Visualizing Fuzzy Overlapping Communities in Networks
Corinna Vehlow, Student Member, IEEE, Thomas Reinhardt,
and Daniel Weiskopf, Member, IEEE Computer Society
Fig. 1. Fuzzy overlapping communities in a weighted undirected graph of 254 co-appearances of 77 characters in Victor Hugo’s novel
“Les Mis
´
erables” [32] shown at different levels of detail. In the fully aggregated graph, the uncertainty of a community is mapped to the
stars’ depth of jags. In the partially aggregated graph, all aggregated vertex subsets are completely certain as they are represented
by circles instead of stars. In the partially aggregated and the original graphs, the fuzzy community membership for single vertices is
mapped to the lightness and saturation of nodes using a radial color gradient.
Abstract—An important feature of networks for many application domains is their community structure. This is because objects within
the same community usually have at least one property in common. The investigation of community structure can therefore support the
understanding of object attributes from the network topology alone. In real-world systems, objects may belong to several communities
at the same time, i.e., communities can overlap. Analyzing fuzzy community memberships is essential to understand to what extent
objects contribute to different communities and whether some communities are highly interconnected. We developed a visualization
approach that is based on node-link diagrams and supports the investigation of fuzzy communities in weighted undirected graphs at
different levels of detail. Starting with the network of communities, the user can continuously drill down to the network of individual
nodes and finally analyze the membership distribution of nodes of interest. Our approach uses layout strategies and further visual
mappings to graphically encode the fuzzy community memberships. The usefulness of our approach is illustrated by two case studies
analyzing networks of different domains: social networking and biological interactions. The case studies showed that our layout and
visualization approach helps investigate fuzzy overlapping communities. Fuzzy vertices as well as the different communities to which
they belong can be easily identified based on node color and position.
Index Terms—Overlapping community visualization, fuzzy clustering, graph visualization, uncertainty visualization
1 INTRODUCTION
Relations among objects are usually modeled as a graph consisting of
a set of vertices connected by a set of edges. Graph visualization is
a useful tool to understand the network properties. Graphs that repre-
sent real systems, e.g., biological or social networks, are not regular,
i.e., the distribution of edges is globally and also locally inhomoge-
neous. Oftentimes, they consist of structural subunits, i.e., highly in-
terconnected sets of vertices, where the density of edges between these
groups is low. Such clusters, in the graph-theoretical sense called com-
munities, can be detected by community detection methods.
The analysis of such community structures is of high importance to
understand the structural and functional properties. Based on the com-
munity structure, we can draw conclusions about the attributes of the
network members because objects within the same community usually
have some properties in common. Unfortunately, vertices may belong
• Corinna Vehlow is with VISUS, University of Stuttgart, Germany. E-mail:
corinna.vehlow@visus.uni-stuttgart.de.
• Daniel Weiskopf is with VISUS, University of Stuttgart, Germany. E-mail:
daniel.weiskopf@visus.uni-stuttgart.de.
• Thomas Reinhardt is with University of Stuttgart, Germany. E-mail:
reinhats@studi.informatik.uni-stuttgart.de.
Manuscript received 31 March 2013; accepted 1 August 2013; posted online
13 October 2013; mailed on 4 October 2013.
For information on obtaining reprints of this article, please send
e-mail to: tvcg@computer.org.
to several communities at the same time. The overlap in networks is
obvious in our everyday life, e.g., nodes in social networks participate
in a multitude of diverse, overlapping contexts, all encoded in a single
network structure [39]. Vertices assigned to more than one community
are usually located at the boundary between clusters, thereby repre-
senting mediates or bridges between these communities. The identi-
fication of such bridges is an important topology-based task for ana-
lyzing the connectivity in graphs [35]. In contrast, vertices that have
a high edge degree within the community take up a central position in
the community and have an important function concerning the stabil-
ity of the group. To analyze communities in realistic systems, the issue
of detecting overlapping communities has become of high interest and
various algorithms have been developed [16].
In general, we can differentiate two types of overlapping communi-
ties: crisp and fuzzy overlapping communities [20]. With crisp over-
lapping community detection methods [19, 40, 44], each object fully
belongs to one or more communities; with fuzzy overlapping commu-
nity detection methods [4, 34, 53], vertices may belong to different
communities to different extent. Which type of overlapping commu-
nities is more suitable, depends on the data.
