Ambiguity-free edge-bundling for interactive graph visualization.

Abstract

Graph visualization has been widely used to understand and present both global structural and local adjacency information in relational data sets (e.g., transportation networks, citation networks, or social networks). Graphs with dense edges, however, are difficult to visualize because fast layout and good clarity are not always easily achieved. When the number of edges is large, edge bundling can be used to improve the clarity, but in many cases, the edges could be still too cluttered to permit correct interpretation of the relations between nodes. In this paper, we present an ambiguity-free edge-bundling method especially for improving local detailed view of a complex graph. Our method makes more efficient use of display space and supports detail-on-demand viewing through an interactive interface. We demonstrate the effectiveness of our method with public coauthorship network data.

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Ambiguity-Free Edge-Bundling
for Interactive Graph Visualization
Sheng-Jie Luo, Chun-Liang Liu, Bing-Yu Chen, Member, IEEE, and Kwan-Liu Ma, Senior Member, IEEE
Abstract—Graph visualization has been widely used to understand and present both global structural and local adjacency information
in relational datasets (e.g., transportation networks, citation networks, or social networks). Graphs with dense edges, however, are
difficult to visualize because fast layout and good clarity are not always easily achieved. When the number of edges is large, edge
bundling can be used to improve the clarity, but in many cases, the edges could be still too cluttered to permit correct interpretation of
the relations between nodes. In this paper, we present an ambiguity-free edge-bundling method especially for improving local detailed
view of a complex graph. Our method makes more efficient use of display space and supports detail-on-demand viewing through an
interactive interface. We demonstrate the effectiveness of our method with a public coauthorship network data.
Index Terms—Graph Visualization, Network Visualization, Edge Ambiguity, Edge Congestion, Edge Bundling, Detail-on-Demand,
Interactive Navigation.
✦
1
V
sists of nodes and edges where the nodes represent some
entities and the edges represent the relationships be-
tween the entities (nodes). Its aim is to succinctly reveal
the structural and relational information in the datasets.
As the size of a graph grows, with a limited display
space, laying out the graph directly would lead to two
potential problems: edge-congestion and edge-ambiguity.
The former is due to dense edges which may decrease
the readability of the graph, while the latter is due to
inappropriate edge layout, which significantly decreases
the number of possibly and correctly perceivable rela-
tions from the data.
Edge-congestion and edge-ambiguity problems make a
graph difficult to read and use. Several techniques have
been introduced to relieve these problems. Among these
techniques, edge-bundling [1] [2] [3] [4] [5] [6] is a partic-
ularly effective solution, which reduces the visual clutter
caused by dense edges by merging some groups of edges
together to use less screen space for the same amount of
information. It has been quickly adopted by the graph vi-
sualization community, and there are many approaches
to bundle edges, such as Hierarchical Edge Bundling [2],
Geometry-Based Edge Bundling [4], and Force-Directed
Edge Bundling [6] , etc. They all improved the edge
layout to some degree and can help to visualize an entire
graph with excessive edges. Specifically, these methods
provide better visualization of the structural information
of the whole graph.
INTRODUCTION
ISUALIZING relational datasets as graphs is a com-
mon approach. In graph visualization, a graph con-
• S.-J. Luo, C.-L. Liu, and B.-Y. Chen are with National Taiwan University.
E-mail: {forestking,being31}@cmlab.csie.ntu.edu.tw, robin@ntu.edu.tw
• K.-L. Ma is with University of California at Davis.
E-mail: ma@cs.ucdavis.edu
The goals for visualizing a graph in overview and
detail-view are different. Visualizing a graph in overview
seeks to provide a clearly visible high-level edge pattern
in the whole graph. Under this criteria, the perception
of precisely which two nodes are connected is relatively
less important, and thus displaying detailed connectivity
information in local areas of the graph is usually com-
promised. On the contrary, visualizing a graph in detail-
view seeks to provide accurate connections between the
nodes. Precisely discriminating which two nodes are
connected is essential to allow the viewer to trace a path
and answer some specific questions. If the subgraphs
of interest have highly connected nodes, we still need
to address both the edge-ambiguity and edge-congestion
problems in such local areas.
