An Empirical Study on the Impact of Edge Bundling onUser Comprehension of Graphs

Fintan McGeeSchool of Computer Science and Statistics

Trinity College DublinIreland

mcgeefi@cs.tcd.ie

John DinglianaSchool of Computer Science and Statistics

Trinity College DublinIreland

john.dingliana@cs.tcd.ie

ABSTRACTEdges are one of the primary sources of clutter when view-ing graphs as node-link diagrams. One technique to reducethis clutter is to bundle edges together based on a nearbysource or destination. Combined with edge translucency,edge bundling is reported to reduce the clutter and revealhigher-level edge patterns. However there is very little em-pirical data on the impact of edge bundling on user perfor-mance, as well as the impact of graph characteristics suchas edge density and graph size on the effectiveness of edgebundling as a graph-visualization technique. We have per-formed user experiments to evaluate the impact of bundlingon user performance, using a set of randomly generatedundirected compound graphs with varying vertex counts andedge densities. Our results indicate that edge bundling neg-atively impacts user performance at tracing paths betweennodes, both in terms of accuracy and time. They also indi-cate that while edge bundling may provide no clear signifi-cant benefit in terms of accuracy for recognising higher-levelcluster connectivity, it does provide a significant improve-ment in user response time.

Categories and Subject DescriptorsH.5.2 [Information Interfaces]: User Interfaces—Evalua-tion/Methodology, Benchmarking ; G.2.2 [Discrete Math-ematics]: Graph Theory—Graph Algorithms

KeywordsEdge Bundling, Edge Density, Graphs, User Evaluation

1. INTRODUCTIONClutter is defined by Rosenholz et al. [17] as “the state

in which excess items, or their representation or organiza-tion, lead to a degradation of performance at some task”.Edges are one of the primary sources of clutter when view-ing graphs as node-link diagrams. As the edge density ofa graph increases, the resulting clutter negatively impacts

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comprehensibility. Edge crossings and edge length have beenshown to play a role in the readability of graphs[14]. Edgebundling [6, 3, 8, 5], sometimes referred to as edge cluster-ing, has recently become a common approach to reducingthe clutter of dense graphs. For graphs with a hierarchicalstructure, edges between adjacent nodes can be routed usingbundles of B-splines, using hierarchy nodes as control points.It is also possible to bundle edges based on edge geometryand the graph layout. The colouring and translucency ofedges can also help the comprehensibility of nodes, indicat-ing direction or accumulating to indicate local edge density.Bundling can also be implemented at different strengths,with the weaker levels resulting in edge bundles containingmany distinct strands and the strongest resulting in bun-dles which are so tight that most edges actually overlap fora large proportion of their length. However there is verylittle empirical data on the effectiveness of edge bundling asa graph-visualization technique. We aim to experimentallyevaluate the impact of bundling under different constraintsfor graph size and density, as well as with different levelsof bundling on the edges of a compound graph. Our pri-mary hypotheses are that bundling improves user compre-hension of low-level connectivity tasks and also user compre-hension of higher level trends. Our secondary hypotheses arethat bundling will improve user performance at both levelsas graph size and edge density increase, relative to usingstraight line edges. We have performed two user experi-ments to this end. The first examines user performance at apath tracing task to evaluate user comprehension of the con-nectivity between individual nodes. The second examinesuser understanding of higher level inter-cluster connectivitytrends, by asking the user to identify which cluster is moststrongly connected to a highlighted cluster. We performedour experiments using compound graphs of various size anddensity, with different levels of edge bundling and laid outusing a simple balloon tree layout.

1.1 MotivationThe purpose of this work is to understand the impact of

bundling on user comprehension. In particular we are look-ing at dense graphs, where edge bundling becomes more ap-pealing as a technique to reduce clutter. Purchase [14] hasdemonstrated how the crossing of edges is the graph aes-thetic which affects most human understanding of graphs.Unfortunately in large dense graphs edge crossings are un-avoidable, as once a graph has more than 3|V | − 6 edges,where |V | denotes the number of vertices, it is mathemat-ically impossible to lay out the graph in a planar fashionsuch that no edges cross. Weidong et al. [20] have shown

Figure 1: An experiment graph with dl = 5.26

that the maximising of angles where edges cross also helpsincrease comprehensibility. In graphs within which edges arebundled, edges can frequently overlap and cross paths andintersect at acute angles. Bundling also introduces bendsto edges which have been shown to have a significant effecton user errors and an approaching significant effect on userreaction times for low-level relational tasks [14].