In this paper, we consider two typical application domains: bio-
logical and social networks, in which crisp as well as fuzzy over-
lapping communities occur. Hence, there is a need for visualization
approaches for fuzzy overlapping communities facilitating the analy-
sis of those. Previously available visualization techniques mainly al-
low us to visualize crisp overlapping or disjoint communities but do
2486
1077-2626/13/$31.00 © 2013 IEEE Published by the IEEE Computer Society
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 19, NO. 12, DECEMBER 2013
not support analyzing uncertainty of communities described by fuzzy
community memberships. We present a visualization approach that
enables users to analyze fuzzy overlapping communities in weighted
undirected graphs at different levels of detail (see Figure 1). Compared
to existing visualization approaches for crisp overlaps, our approach
benefits from its advanced visual mappings and hierarchical structure.
Our approach is based on node-link diagrams, where the fuzziness of
the community memberships is encoded in the nodes’ positions as well
as their color gradients or shapes. The former is achieved by a layout
approach that incorporates the fuzzy memberships. Due to the hier-
archical structure of our approach, users can use it to investigate the
network at the community level, the vertex level, as well as all inter-
mediate states in between by partially aggregating subsets of vertices
with a maximum fuzziness. Therefore, our approach allows users to
concentrate on shared nodes of minimal fuzziness only. Finally, the
details-on-demand option facilitates the investigation of the strength
with that each vertex contributes to each of the communities, which is
defined by the belonging coefficients of the respective vertex.
2 RELATED WORK
The most prominent visual representations of relational data are node-
link diagrams and adjacency matrices. Node-link diagrams are intu-
itive and effective for perceiving relations between objects because
they exploit Gestalt principles [33] of closure and good continuation.
When using node-link diagrams, vertices are mapped to geometric
forms such as circles or squares and relations among them are ex-
pressed by straight or curved links. A challenge is the computation of
an aesthetic layout. Common graph layout algorithms, such as force-
directed, orthogonal, or hierarchical layout algorithms, aim at optimiz-
ing a set of aesthetic criteria for graph drawings [13]. Force-directed
layout algorithms, such as the Fruchterman-Reingold model [18] or
the Kamada-Kawai model [31], can reveal clusters. Due to the com-
bination of repulsive spring forces between all nodes and attractive
forces between adjacent nodes, vertices that are highly connected are
positioned close to each other. Thereby, clusters emerge visually but
roughly, although they have not been extracted explicitly. However,
this “visual clustering approach” does not extract fuzzy memberships
as it does not become apparent to which communities a vertex be-
longs with what extent. We use a force-directed approach to produce
an aesthetic global layout for aggregated graphs representing relations
between communities.
Clusters or communities can also be derived algorithmically based
on community detection algorithms. Again, layout algorithms can
help reveal cluster structures in the graph, e.g., using a divide-and-
conquer approach in which each cluster is laid out separately before
the clusters are composed to form the graph [8, 52]. Besides spatial
proximity, color can be used to convey to which community a vertex
belongs [3]. In contrast, hierarchically clustered graphs are usually
visualized as recursively nested regions in the plane [14]. Groups of
vertices are thereby surrounded by 2D [3] or 3D (semi-transparent) [7]
convex hulls. Alternatively, hierarchically clustered graphs can be vi-
sualized using multi-level representations that visualize graphs at dif-
ferent abstraction levels by aggregating subgroups of vertices (clus-
ters) and edges [14]. They are usually realized by drawing each level
on a plane at a different z-coordinate and with the clustering struc-
ture drawn as a tree in the third dimension. In addition, there is much
previous work on using hierarchical clustering to control the extent
of represented information and to navigate through the graph using
graph aggregation methods [2, 15, 49, 51]. The system developed by
Archambault et al. [6] even supports the investigation of several hier-
archies of the same data based on different clustering criteria. In com-
parison to the approaches mentioned before, Ham and Van Wijk [49]
combine their degree of abstraction function with the fisheye approach
to show detailed information for a specific section and fewer details
for its surrounding. This degree of abstraction function is based on the
topology-based hierarchical clustering they extract in advance based
on a force-directed layout using a distance measure. Of course, also
non-hierarchically clustered graphs can be visualized on an abstract
level, e.g., CFinder [3], which provides users with a graph view of the
communities. However, all these methods were developed for non-
overlapping communities.
