3D Graph Drawings: Good Viewing for Occluded Vertices

Tariq O. Fadl Elsid Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 9, (Part - 3) September 2015, pp.19-23
www.ijera.com 19|P a g e
3D Graph Drawings: Good Viewing for Occluded Vertices
Tariq O. Fadl Elsid1
,Samani A. Talab2
1
F. of Science & Technologies,Nile Valley University, Atbara, Sudan,
2
F. of Computer Science& Information Technologies,Al-Neelain University, Khartoum, Sudan
Abstract
The growing studies show that the human brain can comprehend increasingly complex structures if they are
displayed as objects in three dimensional spaces. In addition to that, recent technological advances have led to
the production of a lot of data, and consequently have led to many large and complex models of 3D graph
drawings in many domains. Good Drawing (Visualization) resolves the problems of the occluded structures of
the graph drawings and amplifies human understanding, thus leading to new insights, findings and predictions.
We present method for drawing 3D graphs which uses a force-directed algorithm as a framework.
The main result of this work is that, 3D graph drawing and presentation techniques are combined available at
interactive speed. Even large graphs with hundreds of vertices can be meaningfully displayed by enhancing the
presentation with additional attributes of graph drawings and the possibility of interactive user navigation.
In the implementation, we interactively visualize many 3D graphs of different size and complexity to support
our method. We show that Gephi Software is capable of producing good viewpoints for 3D graph drawing, by
its built-in force directed layout algorithms.
Keywords:3D Graph Drawings, 3D Graph Visualization, Vertex-Vertex Occlusion, Good Viewpoint, Gephi.
I. Introduction
The field of graph drawing is a basic
visualization tool. For graphs of up to hundreds of
nodes(vertices) and edges, there are many effective
techniques available. The visualization of relational
information as a drawing of a graph in space.
Traditionally, graph drawing research has
concentrated on creating two-dimensional drawings.
However, in this decade, researchers have begun to
explore and analyze the possibilities offered by three
dimensional data are often made by means of
dimension reduction techniques, that means three-
dimensional graph drawing is mapped to a two-
dimensional image via a projection.
Experiments suggest that people can comprehend
information more efficiently when it is presented in
three dimensional representations because it convey
information, provide a meaningful mapping of data in
such low dimensional spaces is the issue [5], [16].
Unfortunately, the past ten years of 3D graph
drawing research has had very little impact on the
graph drawing industry. Even though these 3D
algorithms are theoretically significant, none of them
have been adopted by the commercial graph drawing
software providers. However, achieving good 3D
visualization is, in fact, quite a challenging problems
due to the occlusion and navigation problems
involved, these problems often dissolve the
effectiveness of the three dimensional graph
drawings [12].
We focus on the problem of drawing graph with
straight-line edges. By using an approach of force-
directed algorithms which are the most effective
techniques for handling undirected graphs, the
algorithms are based on virtual physical models.
Force-directed approach introduces a heuristic
and an energy functionto map the graph drawing, by
achieves its local minimum when the layout is nice.
Variants of this approach differ in the definition of
the energy, and in the optimization method that finds
its minimum.
The final resulting layout of the force-directed
approach brings the system to
equilibrium(conversion), where the total forces on
each vertex is zero, or equivalently, the potential
energy is locally minimal with respect to the vertex
positions. Regarding the drawing standard, force-
directed methods draw the edges as straight-line
segments, so the whole issue reduces to the problem
of positioning the vertices. For some known
algorithms are those of [15], [4], [2]. Major
advantages of force-directed methods are their
relatively simple implementation and their flexibility
(heuristic improvements are easily added), but there
are some problems with them too. One severe
problem is the difficulty of minimizing the energy
function when dealing with large graphs. The above
methods focus on graphs of up to 100 vertices. For
larger graphs the convergence to a minimum, if
possible at all, is very slow [10].
In this paper, we propose a model that is based
uponFruchterman manReingoldalgorithm [4] to
address the problem of vertex-vertex occlusions of
three dimensional graph drawings. Fruchterman and
Reingold algorithm is a force directed layout
algorithm. Our model produces good viewpoints of
three dimensional graph drawings without vertex-
RESEARCH ARTICLE OPEN ACCESS

