Cluster-based Edge Bundling based on a Line Graph

Takafumi Yamashita and Ryosuke Saga

Graduate School of Engineering, Department of Computer Science and Intelligent Systems, Osaka Prefecture University,

1-1 Gakuen-cho, Naka-ku, Sakai, Osaka, Japan

Keywords: Information Visualization, Edge Bundling, Edge Clustering, Line Graph.

Abstract: Information visualization enables simple and intuitive understanding of data. Edge bundling is a visualization

technique and is beneficial for visual analysis. By transforming data into a network diagram, the relationships

among data can be recognized intuitively. In such situation, edge bundling reduces the visual clutter by

bundling the edges on the basis of several approaches. Results show the bundles of edges are organized in a

few relationships. In other words, the bundles can be regarded as clusters of edges. Therefore, we propose a

new bundling method based on edge clustering. By changing a network into a line graph, edges can be

regarded as nodes such that several node clustering methods can be applied to edge clustering. We bundle

edges on the basis of the result of edge clustering. This approach is a novel concept of edge bundling and edge

clustering. Using the proposed method, most edges are clearly bundled whereas a few edges belonging to

different clusters are not bundled.

1 INTRODUCTION

Network diagram is a common technique of

information visualization (Gansner et al., 1993). This

approach can simply represent the relationships

among data through links between nodes such that

observers can recognize these features intuitively.

Network diagram is increasingly being investigated

because of the growth of social networks, such as

Twitter and Facebook. Network diagram can be

applied to social networks and enables observers to

find features by network analysis, such as community

detection (Newman et al., 2004). However, when the

number of nodes and links is large, the visibility of

the graph decreases because of the visual clutter

caused by the complicated structure of the graph. This

issue must be resolved.

To reduce the visual clutter, graph layout

approaches have been proposed (Mueller et al., 2006;

Kamada et al., 1989; Fruchterman et al., 1991;

Michael et al., 2004; Archambault et al., 2007). This

approach rearranges nodes correctly such that the

visibility of the graph increases to a certain degree.

However, this approach cannot solve the problem

when the graph contains many edges.

To address this issue, a new approach called edge

bundling has been proposed (Holten, 2006; Zhou et

al., 2008; Telea et al., 2010). This method enables

observers to easily find the relationships among data

through the mainstream of the edge bundles. The

methods mentioned above are based on several rules,

such as hierarchical structure of nodes, parallel

coordinates, and mechanical models. The model-

based bundling methods presented in previous works

have improved the visibility of the graph based on

each concept.

Bundles are several convergences of edges and

can be classified as clusters of edges. A similar

concept has been proposed (Cui et al., 2008; Telea et

al., 2010); however, this approach only focuses on

geometry-based edge cluster and only bundles edges

to find edge clusters. In other words, this approach

regarded merged edges as a cluster. In the big data era,

data do not always present location information. The

edges must also be classified using other information.

In this paper, we propose a novel concept of edge

bundling. This method is called cluster-based edge

bundling (CBEB). CBEB bundles the edges based on

the cluster information of edges. To detect the clusters

of edges, we replace the problem with community

detection using line graph. Edges can be regarded as

nodes on the line graph such that the community

detection method can be adapted to the edge cluster

detection.

Yamashita T. and Saga R.

Cluster-based Edge Bundling based on a Line Graph.

DOI: 10.5220/0006236003110316

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 311-316

ISBN: 978-989-758-228-8

311

Figure 1: Framework of our CBEB.

The contributions of the study are as follows:

1. A novel concept of edge bundling is

proposed, which can find topological edge

cluster.

2. Edge cluster detection is replaced with

conventional community detection using

line graph.

3. The proposed concept is suitable to other

bundling algorithms or community detection

algorithms.

2 RELATED WORKS

Holten et al. (2009) proposed the force-directed edge

bundling (FDEB) method. FDEB has been applied to

undirected- and single-edge-type graphs. In this

method, the edges are considered a spring with

several control points and are bundled by the spring

force based on Hooke’s law and the electrostatic force

as the attractive force among the points. The bundling

methods can be used to reduce the computational

complexity from O(E

) to O(E

C), where E is the

number of edges and C is the number of control

points.

However, when the forces are excessively strong,

the edges are also excessively bundled and the node–

link diagrams present incorrect relationships. To

solve this problem, Holten et al. (2009) introduced a

compatibility measure that works for the force among

the incorrect pairs of edges in consideration of length,

position, angle, or projection overlap (called

visibility); they then filtered the incorrect pairs by a

threshold. Of course this method ignores edge

clusters.

Telea et al. (2010) proposed another concept of

edge bundling, which is called the image-based edge

bundling method. This method helps observers

recognize the coarse-level bundle by emphasizing

such bundles. This approach allows any layouts of

bundles. Given the layout of input graph, this method

classifies edges into some clusters. After clustering,

for each cluster, a compact shape is computed. Based

on each shape and its skeleton, a cushion-like shading

profile is constructed. Finally the graph is drawn by

these process.