Existing visualization approaches mainly address the visualization
of flat disjoint communities and crisp overlapping communities. The
results of crisp overlapping community detection algorithms are so
far visualized by highlighting shared nodes in a different color [3, 4].
Besides, overlapping sets of vertices or elements in general are com-
monly represented using Euler-like diagrams [37, 43, 46], overlapping
convex hulls [22, 34, 40], or bounding isocontours [12]. While some
approaches use only color and transparency of the contours or shapes
to convey set membership [12, 34, 40, 43], Simetto et al. [46] addi-
tionally make use of texture. Alper et al. [5] use curves of different
color connecting all elements of a set instead of surrounding them to
represent set memberships. Only few papers included visualizations
of fuzzy overlapping communities [4, 50], where the membership dis-
tribution, i.e., the belonging coefficients, of a vertex are represented by
a pie chart. In the approach by Itoh et al. [24], pie charts are used to vi-
sualize the multiple categories to which a node belongs and which are
divided into equally sized segments. As pie charts should have a min-
imum size to allow for the differentiation of membership degrees, the
display becomes cluttered easily with increasing size of nodes. How-
ever, most of the visualization approaches are only for crisp and not
for fuzzy overlapping communities.
Whereas the use of convex hulls does not support the investigation
of fuzzy overlapping communities, some of the other approaches can
be adapted to that effect. Similar to existing approaches, we use spatial
proximity [52] and color [3] to visualize the memberships but indeed,
with respect to their fuzziness. In particular, the node-link diagram at
the vertex level is laid out based on a divide-and-conquer approach that
includes the fuzzy membership into the calculations of the vertex po-
sitions. Moreover, our approach supports the user with a graph view
of the communities, similar to the work of Adamcsek et al. [3], and
is furthermore related to multi-level graph drawings [14], where only
the graph view of one particular level is displayed at a time instead of
showing a three-dimensional stack of drawings. The multiscale hierar-
chy does in this case not result from a hierarchical clustering approach
but is generated based on the fuzzy memberships.
To analyze fuzzy overlapping communities, the fuzziness of the ver-
tices memberships needs to be visualized. The quantification and vi-
sualization of uncertainties within data has been recognized as one of
the most important issues in scientific visualization [25]. Here, uncer-
tainties originate during the clustering process because vertices cannot
be allocated to one single community due to their topology. This un-
certainty, i.e., the fuzziness of the vertices memberships, is described
by scalar values, which could be directly visualized using one of the
possible approaches: adding glyphs or geometry to the rendered scene,
modifying geometry, modifying attributes of the geometry, animation,
or by addressing other human senses [41]. A common approach is
the modification of geometry using visual attributes, like color, size,
position, shape, transparency, and so on, or the plotting of discrete
data points as glyphs (e.g., box plots or quartile plots) with specific vi-
sual attributes. Concerning color, in particular, the lightness or trans-
parency is commonly used to visualize uncertainty. When using the
deformation of geometry, usually the degree of bumpiness is used to
represent uncertainty, where smooth shapes imply certainty. We de-
cided to use two different mappings: the lightness of a node to repre-
sent the fuzziness of individual vertices and the shape of the node to
represent the overall fuzziness of aggregated vertex sets.
While there is extensive research on visualizing uncertainties of
flow fields and surface representations [21, 26], only little work has
been done on visualizing uncertainties within graphs, i.e., the uncer-
tainties of attributes of the graph. Collins et al. [11] visualized un-
certainties of translations using lattice graphs that show multiple lin-
ear paths for a translation. They mapped the likelihood of a trans-
lation to the fuzziness and hue as well as on the vertical positioning
of nodes. Cesario et al. [9] visualized uncertainties of multiple static
node attributes using a spatial layout and multiple linked views, e.g.,
the bullseye. Within the bullseye, nodes are plotted at a specific angle
and radius within a circle subsector to visualize the attribute value and
2487VEHLOW ET AL: VISUALIZING FUZZY OVERLAPPING COMMUNITIES IN NETWORKS
v
7
v
1
v
6
θ
mv
0.0
1
mv
0.33
1
mv
1.0
1
MV
1.0
1
→C
1
1.0
0.0
f
e
e
e
e
e
me
0.0
me
0.0
me
0.33
me
0.33
e
e
→ vertex set of
the core graph G
core
1
v
6
v
6
v
7
v
1
v
2
v
3
v
5
v
4
Fig. 2. Meta-graphs G
θ
describing different aggregation levels of a
graph, where only the section of community C
1
with |C
1
| = 7 is repre-
sented. Vertices v
i
and meta-vertices mv
θ
1
are saturated with respect to
their highest belonging coefficient f
i
max
and average belonging f
1
avg
.