In this paper, we present an ambiguity-free edge-
bundling method coupled with an interactive (un-
)grouping interface. This method was especially de-
signed to reduce the confusion caused by the clut-
tered edges in the detail-view. Our method improves
users’ ability to accurately perceive individual relation-
ships since the edges with the same target or source
node(s) are merged and curved and the curvature is
automatically adjusted to avoid ambiguities. We employ
a quadtree structure to overcome the time complex-
ity issue, so users can freely interact with the graph.
Through the easy-to-use, interactive detail-on-demand
interface, the users can select some local areas to focus
on by simply enclosing the corresponding subgraphs
with a stroke. Our method effectively removes the short-
comings of the original edge-bundling layout method
and, in particular, enhances the visualization of graphs
with locally dense edges. As shown in each example
presented in this paper, the subtle difference that our
method can be critical in correctly comprehending and
dissecting a graph.
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(a)(b)
Fig. 1.
or even through some unrelated nodes causes incorrect
perception of relationships. (b) The result of routing and
curving the edge away from the unrelated nodes.
(a) The case in which an edge passes near
(a) (b)(c)
Fig. 2. (a) The original graph. (b) The result of an existing
edge-bundling method, and there are some incorrect rela-
tions due to the edge-bundling. (c) The result of our edge-
bundling while considering to avoid the edge-ambiguity
problem.
1.1 Edge Ambiguity
Most of the edge-ambiguity problems happen when an
edge overlaps unrelated nodes. As illustrated in Fig. 1(a),
the center edge, according to the real dataset, actually
connects Node A and Node B while Node C is only inciden-
tally located on the path of the edge. This case may cause
users to incorrectly perceive that Node A - Node C and
Node B - Node C are connected, respectively. Moreover,
even though the edge does not actually overlap Node
D, the geometrical nearness could still confuse a user’s
visual perception of the connections and then cause an
ambiguity problem. The basic idea of solving the edge-
ambiguity problem is illustrated in Fig. 1(b). Obviously,
the inappropriate overlapping and nearness problems
are avoided. The routed edge uses a smooth curve to
bypass the unrelated nodes. The smoothness improves
the users’ ability to trace the curve. Furthermore, a
routed edge can always choose a bypassing path that
minimizes the number of crossings.
Although edge-bundling is usually used to reduce
the visual clutter problem, it may sometimes introduce
its own edge-ambiguity problem. Most edge-bundling
methods merge the edges together if those edges are
both geometrically close and in similar direction as illus-
(a) (b)
Fig. 3. (a) The graph contains several dense nodes and
edges in some local regions, where high-degree nodes
gathering in the local regions decrease the readability
of the graph. (b) Bundling the edges, which connect to
the high-degree nodes, together can relieve the local
congestion.
trated in Fig. 2(b). According to the actual relationship
shown in Fig. 2(a), Node A - Node C and Node B - Node
D are connected, respectively, and these two relations
are represented by two individual edges. However, since
the two edges are close and in similar direction, if we
bundle these two edges together as shown in Fig. 2(b),
it becomes uncertain whether Node A is connected by
either Node C or Node D or both. Using edge-bundling for
showing a graph’s overview is reasonable since its goal is
to provide a clearly visual edge pattern. However, when
it comes to the detail-view, any perception problem
becomes crucial, since precise and accurate relationships
between nodes is usually necessary. With our improved
edge-bundling method, the ambiguity problem is ad-
dressed. Hence, our result has no such edge-ambiguity
problem as shown in Fig. 2(c).
1.2Edge Congestion
Most edge-congestion problems in the detail-view
are caused by high-degree nodes, which connect to
many other nodes. These high-degree nodes are usu-
ally clustered into one small region. This is because
most relational datasets have small-world characteris-
tics [7] [8] [9], which tend to make the graphs contain
more subgraphs with high-degree nodes gathering in a
few local regions and the edges in such regions usually
have relatively short lengths as is shown in Fig. 3(a).
Our ambiguity-free edge-bundling method condition-
ally bundles the edges to relieve the edge-congestion
problem in local regions. Ambiguities in the edge-
bundling process are considered, and only the edges
which share a common node can be merged. Fig. 3(b)
shows our result. Comparing it to Fig. 3(a), the bundled
edges better utilize the screen space which helps users
to discriminate the connections of the high-degree nodes.