1.2 Edges and Dense GraphsIn their taxonomy of clutter reduction Ellis and Dix de-

scribe clutter as the result of “too much data on too smallan area of the display”[4]. In a dense graph, edge congestionis the primary source of clutter. The links in a node-link vi-sualisation convey important information. However if theybecome too dense the graph becomes less comprehensible,resulting in nodes and other links becoming obscured. Interms of graph theory, the density of a graph is usually con-sidered the ratio of edges to the maximum possible numberof edges in the graph[2]. For an undirected graph this canbe described as

d =|E|

|(V |(|V | − 1)/2).

A graph is then considered dense if this ratio approaches1.0. However in practical real world examples of informationvisualisation such dense graphs are rarely seen. Anothercommon measure of the density of a graph is the ratio ofedges to nodes, referred to as the linear density

dl =|E||V | .

Most real-world graphs have a value of dl <= 10 [12], how-ever this is still enough to cause a large amount of clutter.Given the frequency that dense graphs are encountered inthe real world it is important to include edge density as partof our analysis. One of the more dense graphs used in ourexperiments can be seen in figure 1.

2. RELATED WORK

2.1 Edge RoutingDeveloped by Holten, hierarchical edge bundling [6] is one

of the earliest forms of edge bundling, however it is limited

to compound graphs which are laid out in the form of a hier-archy. In the basic form, the nodes of the graph are laid outusing a hierarchical layout and the edges between the leafnodes of the hierarchy are routed using piecewise B-splines.These edges are referred to as adjacency edges to discernthem from the edges of the hierarchy (which are not drawn).The set of control points for the edges is the set of nodes onthe shortest path between the source and target nodes inthe hierarchy. The authors use a straightening parameter toprevent splines which share a common path through the hi-erarchy from overlapping completely. Holten also uses alphablending of the edges to help convey the density of bundles,and to help pick out bundles where large numbers of edgesoverlap. The edges of the graph are directed edges with thedirection indicated by a changing colour gradient. Gansnerand Koren [5] use edge bundling to reduce clutter as part oftheir improved circular layout. Edges are bundled tightly,merging to a single line, leaving the connectivity of nodepairs to be inferred by the user based on the order of nodesat the bundle ends. Holten and van Wijk [8] also devel-oped an edge bundling approach which has no need for ahierarchy, as edges are routed using a force directed algo-rithm. In this approach, edges are subdivided into pointsthat interact with each other in a manner similar to a forcedirected layout, under constraints based on edge compati-bility. The resulting edges are then smoothed using a Gaus-sian kernel to adjust the position of the internal points ofthe edges. The strength of the bundling can be adjusted inmanner similar to the bundles in [6]. The resulting colourof the edges is determined from a gradient scale related tothe number of edges intersecting a specific pixel. Cui et al.developed a purely geometric approach to bundling edges[3],which is referred to as geometric edge clustering. The edgesare routed using control points derived from a control meshwhich can be generated automatically or by the user. Thereis no dependency on a clustering hierarchy and the bundlingproduced is a result of the position of nodes and the meshgenerated. Hong et al. [21] generate a hierarchy of edgesin their approach which they term energy-based hierarchi-cal edge clustering. The control points for routing the edgesare generated by performing a Delaunay triangulation onthe vertices of the graph. This approach differs to [6] and[3] as when edges share a common path they can overlapentirely. Lambert et al. [10] also use a geometric approach,which they term winding roads. This approach takes intoconsideration node edge overlaps, resulting in an even fur-ther reduction of clutter not present in many of the above.Lambert et al. also enhance their bundles using colour aswell as bump-mapping to improve the users perception ofbundle density.