Ersoy et al. (2011) proposed the skeleton-based

edge bundling, which is expansion of the image-

based edge bundling. Image-based method utilizes

skeletons, or centerlines of shapes, only to shade

bundles. In this improved method, edges are

iteratively attracted towards its skeleton of the shape

using a feature transform.

These two method adopts edge clustering. Each

edge has a feature vector, which comprises sampling

points of the edge and edge type. All edges are

clustered by the clustering framework for gene data

using these feature vectors. Telea et al. (2010) stated

that the mixing positions and types in one distance

metric can lead to undesirable results even when

feature vectors can include multiple dimensional

types.

Apart from these methods, other concepts of edge

bundling are also available (Lambert et al., 2010;

Zielasko et al., 2016; Bourqui et al., 2016). A few of

them also treat edge cluster or multilayer graph;

however, a method that focuses on topological edge

cluster is unavailable. We assume that the result of

bundling already shows geographical clusters

because edges are attracted based on geographical

information. Therefore, what we have to extract

before bundling is topological clusters of edges.

Certainly, geographical cluster can clearly visualize

data, but from the viewpoint of data-oriented

visualization, detecting topological edge clusters is

more effective.

3 CLUSTER-BASED EDGE

BUNDLING (CBEB)

3.1 Overview

The overall procedure of our CBEB is shown in

Figure 1. First, we convert an input graph into a line

graph. Edges are then converted into nodes and are

linked if a pair of original edges shares the endpoint

with them. After converting the graph, we apply a

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community detection method to the line graph. The

detected communities on the line graph are the

clusters of edges in the original graph. Finally, the

edge bundling method is applied to the original graph.

Using cluster information, the algorithm for bundling

edges can be weighted. Edges belonging to the same

cluster are tightly bundled, and edges belonging to

different clusters are loosely bundled or repelled.

3.2 Line Graph

Line graph is another form of a given graph and is a

simple concept (Harary, 1969).

We consider a line graph L(G) of a given graph G.

In L(G), each vertex represents an edge of the original

graph G. If two edges in G share a vertex, then the

corresponding vertices in L(G) are adjacent. An

example of a line graph is shown in Figure 2.

In L(G), the original edges are regarded as nodes

to solve a problem. Node clustering on line graph is

equivalent to edge clustering on an original graph

according to the definition of line graph. That is, we

attach importance to not geographical but topological

relationships. In addition, we assume that edges that

are not connected through any path or are connected

through many edges do not belong to the same cluster.

In this situation, node clustering method is useful

mentioned next.

Figure 2: Example of a line graph.

3.3 Modularity-based Clustering on

Line Graph

Node clustering on line graph is beneficial for edge

cluster detection. Several node clustering methods are

available, such as clique percolation method (Palla et

al., 2005). Among these techniques, we use the

modularity-based clustering method in consideration

of runtime.

Newman and Girvan (2004) proposed modularity,

which is a metric of graph clustering. Generally, a

cluster is “good” when it contains many intra edges

and a few outer edges. Modularity can evaluate

whether the clustering result is good or not according

to this concept. Modularity  is presented as follows:



∑















(1)

where 



is the fraction of edges in the network that

connects vertices in the same community, and 





the expected value of the same quantity in a network

with the same community divisions but random

connections between the vertices. Based on this

metric, Newman (2004) proposed the agglomerative

community detection method. This method uses ∆,

which is the increment of modularity when two

clusters are combined. ∆ is presented as follows:

∆  







2







2











,

)

where 



is the number of edges between community

 and , and 



is the number of edges in community

.

First, all nodes on the line graph are regarded as a

cluster, and the pair of nodes with the highest value

of ∆ are combined. After combining the pair, each

∆ between every pair of nodes are recalculated.

Finally, all communities are detected by repeating

this process until all values of ∆ become negative

numbers.

If the detected clusters on a line graph show

connected original edges, then these clusters contain

edges that share only one of the endpoint of them. An

example of this case is shown in Figure 3. If all edges

in several subgraphs share one node, then the line

graph of the subgraph is a complete graph. A

complete subgraph is treated as a cluster in several

methods such that node clustering on a line graph can

detect edge clusters.

Figure 3: Example of an edge cluster.

3.4 Cluster Compatibility

After clustering on line graph, edges are bundled by

several forces based on a particular model. To bundle

the edges based on cluster information, we now

define an additional compatibility called cluster

compatibility. The cluster compatibility 



between

two edges  and  is defined as follows:







,









 











(3)

where 







and  is the cluster which  and 

belong to, and  is the number of clusters. If  and 

belong to the same cluster, then they are bundled

Cluster-based Edge Bundling based on a Line Graph

313

tightly. However, if two edges belong to different

clusters, then the force is weighted low. This

compatibility eases the tight bundling of edges

classified to the same cluster.

