Information Technology Reference
In-Depth Information
Ta b l e 2 .
Mean (
μ
) and standard deviation (
˃
)of
Accuracy
(in percentage). Statistically significant
differences between completion time in layouts with the low and highnumber of edge crossings
are highlighted.
graphs
the number of crossings
t
-test results
p
-value
t
-value
low
high
small
μ
=94
.
1%
˃
=4
.
3
μ
=89
.
4%
˃
=4
.
4
p<.
05
t
(15) = 2
.
8
large
μ
=86
.
3%
˃
=3
.
4
μ
=83
.
1%
˃
=4
.
0
p
=
.
06
t
(15) = 2
.
0
small sparse
μ
=93
.
7%
˃
=6
.
4
μ
=92
.
9%
˃
=6
.
3
p
=
.
77
t
(15) = 0
.
2
small dense
μ
=94
.
5%
˃
=7
.
8
μ
=85
.
9%
˃
=13
.
5
p<.
05
t
(15) = 2
.
2
largesparse
μ
=89
.
1%
˃
=11
.
1
μ
=89
.
0%
˃
=9
.
0
p
=
.
81
t
(15) = 0
.
2
large dense
μ
=83
.
5%
˃
=7
.
5
μ
=77
.
3%
˃
=13
.
1
p<.
05
t
(15) = 2
.
4
It is somewhat surprising to see that increasing the crossings affects different task in
markedly different ways. It is particularly unexpected to see a statistically significant
positive impact on accuracy, with the increase of edge crossings, for Task 4 in large
graphs! It is also worth noting that with the increase of edge crossings, the average ac-
curacy increases for Task 3 in small graphsforTasks3and4inlarge graphs. This might
be due to participants paying more attention in the cases where the problem was more
difficult, possibly related to the “chart junk” effect [2]. But it is also possible that edge
crossings may not be as bad as we normally think, as indicated by Huang et al. [15],
who found that crossingshavenegative effect only on some of their tasks.
There are good indications that density plays a possibly independent role, especially
on accuracy. Note that we only considered two density settings(1
.
5 and 2
.
5), both of
which are relatively low. Yet, together with increased number of crossings, the high
density settingsresulted in statistically significant decrease in accuracy both for small
and large graphs. It is probably worth exploring further the nature of the interactions
between size (number of vertices), density (ratio of number of edges to number of ver-
tices) and edge crossings upper limit of density.
4
Edge Crossings and Other Aesthetic Criteria
As mentioned earlier, several traditional methods for drawing large undirected graphs
are based on the assumption that minimizing asuitably-defined energyfunction of the
graph layoutresults in aesthetically pleasant drawing.Butdosuch methods also (pos-
sibly indirectly) optimize some of the standard aesthetic criteria? Next we qualitatively
analyze layouts produced by
fdp
(force-directed) and
neato
(MDS-based), with re-
spect to three commonly used and well-defined quality measures: the energyofthe
layout, the number of crossings, and the angles between pairs of crossing edges.
In a number of studies, the energyofalayout is defined as the variance of edge
lengths in the drawing, known as
stress
[18]. Assume a graph
G
=(
V,E
) is drawn
with
p
i
being the position of vertex
i
∈
V
. Denote the distance between two vertices