2.2 Previous Experimental approachesEmpirical evaluations of graph approaches frequently use

a simple low-level relational task, such as path tracing ora similar variant, as a basis for the evaluation [14, 13, 19,20, 18, 7]. In some cases, such as Risden et al.’s work [16],a high level task is used, such as determining where filesshould be placed in a directory structure. However this typeof high level task only applies to a visualisation which hasbeen designed for that specific task and does not necessarilygeneralise to a wider category of visualisation. While pathtracing is a common simple task, it is also very low-levelspecific and it is not the only way in which a graph can be

Figure 2: A less-dense experiment graph with dl = 1.6

read. For example, if a graph is clustered using a hierar-chy, higher level trends of connectivity can be spotted byanalysing the links between clusters / hierarchy branches.Knowing that node A is connected to node C by a graphdistance of two is useful, but also knowing that node A andnode C are in two very strongly connected clusters may alsobe useful. High level trends are important in graphs partic-ularly when graph sizes and edge density become significant,and the analysis of individual paths contributes less to theunderstanding of the graph as a whole. Other approachesthan using path tracing tasks exist. Huang et al. [20] investi-gate the impact of crossing angles using specifically designeddiagrams containing edge crossings. Techniques such as eye-tracking and user surveys can also be used[9]. These othertechniques often provide information on the how and whyof a specific effect, however this paper is concerned with theconsequences of bundling, not the how and why. Thereforethe measures by which we rate our user’s performance atexperimental tasks are the accuracy of the user’s responseand the time taken to respond.

3. EXPERIMENTAL APPROACHWe adopt Holten’s [6] hierarchical approach to edge bundl-

ing for our experiments. This approach does not necessitateany techniques to smooth the curves once they are drawnand allows us to generate test graphs which are consistentin bundling for different node sizes and different densities.The level of bundling can also be easily controlled as an inputparameter to the experiment. The clusters of the graphsare also well defined as a result of the use of the clusteringhierarchy, reducing ambiguity in what the user perceives as acluster and what the experiment design defines as a cluster.

Experiment Graphs.We used context free procedurally generated graphs for

our experiments, as work by Purchase et al.[15] has indicatedthat there are additional semantic issues that need to be con-sidered when a layout algorithm is used in a domain-specifictool. Our generated graphs do reflect real-world densities.However the focus of our experiments is on the impact ofedge bundling, particularly in terms of how well users readbundled inter-cluster edges. Therefore it is the inter-clusteredge count that must scale between different edge densities,not just the general edge density. As mentioned by Lanci-chinetti et al. [11], cluster size and edge distribution varyin real world graphs. We distribute our inter-cluster edgesrandomly between clusters, however the inter-cluster edge

count for the clusters is determined using a power law sothere is a realistically wide range of inter-cluster edge countsdistributed among the clusters. Adjusting the cluster sizesin a similar manner would result in some very large and verysmall clusters, providing extra information to the user whendetermining shortest paths or determining which clusters aremost strongly connected. Therefore we distribute the nodesamongst the clusters evenly. Undirected compound graphswere used as the basis for the experiment. A clustering hi-erarchy 3 layers deep is used. The first layer contained 4child nodes, the second layer contained 3 children for eachof the 4 parent nodes. The actual graph nodes were assignedevenly to the 12 third tier hierarchy nodes. The number ofnodes in the graph is always proportional to the number ofleaf-level clusters (12) within the graph, resulting in an evennumber of nodes in each cluster. The graph area for each isdirectly proportional to the number of nodes in the graphto reduce overcrowding of the display space for the largergraphs. Each graph is displayed extending the full heightof the screen. An example of the resulting graphs gener-ated can be seen in figures 2 and 1. For our test graphs wehave small, medium and large graphs, using node set sizesof |V | = 60, 120, 180. Larger sizes than this resulted in thedisplay becoming too crowded, and would have excessivelyimpacted user performance, independently of bundling.