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The smoothly curved-edges also help the users to trace
the edges. Inappropriate overlapping and nearness prob-
lems, which often occur in traditional edge-bundling
methods, are avoided.
2RELATED WORK
Correctly perceiving individual relations between nodes
is often the goal of requesting and examining an detail-
view of a graph in graph visualization. Several tech-
niques have been introduced to address this require-
ment. van Ham used a matrix-based representation to
reduce the visual clutter caused by edges [10]. However,
the matrix-based representation is often less intuitive
than using node-link graphs. Eades et al. proposed a
method that only draws the edges between node clus-
ters [11] [12]. Becker et al. proposed half-lines [13] to
visualize a directed edges, which reduces the visual
clutter by drawing only the first half part of the edge
between two nodes. Existing popular methods that re-
duce the edge clutter may be classified into two ma-
jor types, which are interactive approaches and edge-
routing approaches. Interactive approaches refer to the
ability to discriminate additional information by inter-
actively selecting or filtering out current information.
Fisheye views [14], a distortion-oriented method, is one
of the examples, which distorts the content while still
maintaining the user’s mental map model [15]. Many
extensive methods have been presented in the fields
of information visualization [16] [17] [18] [19] as well
as human-computer interaction [20] [21]. Through this
method, a user can interactively manipulate a fisheye-
like view to enlarge some local areas of a graph/network
and examine the details of the confusing subgraph.
Another group of methods under the interactive ap-
proaches can be treated as the edge-dispersing tech-
niques, which disperse the edges away from one local
region, so that the underlying pattern can be revealed
and edge-ambiguity can be temporarily relieved. For ex-
ample, EdgeLens [22] allows users to interactively bend
the edges away from one’s focus without modifying the
nodes’ positions. The lens open up sufficient space to
disambiguate the relationship between the underlying
nodes. Edge Plucking [23] is another technique that
allows users to pull edges away from the region of
interest. The two methods also help users to clarify
the clustered edges without edge-ambiguity problems.
However, they require a lot of efforts from the users
to interact with the local regions and to memorize the
temporal discrimination between the edges.
On the other hand, edge-routing approaches tend to
use a mathematical model to reduce edge crossings,
which cause many of the edge-congestion and edge-
ambiguity problems. Topology preserving constrained
graph layout [24] treats the edges as rubber bands, which
is a constrained graph layout algorithm that supports
polyline edges and clusters where clusters/nodes/edges
may not overlap. However, this algorithm only adapts
well if the number of edges in the graph is relatively
small and recognizable node displacements are pro-
duced.
Recently, many methods were proposed to provide
clearer visual edge patterns even with an excessive num-
ber of edges. The ability to perceive individual relations
is compromised, since perceiving the overall structure of
the graph is the primary goal of visualizing the whole
graph. In spite of this, the idea of relieving the edge-
congestion problem can still be extended for visualizing
the detail-view of the graph if we carefully consider
the perception of individuals. Carpendale et al. first
proposed an edge-displacement technique [25] by using
a distortion-oriented formula to curve the edges that are
passing by the circular region of interest. Dickerson et al.
proposed a technique called confluent drawing [26] to
visualize non-planar graphs in a planar way. The idea
is to allow groups of edges to be merged together.
Phan et al. presented the flow map layout [1] to visualize
flow data. They automatically generate a flow map that
depicts the backbone of a graph in an overview. How-
ever, it may cause the edge-ambiguity problem in the
detail-view. Force-directed edge bundling [6] used a self-
organized approach and modeled the edges as flexible
springs, which can be pulled together to form bundles.
In the detail-view, this method may also cause the edge-
ambiguity problems due to inappropriate node-edge
overlapping and bundling of unrelated edges. Hence,
users may likely perceive incorrect relations, like the
result shown in Fig. 2(b).