confidence respectively. Their approach is designed to compare two
static graphs and their attributes. Both approaches make use of the
position to encode uncertainty but could not be extended to visualize
fuzzy (and hence uncertain) overlapping communities.
3 VISUALIZATION TECHNIQUE
Our visualization approach for fuzzy overlapping communities in
weighted undirected graphs is based on node-link diagrams. The
fuzziness of community memberships is visually encoded by the node
positions as well as the attributes shape and color of a node. We de-
cided to combine different mappings because using either the position
or color of nodes leaves ambiguities concerning the communities to
which a vertex belongs. After introducing our data model, we will
present our hierarchical layout approach that considers the fuzzy mem-
bership distributions. In the following subsections, we will introduce
our layered visualization model and show how the fuzziness of ver-
tices or aggregated sets of vertices as well as the edge weights can be
mapped to visual attributes of nodes and links, respectively. Finally,
we will explain how users can interact with our visualization approach.
3.1 Data Model
We model an undirected weighted graph G = (V,E) as a set of ver-
tices V and a set of edges E ⊆V ×V , where each edge e
j
∈E, 1 ≤ j ≤
m, m = |E| is assigned a weight w
e
j
∈ R. We define the overlapping
community structure of the graph as set of communities {C
1
, . . . ,C
K
},
where each C
k
⊂V and K denotes the number of clusters. The com-
munities are not necessarily disjoint, i.e., there are at least two com-
munities C
k
1
and C
k
2
with C
k
1
∩C
k
2
6= /0. Then, fuzzy overlapping
communities, also referred to as covers, can be described by the cover
matrix F [38]. In particular, a belonging coefficient, also called fuzzy
membership degree, f
ik
describes how strongly v
i
belongs to the k-th
cluster C
k
. This strength is usually expressed by values f
ik
∈ R with
0 ≤ f
ik
≤ 1 such that for each v
i
,
∑
K
k=1
f
ik
= 1 [20]. Using this mea-
sure, none of the vertices can strongly belong to several communities.
Vertices v
i
whose membership degrees are equally distributed across
communities, e.g., f
i1
= 0.5 and f
i2
= 0.5, represent perfect bridges
between these communities [38].
We define the predominant community C
i
k
of a vertex v
i
to be that of
the highest membership degree f
i
max
. The fuzziness f
i
fuz
of a vertex
v
i
is then described by 1 − f
i
max
. Vertices whose fuzziness is greater
than 0.5 are considered as extremely fuzzy vertices because they do
belong to their predominant community with less than 50% certainty.
If a vertex belongs to several communities equally, e.g., f
i1
= 0.5
and f
i2
= 0.5, the predominant community is selected randomly from
them. Based on the predominant community structure, we construct
aggregated graphs, also called meta-graphs, G
θ
= (V, E,V
meta
, E
meta
)
by collapsing subgraphs G
θ
k
= (MV
θ
k
, ME
θ
k
) with MV
θ
k
⊆ C
k
and
ME
θ
k
⊆ MV
θ
k
×MV
θ
k
into meta-vertices mv
θ
k
∈ V
meta
and by trans-
forming inter-cluster-edges e
j
of G into meta-edges me
θ
∈E
meta
. The
weight w
me
θ
of a meta-edge me
θ
(mv
θ
1
, mv
θ
2
) is equal to the sum of
weights of all the inter-cluster-edges between vertices of the two sub-
sets MV
θ
1
and MV
θ
2
. Of course, meta-edges can also connect a meta-
vertex mv
θ
k
with a single vertex v
i
, aggregating the edges between
those.