Choice of Graph Density.The focus of our experiments is on the impact of edge

bundling, particularly in terms of how well users read bun-dled inter-cluster edges. It is not appropriate to simply as-sign edges to nodes at random, as such an approach will notnecessarily result in an even scaling of the number of inter-cluster edges along with graphs size and density. The num-ber of inter-cluster edges is related to the number of intra-cluster edges and hence cluster size. We denote the numberof intra-cluster edges |Eint| (internal) and the number ofinter-cluster edges is denoted |Eext| (external). The numberof intra-cluster edges is a result of the internal cluster den-sity of the graph which we refer to as d(Eint). We refer tothe inter-cluster density as the external density, d(Eext). Forpath-tracing in particular, if the intra-cluster edges are sodense that the individual cluster edges form cliques, the ex-periment participant will be able to infer connectivity with-out considering the internal edge structure of the cluster.The same can be said if the cluster’s nodes form disjoint sets,with no edges between them. For the experiment to be validthe user will need to be forced to trace edges within a cluster.Towards this end, we fix the intra-cluster density such thateach edge is connected to one third of the other nodes withinits cluster, setting the clusters to have an internal graphtheoretic density d(|Eint|) = 0.333. Using the linear defi-nition for density dl = |E|/|V | would result in intra-clusteredge count having a differing impact on the likelihood of anedge being traceable through a cluster depending on clus-ter size. This leaves the inter-cluster graph density as themeans by which edge density will change between graphs.Given a graph with leaf level clusters forming a clusteringC = (c0, c1, ...cn−1) where |C| = n, a cluster i has its inter-nal density defined by d(Eint)Ci = |E(ci)|/E(ci)max whereE(ci)max denotes the maximum number of edges possiblein the cluster ci. The maximum possible number of intra-cluster edges is the difference between this and the maximum

number of edges for the graph as a whole.

|Eext|max =|V |(|V | − 1)

2−

x=|C|−1∑x=0

|cx|(|cx| − 1)

2

If we assume, as is the case for our experiment graphs, thateach cluster is evenly sized, i.e. |V | = α|C|, α ∈ I then|cx| = |V |/|C| and

|Eext|max =|V |2(|C| − 1)

2|C|

Based on the graph theoretic density, we define the graphexternal density as d(Eext) = |Eext|/|Eext|max. From above,it is clear the graph theoretic density is proportional to thesquare of the number of vertices in the graph, which willresult in graphs of a different size having wildly differentinter-cluster edge counts. We would prefer a more linearscaling of density between graphs, therefore we define thelinear inter-cluster edge density as dl(Eext) = (|Eext|/|V |)2.For our path tracing experiments we used three differentlevels of density. For the cluster connectivity experimentwe added a fourth level of density, which was not used forthe path tracing experiment due to the difficulty of the pathtracing in such a dense graph, and the constraint of havinga reasonable duration of experiment. For our inter-clusteredge densities we chose dl(Eext) = 1, 5, 10, and 20, which re-fer to as low, medium, high and very high. A higher densitywould not be frequently encountered in real world situationsand results in a very crowded presentation of graphs. A den-sity of less than 1 results in a low necessity for edge bundlingfor our test graphs. Our test graph sizes and the resultingdensity values can be seen in table 1.

Graph Layout and Display.It is very difficult to rate one layout algorithm as being

better than another in terms of simple relational tasks[13].For our layout we use a simple balloon tree layout, which isessentially a projection of a cone tree layout onto a 2D plane[1] as seen in figures 2 and 1. This allows us to use a visuallyconsistent layout for each of our test graphs. One impact ofusing this layout is that nodes which are close together interms of graph distance will not necessarily be closely re-lated in terms of geometric distance. Two nodes which areneighbours may appear on opposite sides of the graph. Ifwe were to use a force directed layout, related nodes wouldappear geometrically closer together, however the graph lay-out would be significantly different between graphs addingan even more significant confounding factor when trying todetermine the impact of bundling on rendering edges. Ourexperiment graphs were rendered onto a 24” wide screen dis-play, with full screen anti-aliasing enabled. All of the graphrenderings are static, the user was not able to manipulatethe graph or alter their view of the graph (e.g. using pan orzoom functionality). Per Holten [6], we use alpha blendingto help allow individual edges to be more easily discernedwithin the bundles, by drawing shorter curves at a higherlevel of opacity. Edges within the unbundled drawings arealso blended based on length. Some of the previously de-scribed approaches use colour hue to indicate edge direc-tion[6, 3] or edge density [8, 10]. We feel that this mayadd an extra confounding factor to the graphs, therefore alledges are shaded with no hue. The shortest edges are black,blended to grey with the white background for longer edges.