Cui et al. proposed an edge-bundling method [4] for
revealing the high-level edge pattern based on the nodes’
geometrical information. They used a uniform grid struc-
ture to sample the excessive edges, and then generate
a control mesh to guide the edge-curving by bundling
the edges that are geometrically close and directionally
similar. The result of this method also provides a clear
visual pattern of dense edges. However, the effectiveness
of this method highly relies on the quality of the control
mesh, and there is no guarantee that an effective mesh
can always be obtained automatically. If this method is
applied directly to visualize the detail-view of a graph,
the result also shows that the curved-edges are forced
to pass close to all nearby nodes. Thus, the relationship
ambiguity for the bundled edges frequently happens
because only the edges’ geometries are taken into con-
sideration. Hierarchical edge bundles [2] was designed
to visualize the data that contains both adjacency re-
lationship and hierarchical structure. To draw an edge
linking two tree leaves, the edge is curved according
to the path connecting the two nodes on the hierarchi-
cal tree structure. The algorithm bundles these edges
together if they share a common path segment on the
hierarchical tree. This method reduces the visual clutter
and also visualizes the hierarchical pattern embodied in
the straight-line graph. It performs well if the dataset
contains hierarchically structured nodes.
There is another approach to deal with the edge-
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Fig. 4. An overview of our ambiguity-free graph visualization process.
congestion problem called the node-adjustment ap-
proach, which adjusts the nodes’ positions instead of
curving edges. To avoid content confusion, the approach
rearranges the nodes’ positions in such a way as to mini-
mize the edge density, crossing, and occlusion. However,
this approach is difficult to obtain an appropriate rear-
rangement with dense edges [27] [28]. Furthermore, it
may not be suitable for the graphs, which the nodes’
positions have semantic meanings, e.g., the nodes repre-
sent the cities in a map.
3AMBIGUITY-FREE EDGE-BUNDLING
Fig. 4 illustrates an overview of our design. First, we
perform a force-directed layout algorithm to place nodes
in the display space. Then we use a quadtree [29] struc-
ture to decompose the two-dimensional display space
according to the nodes’ positions in the input graph, and
this hierarchical data-structure provides a subdivision-
based approach for further enhancing the efficiency in
the following six major steps:
1) Space partitioning (Sec. 3.1).
2) Occupancy detection (Sec. 3.2).
3) Edge-ambiguity avoiding (Sec. 3.3).
4) Curved-edge bundling (Sec. 3.4).
5) Rendering (Sec. 3.5).
6) Interaction (Sec. 3.6).
The space partitioning step constructs a quadtree struc-
ture by using the nodes’ positions. During this process,
it also identifies which quadtree cell is occupied by
which node(s) in the graph. The occupancy detection step
detects which quadtree cell is occupied by which edge(s)
in the straight-line mode. The edge-ambiguity avoiding
step detects if there exists edges passing near one or
more unrelated nodes. If such instances are detected,
the algorithm creates some control points for routing
these edges away from the unrelated nodes. The curved-
edge bundling step geometrically bundles some matched
edges together while conforming to the edge-ambiguity
requirement. It further enhances the visual discrimina-
tion of the bundled edges between the related nodes
and unrelated ones by separating them when necessary.
Finally, the rendering and interaction steps provide the
final visualization and interaction controls for users.
3.1Space Partitioning
In order to achieve realtime interaction, an efficient
data structure, quadtree, is first deployed to partition
the space. The quadtree is subdivided according to the
nodes’ positions. For some applications, if the nodes in
the dataset have no exact geometric information in ad-
vance, e.g., social network data, a force-based model [30]
is first applied to compute the initial nodes’ positions.
A quadtree cell is recursively subdivided into four chil-
dren, and the subdivision of a cell stops when either no
node or exactly one node is in that cell. Fig. 5 illustrates a
simple example with only seven nodes. For identification
purposes, we define two types of cells in the quadtree
structure, which are Node Cells and Empty Cells. A Node
Cell contains exactly one node in its local region, and
an Empty Cell contains no node and represents a usable
area for later steps. Unfortunately, the size of a Node Cell
may be too large, which would waste valuable screen
space. Therefore, after constructing the quadtree, we
further subdivide large Node Cells until they fall below
a threshold size U, releasing more Empty Cells in the
process. The result of this subdivision is shown in Fig. 5,
where the nodes are all put into sufficiently smaller Node
Cells. Note that, after this subdivision, Node Cells have a
maximum size of U, but no minimum size, and Empty
Cells have neither a maximum nor a minimum size.