The aggregation level of the graph depends on the threshold θ ∈
R [0, 1], as only vertices v
i
with f
i
max
≥ θ are aggregated into the
respective meta-vertex mv
θ
k
of their predominant community C
i
k
(see
schematic illustration in Figure 2). This threshold is used as deter-
mining parameter for our degree-of-interest function. An aggregated
graph G
θ
may therefore consist of vertices v
i
and meta-vertices mv
θ
k
,
where G
0
contains meta-vertices only. G
1
represents a second special
case, as it aggregates only non-fuzzy vertices ( f
i
fuz
= 0). Finally, G
>1
is equal to the original complete graph G containing vertices only (in
the following the superscript
>1
will be dropped for G and commu-
nity subsets MV
k
). We define the certainty of a meta-vertex mv
θ
k
as
the average belonging coefficient f
k
avg
of all its vertices v
i
∈ MV
θ
k
:
∑
f
i
max
|MV
θ
k
|
representing how strongly the members belongs to the com-
munity, where f
k
avg
= 1 for a completely certain community. The
certain community subsets MV
1.0
k
with f
i
max
= 1.0 for all v
i
∈C
k
are
referred to as the core communities.
3.2 Graph Layout
As mentioned in Section 2, force-directed layouts can help reveal com-
munities because vertices that are highly connected are often—but not
always—positioned close to each other (see Figure 3(a)). Our visual-
ization approach uses the Gestalt principles of closure (spatial proxim-
ity) as well as texture patterns [33] to encode the community member-
ships using a hierarchical layout approach. In particular, we employ a
regular and highly symmetric disk-like layout for the core communi-
ties to visually differentiate their (certain) vertices from other (fuzzy)
vertices (see Figure 3(c)). For background literature on texture per-
ception, we refer to Julesz) [29, 30]. Mirror symmetry is known to be
recognized preattentively [23, 48]. Therefore, our layout is designed to
show mirror symmetry; in fact, the regular disk-shaped layout shows
mirror symmetry along multiple axes to facilitate preattentive percep-
tion and hence effortless and efficient differentiation between certain
and fuzzy community members. Regularity is one of the primary tex-
ture dimensions [42]. Therefore, we generate layouts with regular pat-
terns for community cores in order to make their texture appearance
different from that of fuzzy vertices, adding a visual cue on top of
mirror symmetry. Furthermore, the distance of fuzzy vertices to their
community gives some indication of their fuzziness.
Our hierarchical approach is related to the divide-and-conquer ap-
proach by Wang et al. [52], in which each cluster is laid out separately
before the clusters are composed to form the graph (see Figure 3(b)):
Algorithm 1 Calculate the layout of G
1: L ← layout(G
0
) {lay out aggregated graph}
2: for k=1 to K do
3: L
k
← sublayout(G
k
) {lay out community subgraph}
4: integrate L
k
into L
5: end for
Similar to the approach by Wang et al., the global layout (line 1 of
Algorithm 1) is derived based on a layout algorithm that considers the
size of the sublayouts, which is proportional to the number of vertices
2488 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 19, NO. 12, DECEMBER 2013
(a) (b) (c)
Fig. 3. Comparison of different layout approaches for an example graph: a personal friendship network of a faculty of a UK university, consisting of
81 vertices (individuals) and 817 weighted connections [38]. The network was clustered using the algorithm by Gregory [20] for fuzzy overlapping
communities. (a) The graph is laid out with the force-directed Fruchterman-Reingold model [18] regardless the graph’s community structure. (b) The
layout is derived using a divide-and-conquer approach, where each cluster is laid out separately using the force-directed Fruchterman-Reingold
model before the clusters are composed to form the graph. Shared (fuzzy) nodes are thereby included in the subgraph of their predominant
cluster. (c) The layout is derived using our extended divide-and-conquer approach that includes the fuzziness of nodes, i.e., the different belonging
coefficients, in the calculation. Vertices with certain cluster membership are visually encoded by regular and symmetric disk-shaped layouts.
the subgraphs G
k
contain. In contrast to their approach, we incorporate
an additional division step to derive the sublayouts (both alternatives
are compared in Figures 3(b) and 3(c)):
Algorithm 2 Calculate the sublayout of G
k
1: MV
1.0
k
← all v
i
∈C
k
with f
i
max
= 1.0 {extract core-vertices}
2: G
core
k
← (MV
1.0
k
, MV
1.0
k
×MV
1.0
k
) {extract complete core graph}
3: L
core
← layout(G
core
k
) {lay out core graph}
4: L
k
← layout(G
k
,L
core
) {lay out subgraph}
5: return L
k
To achieve a regular and symmetric disk-shaped layout for a com-
munity core, the core-vertices are positioned based on a layout de-
rived for the complete core-graph G
core
k
(lines 1–3 of Algorithm 2).