|V | |E| |ci| |Eint| |Eext| dl(Eext) dl Task60 96 5 36 60 1 1.6 1,260 170 5 36 134 5 2.83 1,260 226 5 36 190 10 3.77 1,260 304 5 36 268 20 5.07 1120 282 10 162 120 1 2.35 1,2120 430 10 162 268 5 3.58 1,2120 541 10 162 379 10 4.51 1,2120 699 10 162 537 20 5.83 1180 558 15 378 180 1 3.1 1,2180 780 15 378 402 5 4.33 1,2180 947 15 378 569 10 5.26 1,2180 1183 15 378 805 20 6.57 1

Table 1: Experiment graph properties

Bundling Strength.The different levels of edge bundling offered by the imple-

mented approach do not scale linearly in their visual impact(see figure 4). A bundling level of 0.25 offers relatively lit-tle difference when compared to a bundling level of 0.0 (nobundling). We have chosen to bundling levels of β = 0.0, 0.7and 0.9 for the path tracing experiment, and the same levelswith an additional level of β = 1.0 (the maximum tightnessof bundles) for the cluster connectivity experiment. Theβ = 1.0 level of bundling is not used in the path tracing ex-periment as edges will frequently overlap as one line, makingpath tracing impossible in all cases.

3.1 Experiment MethodologyFor each experiment the participant was shown a sequence

of graphs on screen and was asked to perform a task spe-cific to that experiment. To ensure the user comprehendedthe task and to reduce the impact of any training effect,the initial six graphs displayed were training graphs and theparticipant’s answers for these graphs are not used in theanalysis of the results. The actual experiment graphs dis-played after the training graphs were shown in a randomorder. In order to ensure a rapid response from the par-ticipant and to avoid excessive experiment durations, eachgraph was only displayed for at most 20 seconds. The expi-ration of the time limit is considered a wrong answer. Theparticipants were told to answer the question as accuratelyas possible, however if the answer is not clear, to answer withwhat they believe to be the most likely correct answer. Theexperiments were ordered so that half of the participantsbegan with experiment 1, the other half with experiment 2.Prior to each experiment the user was given an informationsheet describing the task for the graph, and was also verballyinstructed on the task. All experiment participants, bar 2from the total of 21, had a background in computer scienceand had some level of familiarity with the node-link displayof graphs. Due to the comparative nature of the experiment(showing the same data under different levels of bundling)it was felt that the use of people familiar with node linkdiagrams would not negatively impact the evaluation andwould reduce the amount of learning required by users forthe experiment tasks. For the path tracing experiment theuser was shown a total of 81 experiment graph renderings.The 81 experiment graph renderings consist of the nine testgraphs, displayed under three different levels of bundling.Each of the graphs under each of the bundling conditions isdisplayed to the user 3 times over the course of the experi-

(a) β = 0.25 (b) β = 0.5 (c) β = 0.7 (d) β = 0.9 (e) β = 1.0

Figure 3: Illustration of the visual impact of different levels of bundling strength β

ment. Each time the graph is displayed two nodes are high-lighted and the user must indicate the shortest path betweenthe highlighted nodes, by pressing the corresponding key onthe keyboard. The shortest path length between the high-lighted nodes is either 1, 2 or 3 and each path length is usedfor one of the three times the graph is displayed. The exper-iment graphs are displayed in a random order. The nodesare highlighted by being coloured blue and drawn with asquare glyph, while the rest of the nodes are red and drawnwith a circular glyph. The highlighted nodes are selectedat random from the list of all node pairs that have the re-quired shortest path distance between them. An additionalconstraint is placed so that the nodes are contained in sepa-rate clusters, to avoids cases where participants would onlyhave to trace edges within a single cluster.