The thresholding value U used as the stopping cri-
terion plays an important role in constructing the
quadtree. In our observations, if the value of the thresh-
old U is too small, the following edge-bundling mecha-
nism would produce a frequently-turning curve layout,
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Fig. 5. The comparison between the space partitioning
without (left) and with (right) further subdivision. The red
cells (Node Cells) are the regions occupied by the nodes,
and the green cells (Empty Cells) are free regions.
a.k.a. the zigzag effect. The zigzag curves change their
edge direction frequently, which makes it more difficult
to perceive the connections between the nodes. More-
over, the meshes in the quadtree will become much more
dense. A dense space partition significantly decreases
the performance of edge-bundling because the widths
of the Empty Cells also become smaller. On the other
hand, if the value of the threshold U is too large, the
edge-ambiguity avoiding mechanism might not be able
to properly find enough suitable Empty Cells for rerout-
ing the ambiguous edges. With less Empty Cells being
identified, the result of edge-bundling and searching the
Empty Cells for avoiding ambiguity will become worse.
In our experience, the value of the threshold U should
be decided by considering both the number of nodes on
the display and the area of the display space. Hence,
we suggest the threshold U to be defined using the
following equation:
U =
?
A/4N,
(1)
where A is the area of the display space and N is the
number of nodes on the display.
3.2Occupancy Detection
Quadtree-based space partitioning provides the ability
to quickly identify any query related to node positions.
If the available display space is sufficient for a standard
edge layout, we consider straight-line edges as the best
drawing style due to their simplicity. However, ample
amounts of display space is not always enough to solve
the visual clutter problem. In our design, the idea is
to improve the edge style from the straight-lines to
curves if the straight-line edge style has edge-congestion
or edge-ambiguity problems. Thus, it is necessary to
analyze which regions are occupied by which edge(s) in
the straight-line style, and then use this information to
detect the regions which have edge-congestion or edge-
ambiguity problems.
To detect the edge-occupied cell, we propose a divide-
and-conquer collision detection method based on the
quadtree data structure. For each edge, the system finds
two Node Cells C1 and C2 that contain two vertices of
the edge. Then, the parent cell CP in the lowest level of
?
?
?
?
?
?
?
Fig. 6.
searching Empty Cells which are passed by specific
straight-line edges.
Cells colored in sky-blue indicate the result of
the quadtree that contains both C1and C2is found. The
collision detection is performed recursively from CP. It
maintains a queue of cells that need to perform collision
detection and adds CP in the beginning of the queue.
At every step, it takes a cell C from the queue, detects
the collision of the edge and the four children of C, and
adds the children that are collided by the edge back to
the queue. Detection of edge-cell collision is done by
computing the intersection of the edge and the cell’s four
boundaries. If the cell C is a leaf of the quadtree and is
not a Node Cell, then C is intersected by strait-line edges
(the sky-blue cells in Fig. 6). The process stops when the
queue is empty.
The hierarchical directory of the space partition forms
a cell map that allows us to quickly find which cells are
occupied by which straight-line edge(s). If an edge on the
cell map only passes through Empty Cells, there is a very
low possibility that the edge will cause edge-ambiguity
problems. On the other hand, if it passes through one
or more Node Cells, the edge-ambiguity problems might
occur in those Node Cell regions.
3.3Edge-Ambiguity Avoiding
The edge-ambiguity problem in graph visualization of-
ten leads users to perceive incorrect relationships be-
tween nodes or requires considerable effort to read
the graph. In the previous step, most possible edge-
ambiguity locations are identified by detecting the Node
Cells that intersect with straight-line edges whose desti-
nation or source is not one of the nodes contained in that
cell. After determining all such Node Cells, which may
contain an inappropriate overlap or nearness problem,
we then calculate the control point(s) for every edge
and curve the local part of the involved edge(s) to
pass through the control point(s). Thus, the involved
edge(s) avoid any inappropriate overlap or nearness
problem. The position of the control point for bypassing
is determined by the following three crucial conditions:
• Whether this edge-ambiguity problem is solved or
not.
• Whether additional edge-ambiguity problem(s) will
occur or not by bypassing this location.
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