Density Functions for Visual Attributes and Effective Partitioning in Graph
Visualization
Ivan Herman M. Scott Marshall Guy Melançon*

Centre for Mathematics and Computer Sciences (CWI)



Abstract
Two tasks in Graph Visualization require partitioning: the as-
signment of visual attributes and divisive clustering. Often, we
would like to assign a color or other visual attributes to a node or
edge that indicates an associated value. In an application involv-
ing divisive clustering, we would like to partition the graph into
subsets of graph elements based on metric values in such a way
that all subsets are evenly populated. Assuming a uniform distri-
bution of metric values during either partitioning or coloring can
have undesired effects such as empty clusters or only one level of
emphasis for the entire graph. Probability density functions de-
rived from statistics about a metric can help systems succeed at
these tasks.
CR Categories and Subject Descriptors: I.3.6 [Computer
Graphics]: Methodology and Techniques – Interaction Tech-
niques; I.3.8 [Computer Graphics]: Applications
Additional Keywords: graph visualization, graph navigation,
metrics, clustering
1. INTRODUCTION
A key issue when visualizing graphs in information visualization
is the size of the data. Many applications of graph visualization
require analysis of graphs with several thousand nodes and edges.
Innovative techniques are needed to navigate, to filter, or to create
abstractions from these graphs in order to make them usable in
practice. Many interesting results have been published in the past
few years and this area of research is still very active (see, for
example, the survey on graph navigation[1]).
The use of metrics is one of the interesting techniques in this
area. The concept of a metric appears in several places in the lit-
erature[2-6], although the terminology varies. In this paper, we
will use the term to refer to a measure that is associated with a
node or an edge in the graph. The measure can be application-
specific, can be the result of some function (usually combinato-
rial) of the graph structure, or a combination of these. A few ex-
amples of metrics based on graph structure are the degree of a
node (i.e., the number of edges adjacent to the node), the size of a
subtree for a tree, or the measure of the flow of information in a
directed graph. In general, the goal is to define the relative impor-
tance of a node or an edge with respect to some semantics, where
elements with high metric values are considered more interesting
than those with low values.
Metric values that are associated with nodes and edges can be
used to determine visual attributes such as color and saturation in
order to emphasize differences among elements. A technique that
we find useful renders an edge with continuously shaded color
that reflects the metric values of the nodes at its endpoints. In this
approach, higher metric values are considered more interesting
and are assigned higher saturation values for emphasis. The over-
all effect is the emphasis of edges in the graph with the “most
interesting” metric values. This design of graphical attributes
based on metrics has already been discussed in [5, 6]. For exam-
ple, Figure 1 shows this effect when zooming into the details of a
graph; the darker and thicker lines help to navigate towards more
complex areas of the graph (this particular example uses the
Strahler metric, described in Herman et al. [5]).
Another use of metrics is the generation of fisheye views, as
presented in the seminal paper of Furnas[2]1, where he computes
the “degree of interest” of elements in a tree. Elements with low
values are hidden to improve the display of the structure (some-
times referred to as semantic fisheye) and help emphasize the
more important elements in the tree.
Generating such visual cues is not the only way to use metrics.
In a type of divisive clustering, data sets are partitioned according
to metric values, with the metric value determining group mem-
bership. This subdivision helps the user to partition the graph into
subgraphs of manageable sizes. This technique is not only vital to
navigation in large graphs but also helps the user to identify im-
portant relations among elements, thereby making the information
visualization application much more effective (see Section 5 for
an example). Such subdivision procedures become particularly
important if the underlying graph structure is not a tree, where no
“natural” subdivision (i.e., subtrees) exists.
It is important to note that all these techniques are automatic, in
the sense that no further user input is necessary to generate the
visual attributes or the clusters. A straightforward approach is to
apply a simple linear mapping from the metric values to, for ex-
ample, color saturation. This approach can work well when there
is a uniform distribution of metric values. However, experience
shows that more control over this mapping is necessary for cases

1
Furnas used the term “degree of interest” but, in our terminology, his
DOI function could be considered a type of metric.

*P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Email: {I.Herman, M.S.Marshall, G.Melancon}@cwi.nl