis H3: for sparse graph the traditional force-directed layout is more aesthetically appealing than its modification to reduce the visual complexity, although there is some evidence that for real-world graphs the effect is very small and might even be in favor of the layout with small visual complexity. We rejected Hypothesis H4 in general, but rather found that it is typically easier to find the shortest paths between two nodes with the traditional force-directed layout than the modification, though our hypothesis was found to hold for large random graphs. This leaves the possibility open that using few segments can be beneficial for graphs that are more intertwined.

In short, our findings suggest that visual complexity may positively influence aesthetics, depending on the background of the observer, as long as it does not introduce unnecessarily sharp corners. Hence, drawings trees with few segments give a more schematic alternative over the classic drawing style without the risk of harming the aesthetic perception. However, few-segment drawings tend not to improve task performance. It is worth noting that we did not provide training to our participants, as to suggest how the segments can for example help to easily assess the length of a subpath. Providing such clues may have a positive effect on the performance, but at the same time would also result in an unfair comparison, if no training or strategies for the traditional layout were to be suggested.

Acknowledgments

The authors would like to thank all anonymous volunteers who participated in the presented user study and the anonymous reviewers for their valuable input.

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