For the cluster connectivity experiment the user was showna total of 96 graph renderings. The experiment graph ren-derings consist of the nine test graphs from experiment 1 andthree extra graphs with an even higher linear inter-clusteredge density. Each graph was displayed under four differ-ent levels of bundling, the three levels of bundling used forexperiment 1 and β = 1.0, the maximum level of bundling.Each of the graphs under each of the bundling conditionswas displayed to the user twice over the course of the ex-periment. Each time the graph was displayed, one clusterwas highlighted at random and the user was required toleft-click on the cluster that was most strongly connectedto the highlighted one. The most strongly connected clus-ter is that which shares the most edges with the highlightedcluster. The highlighted cluster was coloured blue and thenodes of the cluster were square instead of circular, as canbe seen in figure 1. The rest of the clusters in the graph arecoloured based on a random selection from a list of colours,in order to make the clusters appear more distinct.

4. RESULTSFor each of the experiments, we performed a repeated

measures ANalysis Of Variance (ANOVA) on both the cor-rectness of answers and the response time. When evaluatingthe user’s performance, we score the user 1.0 for a correctanswer and 0.0 for every incorrect answer. For the pathtracing experiment the factors were the graph sizes (threeoptions), edge density level (three options), bundling level(three options) and the path lengths (three options). Eachcombination of factors was displayed to the user once. Forthe cluster connectivity experiment the three factors werethe three different graphs sizes, four different levels of edgedensity and four different levels of bundling. Each combina-tion of factors was displayed twice, with a cluster selected atrandom each time, and the average of the two scores used asthe score in the final analysis. Post-hoc analysis of signifi-cant results was performed using a standard Newman-Keuls

test for comparison of means. The average response timesand the average accuracy across all participants for bothexperiments can be seen in figure 5. For the path tracingexperiment the average number of time-outs was 6.143, witha standard deviation of 1.34. For the cluster connectivity ex-periment the average number of time-outs was 0.1, with astandard deviation of 0.07.

4.1 Experiment 1 - Path TracingFor our path tracing experiment our primary hypothesis

is that bundling improves the participant’s performance attracing graphs. Our secondary hypotheses are that edgebundling will improve the participant’s performance as thegraph edge density increases and that bundling will improvethe participant’s performance as the graph size increases.From figure 4a, it can be seen that users performed signifi-cantly worse with both levels of bundling used for the pathtracing experiment, compared to when the graphs were ren-dered using straight edges. There is little difference in userperformance between the levels of bundling. Clearly can wenot only say that our hypothesis failed, but also that theconverse is true. Bundling hindered the participant’s per-formance at tracing graph paths. This is consistent withPurchase’s work which indicated that straight edges im-prove the comprehensibility of graph [14]. Bundling at eitherstrength also caused a significant increase in the amount oftime taken to answer. Nor do our secondary hypotheses holdtrue. There appears to be a decrease in performance how-ever this is not statistically significant so we cannot say thatthe converses of our hypotheses are true. A table of the note-worthy significant effects can be seen in table 2. The level ofbundling turned out to have a significant impact dependingon the path length, in that the effectiveness of the tighterlevel of bundling improved over the longer paths. This maybe due to the fact that 3 was the maximum path lengthavailable to the user for selection. Regardless, the bundlingstill underperformed the straight edge cases, although notin a statistically significant manner.

4.2 Experiment 2 - Cluster ConnectivityThe significant effects of this experiment can be seen in

table 3. Our cluster connectivity experiment’s primary hy-pothesis is that bundling will improve the participant’s per-formance at determining the relationship between clusters.It can be seen from figure 4c that this is inarguably not thecase when the maximum level of bundling is used. For theintermediate levels of bundling there is no significant dif-ference in terms of accuracy. However bundling does showa significant improvement in response time (see figure 4d),therefore this hypothesis can be considered validated, forthe bundling levels below the maximum strength. Our sec-ondary hypotheses were that bundling will improve user per-

(a) Experiment 1 (b) Experiment 1 (c) Experiment 2 (d) Experiment 2

Figure 4: The impact of bundling on user accuracy and response time (in seconds) for each of the experiments

Measure Effect F-test post-hocAccuracy Bundling F(2, 40)=9.8268,

p=.00034The two levels of bundling significantly degrade performancewhen compared to straight lines. The performance at each ofthe bundling levels was identical

Accuracy Nodes F(2, 40)=6.3469,p=.00404

The medium and high counts of nodes cause a significant dete-rioration in performance when compared to the lowest. Therewas no significant deterioration between the medium and largegraphs