In particular, G
core
k
is laid out using the force-directed Kamada-Kawai
model [31] based on spring forces proportional to the graph theoretic
distances, as this creates very regular layouts.
In the fourth step of Algorithm 2, all non-core-vertices v
i
∈G
k
with
f
i
max
< 1.0 are positioned around the core-layout L
core
using a force-
directed approach of attracting and repulsive forces that depend on the
belonging coefficients f
ik
and are applied to non-core-vertices only.
The repulsive force between any two vertices v
i
is small, where the
forces between a vertex v
i
and the “pseudo-vertices” representing C
i
k
,
positioned at the center of L
core
, depends on f
i
max
, i.e., the weaker
the membership degree of v
i
to its predominant community C
i
k
, the
stronger is the force of repulsion. This produces distances of vertices
v
i
to the center of the core that are nearly proportional to their fuzzi-
ness. Furthermore, the non-core-vertices v
i
are pulled toward those
communities to which they also belong. This is achieved by attract-
ing forces between a vertex v
i
and all its communities C
k
represented
by “pseudo-vertices”, which are again proportional to the respective
membership degrees f
ik
. For example, for a vertex v
i
with the belong-
ing coefficients of f
i1
= 0.5, f
i2
= 0.4, and f
i3
= 0.1, the attracting
force to community C
2
will be much higher than for C
3
but highest for
C
1
. Whereas the repulsive forces guarantee a minimum distance be-
tween the core-graphs and the fuzzy vertices, the attracting forces keep
the vertices within a maximum distance from the core-graph. Vertices
will therefore mostly, i.e., if f
i
max
>> f
ik
for each C
k
6= C
i
k
, be posi-
tioned closest to their predominant community and toward one or two
other communities to which it significantly belongs. In the final step
(line 4 of Algorithm 1), the sublayouts are integrated into the global
layout L to form the overall layout (see graphs at θ > 1 in Figure 4).
We provide several alternatives for the global layout: a force-
directed layout, either the Fruchterman-Reingold model [18] or the
Kamada-Kawai model [31], and a circular layout. For the force-
directed approaches, the size of the sublayout is described by 2D rect-
angular areas. In contrast, for the circular approach it is described
by an arc of particular length. For our case studies, we use the
Fruchterman-Reingold model as force-directed layout, as it provides
layouts that are aestethically more pleasing, at least for these datasets.
3.3 Layered Visualization Model
Our visualization approaches provides users with graphs at different
aggregation levels described by a degree-of-interest function. This
function aggregates nodes of a particular degree of fuzziness described
by the threshold θ . The layout of the original complete graph (see Fig-
ure 4 right) is used as a basis to derive the layout for any aggregated
graph G
θ
. Meta-nodes (mv
θ
k
) are thereby positioned at the barycenter
of the vertices v
i
∈MV
θ
k
they aggregate. Due to our hierarchical layout
approach, meta-nodes are positioned near the center of the core-graph
G
core
k
. Therefore, changing the level of detail by in- or decreasing the
threshold θ , produces a sequence of plots of the graphs G
θ
at different
abstraction levels that preserves the mental map (see Figure 4). This
holds for both layout approaches, the force-directed and the circular
layout approach.
Figure 4 shows the complete sequence of aggregation states for a
small example graph. The graph consists of 12 vertices, 3 communi-
ties, and 3 nodes whose community memberships are fuzzy. Starting
with threshold θ = 0, the graph contains meta-vertices only and rep-
resents the fully aggregated graph. Increasing the threshold to θ = 0.7
separates node e with f
max
= 0.664 from C
3
. A further increase of θ
to θ ≥ 0.91 separates nodes d and i both with f
max
= 0.908 from C
1
and C
2
, respectively, and results in an aggregated graph whose meta-
vertices contain core-vertices only, i.e., f
k
avg
= 1.0 for all three aggre-
gated subgraphs. Setting the threshold θ > 1.0 finally breaks up the
partially aggregated graph to its single vertices v
i
.