vertex occlusions. The model aims to draw a 3D
graph that has a good viewpoints on three
dimensional plane placing vertices and edges of the
graph on the screen by computing forces on vertices,
in order to produce a pleasant arrangement that
allows simplifying the understanding of the graph.
The rest of the paper is organized as follows.
Section 2, reviews related work. In Section 3, we
present our method that based on Fruchterman and
Reingold algorithm [4] for three dimensional graph.
Section 4, discusses the results, and describes the
quality of the resulting drawings on the layout side.
Finally, we give some conclusions and describe
future research in Section 5.
II. Related Work
The problem of the occlusion in graph drawing
(graph visualization) has beenstudied for many years,
in this section, we outline the published work on
force-directed graph drawing.
Affordable high quality 3D graphics in every PC
has motivated a great deal of research in 3D graph
drawing over the last ten years. The proceedings of
theannual Graph Drawing conferences document
these developments [8].
There are many examined with the force directed
methods, the resulting drawing can reduce visual
complexity and occlusion, and ease navigation. For
examples:
Ware[19] designs 2.5D uses depth selectively
and pays special attention to 2D layout may provide
the best match with the limited 3D capabilities of the
human visual system. The model examined by the
PolyPlane methods, the resulting drawing can reduce
visual complexity and occlusion, and ease navigation.
Kamada and Kawai [15] model the forces slightly
differently, with springs following Hooke’s law,
between every pair of nodes. A preprocessing step
sets the strength of each spring proportional to the
graph theoretic distance between its end nodes. From
this they derive an objective function for optimal
placement of a node(vertex) based on the combined
springs from all the other nodes. They solve for each
node in turn using a Newton-Raphson method and
move the node to the optimal position found. The
process moving a vertex changes the forces affecting
vertices, so this algorithm must also be run iteratively
until the layout reaches a converged point. On the
other hand, Fruchterman and Reingold [4] modified
Eades’ algorithm to more closely approximate the
physical analogy of electrostatic repulsive force
between any two nodes u and v, at positions pu and pv
respectively.
III. Force-Directed Approach to Vertex-
vertex occlusion
In this section, we present a force-directed graph
layout method for vertex-vertex occlusion.
Graphlayout methods, such as the Fruchterman and
Reingoldalgorithm [4], are used to place the vertices
and edgesof the graph on a two- (or three-)
dimensional space. In order to address the problem of
vertex-vertex occlusion, we propose a model that
based on Fruchterman and Reingold algorithm, to
draw a graph that has an aesthetically pleasing layout.
3.1 Fruchterman and Reingold algorithm
The algorithm is the force directed layout that
represents each vertex as an electrically charged
element and each edge as a spring linking two
vertices. In this system, vertices with the same charge
repel each other, while opposites attract due to the
springs. This algorithm iteratively computes a
displacement for each vertex determined by the
forces until a convergence (equilibrium) is obtained
[4], [18].
The repulsive force exists between any two
vertices is inversely proportional to the distance
between them. On the other hand, the attractive force
exists only between neighboring vertices and is
proportional to the square of the distance [14].
Let duv denote the Euclidean distance between two
vertices v and u. Then the attractive force, fa, and the
repulsive force, fr, are calculated as follows:
𝑓𝑎 =
−𝑘2
𝑑 𝑢𝑣
Equation (1)
𝑓𝑟 =
𝑑 𝑢𝑣
𝑘
Equation (2)
Where k is a constant proportional to the square root
of the ratio of the area where the graph is laid out to
the number of vertices. The Fruchterman and
Reingold algorithm is applied for several iterations.
During each iteration, the vertices are moved in
proportion to the calculated attractive or repulsive
forces until the desired layout is obtained. Many
implementations of fruse a “temperature” component
that limits the maximum displacement of a vertex. As
the iterations progress, the “temperature” becomes
lower, in effect allowing finer adjustments to the
positions of the vertices.
3.2 Good Viewpoints
Good viewpoints are those for which the abstract
graph of the three-dimensional graph drawing, and
the apparent abstract graph of the resulting two-
dimensional image, are the same. In application of
threedimensional graph drawings, the drawings are
intended to convey information to the user. This
information is encoded in the graph drawing by a set
of primitive relationships. A user recognizes these
relationships by applying cognitive routines that
query their intermediatelevel representation of the
graph drawing, and build up a semantic interpretation
(a mental map)[13], [3], [17].