Accuracy Bundlingx pathlength

F(4, 80)=3.8084,p=.00694

For the 0.9 level of bundling , participants could significantlymore easily identify the paths of length 2 and 3

Time Bundling F(2, 40)=11.641,p=.00010

Both levels of bundling cause a significant increase in timeover straight edges, with no significant difference between thebundling levels

Time Nodes F(2, 40)=16.357,p=.00001

Increasing the number of nodes caused a significant increase inthe amount of time taken, however the difference between themedium and large size graphs was not significant

Time Edge Den-sity

F(2, 40)=4.6931,p=.01476

Increasing the number of edges caused a significant increase inthe amount of time taken, however the difference between themedium and high density was not significant

Table 2: Results for the ANOVA for the user performance at the path tracing task.

Measure Effect F-test post-hocAccuracy Bundling F(3, 60)=19.083,

p=.00000The only significant difference is using the strongest level ofbundling β = 1.0 , which significantly damages user performance

Accuracy Nodes F(2, 40)=16.765,p=.00001

There is a significant difference between the largest node countand the other two.

Accuracy Edge Den-sity

F(3, 60)=27.781,p=.00000

There is a significant difference between all four levels of edgedensity except the medium and high levels.

Time Bundling F(3, 60)=11.478,p=.00000

Bundling significantly improves user response time when com-pared to straight lines. There is no significant difference betweenthe β = 0.7 and β = 0.9 levels of bundling and the significantfurther improvement with the β = 1.0 level of bundling is irrel-evant due to the significant degradation of performance

Time Nodes xBundling

F(6, 120)=2.7381,p=.01576

Significant for (|V | = 60), where each of the bundling levels hadsignificantly shorter response times than the straight line edges.

Table 3: Results for the ANOVA for the user accuracy and time taken at the cluster connectivity task.

formance as edge density and node count increase. As thereare no statistically significant results these hypotheses alsocannot said to be valid.

5. CONCLUSIONS AND FUTURE WORKFrom our experimental results it is clear that within the

context of our experimental setup bundling does not aidusers, and in fact actually hinders them at path tracingtasks. Not only in terms of accuracy but also in terms oftime taken to complete the task. An important caveat forthis result is that no visual enhancements for the bundlingother than translucency were used. As previously men-tioned, colour is often used in edge bundles and this mayhave a significant effect when it comes to distinguish in-dividual edges within bundles. Holten and van Wijk’s [7]user study, on how best to render directed edges, indicatedcolouring was not the best option, however this was notin the context of bundled edges. The possibility of encod-ing relative sense of depth into the bundle visualisations toclarify the paths of bundles and their individual edges, us-ing a three dimensional depth effect with suitable illumi-nation, may prove beneficial. The use of colour and depthto aid in the perception of bundles are certainly a topicworth of further work. Furthermore the positioning of nodeswithin clusters relative to bundles and their associated con-trol points may have a significant impact. An approach sim-ilar to Ganser and Koren’s [5] reduction of internal crossingwithin circular layouts might be applied to node and clusterpositions to reduce crossing of edges within the inter-clusterbundles. Within the experiment graphs, no consideration tothe routing of bundles around clusters or nodes was givenwith the aim of reducing confounding factor between thebundles and straight line graphs. Adjustment to the clusterhierarchy control points, algorithmically or with user inter-vention, could further reduce visual ambiguity that resultsfrom this. For recognising high level patterns, bundling of-fered no significant improvement for intermediate levels ofbundling, and is a hindrance when the tightest possible bun-dles are used (β = 1.0). However bundling does produce asignificant positive effect on reaction times, therefore can berecommended when understanding high level connectivity isimportant. In terms of the level of bundling used we foundno significant differences between the β = 0.7 and β = 0.9levels of bundling for path tracing based task or recognisinghigher level patterns.

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(a) Average user accuracy across all factors for the path tracing experiment

(b) Average user response time across all factors for the path tracing experiment

(c) Average user accuracy across all factors for the cluster connectivity experiment

(d) Average user response time across all factors for the cluster connectivity experiment

Figure 5: Overall Results for each experiment