3.4 Node-Oriented Visual Mapping
Besides the position, our visualization approach uses further visual
attributes of nodes to encode the community memberships and to em-
phasize the degree of fuzziness of shared nodes (see Figure 5). To
differentiate between single nodes v
i
and meta-nodes mv
θ
k
, they are
represented using different shapes: circles and stars for v
i
and mv
θ
k
,
respectively. To allocate nodes and meta-nodes to a cluster C
k
, each
cluster C
k
is assigned a color based on a colormaps created with Col-
orBrewer [1] and nodes (meta-nodes) are colored with respect to their
predominant community C
i
k
(community C
k
). We use two different
approaches, the modification of lightness and geometry of the object’s
shape, to visualize the node’s and meta-node’s fuzziness, respectively,
to differentiate between both types of nodes. Both mappings repre-
2489VEHLOW ET AL: VISUALIZING FUZZY OVERLAPPING COMMUNITIES IN NETWORKS
θ < 0.7 0.7 ≤ θ < 0.91 0.91 ≤ θ ≤ 1.0 θ > 1.0
Fig. 4. Sequence of graphs G
θ
showing the fuzzy overlapping community structure for an example graph with |V |=12 and K=3 at different levels of
detail, i.e., using different thresholds θ , starting with the fully aggregated graph (left) and ending with the detailed graph (right). For comparison,
the sequences are laid out with a force-directed (top row) and circular (bottom row) global layout L. Increasing θ first of all separates the fuzziest
vertex of the graph (e) from its predominant community C
3
followed by the less fuzzy vertices d and i.
r
in
v
r
out
v
f
i
max
(a) Fuzziness of nodes.
r
in
mv
r
out
mv
f
k
avg
(b) Fuzziness of meta-nodes.
Fig. 5. Mapping the community membership fuzziness to visual at-
tributes of the nodes representing (a) vertices v
i
or (b) meta-vertices
mv
θ
k
. For vertices, the inner circle of the node defined by r
in
v
∼ 1 − f
i
max
is rendered with a radial color gradient. Meta-vertices, on the contrary,
are represented by stars, where again the inner circle depends on the
fuzziness r
in
mv
∼ f
k
avg
. The sequences show the visual mappings for
f
i
max
: [0.2,1.0] and f
k
avg
: [0.6,1.0], respectively.
sent common approaches to convey uncertainty [41]: the brighter (the
more distorted) a shape, the higher the uncertainty.
To visualize the fuzziness of the community membership of a vertex
v
i
to C
i
k
, nodes with f
i
fuz
> 0 are rendered with a color gradient instead
of a constant color (see Figure 5(a)). The gradient progresses radially
from the center of the circle to the inner radius r
in
v
, starting with white
in the center and ending with the respective color for C
i
k
at the inner
circle. The annulus defined by r
in
v
and r
out
v
is rendered without gradient
in the community color. The inner radius r
in
v
describes the circle whose
area is scaled compared to the outer circle area using the fuzziness:
r
in
v
=
√
f
i
fuz
r
2
. To emphasize fuzzy nodes ( f
i
fuz
> 0), these are rendered
with a slightly increased radius compared to non-fuzzy nodes ( f
i
fuz
=
0). If a vertex belongs to several communities C
k
with similar extent,
the circle is divided into the respective number of segments whose size
is proportional to the belonging coefficient f
ik
, where each segment is
rendered with the gradient as described before. We use a threshold of
10% for the similarity criterion in the examples of this paper, i.e., two
membership degrees are regarded as similar if f
i
l+1
≥ 0.9 f
i
l
, where
l denotes the index of the descendingly ordered membership degrees
f
ik
of v
i
, starting with f
i
0
= f
i
max
. Similar to the approach by Itoh
et al. [24], the segments and hence colors of the circle are arranged
such that they are closest to the respective communities. This mapping
shows which communities contribute significantly to the fuzziness of a
vertex, where small coefficients f
ik
remain disregarded by normalizing
the significant coefficients.