3.3 Vertex-Vertex Occlusions
Generally, occlusions occur when a projection
maps two three-dimensional points to the same two
dimensional point. We say the front point occludes
the rear point. The concept of occlusion underlies
many models of good viewpoints [13].
A good projection of 3D graph drawings to 2D
graph drawings should prevent overlaps between the
graph elements, avoiding vertex-vertex, vertex-edge,
and edge-edge occlusions [11].
We focus in this paper on one type of occlusions
calledvertex-vertex occlusion, whichoccurs when a
pair of vertices from the three-dimensional graph
drawing map to a single vertex in the two-
dimensional image. The abstract graph of the image
appears to have a single vertex in place of the
original two, and any edges incident to the original
vertices now appear incident to the combined vertex
(see Figure 1).
Figure 1: A vertex-vertex occlusion. [3]
It may be possible for a given 2D (or 3D) graph
to be drawn in a plane (or a volume) without any
vertex-vertex occlusion. Indeed, achieving such a
graph layout is the objective of the Fruchterman and
Reingoldalgorithm [4].
Similar to most force-directed layout algorithms,
our method makes good viewing for 3D drawings
that have vertex-vertex occlusions by assigns a force
to each vertex and aims to minimize the overall
energy of the system. Therepulsive forces affect
adjacent vertices positions and each pair of vertices
that connected by an edge, pushing vertices away in
order to prevent occlusion between vertices and to
spread the vertices out uniformly throughout the
drawing area. On the other hand attractive force pulls
vertices connected by edges closer together. It is
applied to every pair of vertices connected by an edge
(see Figure 2). The repulsive force pushes vertices
apart. It is applied to every pair of vertices (see
Figure 2).
Figure 2: Repulsive and attractive forces of
graphelements (vertices and edges)
The magnitudes of attractive and repulsion
forces are calculated by equations (1, 2),We use
Gephi software,Fruchterman and Reingold approach
for computing these forces, that is found in section 4.
IV. Results and Discussions
4.1 Implementation
We implemented Fruchterman and Reingold
algorithm using the visualization capabilities of the
Gephi application [6],[1]. Our experiments on a
windows system with an Intel Core i5 (3.20 GHz)
processor and 4 GB RAM. The models of 3D graphs
processed in this paper are different types of graphs.
The Gephi software package[6] written in Java
on the NetBeans platform[1], The Gephi Consortium
is a French non-profit corporation which supports
development of future releases of Gephi. Gephi is
members includeSciencesPo, Linkfluence, WebAtlas,
and Quid[9]. Gephi is an interactive visualization
platform for all kinds of graphs, complex systems,
dynamic and hierarchical graphs.
There are controls for the graph drawing speed,
gravity and iteration that aid three dimensional
perception. Gephi application enables us to change
graph drawing attributes like size and colors of the
vertices and edges.
Several 3D graph drawings produced by Gephi,
fruchterman and Reigold approach are included in
section 4.2 for some sets of 3D graphs.
Table1liststhe data of 3D graphs that has been
taken from [7], the graphs have vertex-vertex
occlusions. We use Gephi application to check the
graphsand got the graph properties that are shown in
Table 1, these properties reflect number of vertices,
edges, the connected components, shorted paths,
graph density and average degree of graphs.

Table 1: Statistics taken by Gephi application (Description of test problems)
4.2 Discussions
Experiment results show that the efficiency and
effectiveness Fruchterman and Reingold algorithm in
addressing vertex-vertex occlusions of 3D graph
drawings. Thus, algorithm is good at separating
vertices as much as possible in the drawing space. In
Figure 3(a), a 3D graph drawing (1097 vertices, 1092
edges) which has vertex-vertex occlusion, whereas in
Figure3(b) the graph drawing has no vertex-vertex
occlusion,(vertices do not occlude each other in
general). Another example is 3D graph drawing (400
vertices, 4240 edges), Figure4(a),the graph suffers
from vertex-vertex occlusion, whereas the 3D graph
drawing in Figure 4(b)has good viewing after
applying the algorithm.
(a)With vertex-vertex occlusion (b)Without vertex-vertex occlusion
Figure 3: Snapshots of the drawings of graphs
(a)With vertex-vertex
occlusion
(b)Without vertex-vertex
occlusion
Figure 4: Snapshots of the drawings of graphs
V. Conclusions
In this paper,we propose a method to address the
problem of vertex-vertex occlusions for three
dimensional graphs. Our methoduses Fruchterman
and Reingold algorithm for drawing three
dimensional graphs forsets of 3D graphs that had
been taken from [7]. We also have discussed the
various graph drawings techniques, specifically
forced directed layout approaches which could draw
good viewpoints of three dimensional graph drawings
by minimizing vertex-vertex occlusion. Since good
viewing of graph drawings brings a lot of advantages
Graph Graph Type |V | |E| Connected
Components
Number of
shortest paths
Graph
Density
Average
Degree
phase1_chr1 Undirected 1097 1092 1097 351274 0.002 1.991
DAGmar1 directed 400 4240 400 44559 0.027 10.600
phase1_chr1 directed 1097 1092 1097 1092 0.001 0.995
DAGmar2 Undirected 400 4240 399 159600 0.053 21.200

in understanding and readability of the graph drawing
that conforms to criteria of aesthetics for graph
drawing.
We applied Gephi to real-world graphs data [7].
The resultsshow that the algorithmdraws 3D
drawings by providing the viewer with goodgraph
drawings viewing. The algorithm positions
interconnected vertices into wellseparated positions
without vertices occlusions. The resulting drawings
have good viewings,the exploring of 3D drawings is
useful if combined with user interaction and graph
attributes value such as the colorand the size of
vertices.
Further, we restricted our attention to vertex-
vertex occlusion, but we may consider the same
problem for different classes of graphs.
References
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Retrieved 2015-08-19
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3D Graph Drawings: Good Viewing for Occluded Vertices

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3D Graph Drawings: Good Viewing for Occluded Vertices