If observers aim to significantly tightly bundle

edges or to repel edges using cluster information, the

can adjust the value of 



without using . The

suitable value can be obtained empirically because

the visibility depends on the subjective evaluation of

the observer.

3.5 Exchangeability of Algorithms

Our concept is the entire flow of edge clustering and

bundling. The two algorithms for detecting cluster

and bundle edges are not concrete. In other words,

observers can use any algorithms that are suitable to

their data set. If the data present several attributions

on the edges, then observers can detect clusters in

consideration of the attributions. In such a case,

observers can apply a method that can find attributed

clusters to the line graph, such as the SA-cluster

method (Zhou et al., 2009). If the data satisfy the

conditions, then observers can apply multi-type edge

bundling (Yamashita et al., 2015; Saga et al., 2015)

to the original graph. Our method is advantageous in

that observers can choose appropriate algorithms

according to their data.

4 EXPERIMENTS

4.1 Simple Case Study

We show the result of the simple case study. We

adopt FDEB (Holten, 2009) and modularity-based

community detection method to implement our

method (Newman, 2004).

We create a sample graph that contains 10 nodes

and 8 edges. The graph can be divided into 2

subgraphs. Each subgraph is composed of 5 nodes

that are connected. No edges exist between the 2

subgraphs.

The result is shown in Figure 4. The light lines

denote the beginnings of edges. The edges classified

to the same cluster are presented in the same color.

By converting the original graph to a line graph, the

clusters of edges are detected on the line graph. Each

edge in each subgraph is classified to the same cluster

on the line graph. As a result, edges belong to the

same cluster are obviously tightly bundled and the

unconnected edges are not bundled.

Figure 4: Bundling for sample graph. Left: Result of FDEB.

Right: Result of our CBEB.

4.2 Application Example for an

Editorial Network

In this study, we choose the 2008 editorial articles

from Yomiuri newspaper as the data set for the graph.

We make co-occurrence graphs of keywords using

the data. The keywords are the top 200 with respect

to TF-IDF score. We use the Jaccard index to measure

the co-occurrence degree, and its threshold is set to be

0.25. We then filter the graph to delete subgraphs that

contain less than 5 nodes. Finally, the graph is

composed of 99 nodes, 259 edges, and 4 clusters of

edges. When drawing the graph, we provide color to

the edges according to their cluster. The edges

classified to the same cluster are presented in the

same color like Figure 4. In addition, we use

FRLayout (Fruchterman et al., 1991), which is a

graph drawing algorithm based on the spring-

embedder model.

Figures 5 and 6 show the results of FDEB and our

CBEB, respectively. FDEB bundles all edges in a few

degrees, but FDEB does not consider cluster

information even when the graph contains edge

clusters.

The results in Figure 6 present a few differences

from those in Figure 5. Specifically, the edges of the

same color are bundled more tightly in Figure 6 than

in Figure 5. Figure 6 also shows that the edges of

different colors are not bundled in several areas

because of cluster compatibility. These edges belong

to different clusters such that the edges do not come

in contact with one another. Therefore, the aim of the

method is achieved.

The results show that edges classified to the same

cluster are located near one another. In other words,

edge clusters are compacted. This finding is due to the

clustering algorithm mentioned above. If another

clustering method is applied or cluster information

such as a tag already exists, then clusters can be

placed sparsely.

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Figure 5: Result of FDEB.

Figure 6: Result of our proposed CBEB.

5 CONCLUSION

In this paper, we propose a novel concept of edge

bundling using edge cluster information. We redefine

edge bundles as edge clusters such that the concept of

line graph can be introduced to the edge cluster

detection. We detect the edge cluster by applying the

node clustering method to the line graph. We then

bundle edges on the basis of the cluster information

using FDEB. Consequently, our approach can reduce

the visual clutter based on our proposed concept.

Future works can focus on the following aspects:

Adjusting Cluster Compatibility. In this paper, we

introduce 



to weight the force. However, this

compatibility is not geographical but topological

unlike existing compatibilities. Therefore, the method

for integrating these compatibilities is open to

argument.

Evaluation Index. Saga (2016) proposed the

quantitative evaluation index for edge bundling.

However, this index can only evaluate geographical

information. Although such feature is important, the

method is unsuitable for evaluating our concept. A

metric must be developed to quantitatively evaluate

the result in consideration of the cluster information.

Soft Clustering. An edge with high betweenness

value corresponds to a node with high betweenness

centrality value on a line graph. Soft clustering can

therefore be conducted on a line graph to address this

problem.

ACKNOWLEDGEMENTS

This research was supported by MEXT/JSPS

KAKENHI 25420448 and 16K01250.

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