For meta-nodes mv
θ
k
, the certainty f
k
avg
is represented by the fringe
degree of the star (see Figure 5(b)). The outer radius r
out
mv
of the star
representing mv
θ
k
describes the circle whose area is proportional to the
number of vertices aggregated in mv
θ
k
, i.e., r
out
mv
=
√
|MV
θ
k
|r
out
v
. The inner
radius r
in
mv
depends on the average strength: r
in
mv
= f
k
avg
2
r
out
mv
. The quadratic
mapping of f
k
avg
allows for an enhanced differentiation of community
fuzziness. Using different visual mappings for the fuzziness of ver-
tices and the sharpness of meta-vertices has the advantage that both
types of vertices are clearly distinguishable from each other.
Besides the fuzziness of nodes, also the distribution of membership
degrees f
ik
of individual vertices v
i
should be visualized because de-
termining to what extent a vertex contributes to its communities is an
important task when analyzing fuzzy overlapping communities. We
decided to use pie charts and bar charts in the force-directed and cir-
cular node-link diagram, respectively. To ensure the readability of the
charts, these are rendered not smaller than a user-specified minimum
size, i.e., radius or width and length, respectively. To avoid visual
overload in the force-directed layout, only selected nodes are rendered
as pie charts, because these have a much bigger radius to make the
individual segments clearly recognizable. In contrast, in the circular
diagram, the bars are attached radially to the nodes instead of replac-
ing them. For both charts, the segments are ordered descendingly by
the fuzzy membership degrees f
ik
.
Whereas in the force-directed layout approach nodes are connected
by straight links, in the circular layout approach we use curved links
to produce aesthetically pleasing diagrams. The curvature of a link
connecting two nodes v
1
and v
2
positioned at θ
1
and θ
2
decreases
with increasing angular distance, such that the link is straight for ∆θ =
180
◦
. To differentiate intra-community edges (e(v
1
, v
2
) with C
1
k
= C
2
k
)
from inter-community edges (C
1
k
6= C
2
k
), the former are rendered in
the respective community color for C
k
, while the latter are rendered
in black. The edge weights (weights w
e
of edges e
j
and aggregated
weights w
me
of meta-edges me) are mapped to the width of the link.
3.5 Interaction Techniques
The most important interaction technique of our visualization ap-
proach is the support to continuously drill from the highest aggregation
level (G
0
) down to the most detailed level showing the complete graph
2490 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 19, NO. 12, DECEMBER 2013
![Fig. 7. Shows the same graph G as Figure 6(d) laid out with the forcedirected Fruchterman-Reingold model [18]. Using a force-directed layout approach, coloring of nodes with respect to their community (same color scheme as in Figures 6 is used) to which they belong and highlighting shared nodes in a different color (here black), is a common approach to analyze crisp overlapping communities.](/figures/fig-7-shows-the-same-graph-g-as-figure-6-d-laid-out-with-the-3509jjrm.webp)
![Fig. 3. Comparison of different layout approaches for an example graph: a personal friendship network of a faculty of a UK university, consisting of 81 vertices (individuals) and 817 weighted connections [38]. The network was clustered using the algorithm by Gregory [20] for fuzzy overlapping communities. (a) The graph is laid out with the force-directed Fruchterman-Reingold model [18] regardless the graph’s community structure. (b) The layout is derived using a divide-and-conquer approach, where each cluster is laid out separately using the force-directed Fruchterman-Reingold model before the clusters are composed to form the graph. Shared (fuzzy) nodes are thereby included in the subgraph of their predominant cluster. (c) The layout is derived using our extended divide-and-conquer approach that includes the fuzziness of nodes, i.e., the different belonging coefficients, in the calculation. Vertices with certain cluster membership are visually encoded by regular and symmetric disk-shaped layouts.](/figures/fig-3-comparison-of-different-layout-approaches-for-an-1hoavodr.webp)
![Fig. 5. Mapping the community membership fuzziness to visual attributes of the nodes representing (a) vertices vi or (b) meta-vertices mvθk . For vertices, the inner circle of the node defined by r in v ∼ 1− fimax is rendered with a radial color gradient. Meta-vertices, on the contrary, are represented by stars, where again the inner circle depends on the fuzziness rinmv ∼ fkavg. The sequences show the visual mappings for fimax : [0.2,1.0] and fkavg : [0.6,1.0], respectively.](/figures/fig-5-mapping-the-community-membership-fuzziness-to-visual-gn3luubr.webp)