<s>
Force-directed	B-Algorithm
graph	I-Algorithm
drawing	I-Algorithm
algorithms	O
are	O
a	O
class	O
of	O
algorithms	O
for	O
drawing	O
graphs	O
in	O
an	O
aesthetically-pleasing	O
way	O
.	O
</s>
<s>
While	O
graph	O
drawing	O
can	O
be	O
a	O
difficult	O
problem	O
,	O
force-directed	B-Algorithm
algorithms	I-Algorithm
,	O
being	O
physical	O
simulations	O
,	O
usually	O
require	O
no	O
special	O
knowledge	O
about	O
graph	O
theory	O
such	O
as	O
planarity	O
.	O
</s>
<s>
Force-directed	B-Algorithm
graph	I-Algorithm
drawing	I-Algorithm
algorithms	O
assign	O
forces	O
among	O
the	O
set	O
of	O
edges	O
and	O
the	O
set	O
of	O
nodes	O
of	O
a	O
graph	O
drawing	O
.	O
</s>
<s>
Edge	O
attraction	O
and	O
vertex	O
repulsion	O
forces	O
may	O
be	O
defined	O
using	O
functions	O
that	O
are	O
not	O
based	O
on	O
the	O
physical	O
behavior	O
of	O
springs	O
and	O
particles	O
;	O
for	O
instance	O
,	O
some	O
force-directed	B-Algorithm
systems	O
use	O
springs	O
whose	O
attractive	O
force	O
is	O
logarithmic	O
rather	O
than	O
linear	O
.	O
</s>
<s>
A	O
force-directed	B-Algorithm
graph	O
can	O
involve	O
forces	O
other	O
than	O
mechanical	O
springs	O
and	O
electrical	O
repulsion	O
.	O
</s>
<s>
In	O
drawings	O
with	O
curved	O
edges	O
such	O
as	O
circular	O
arcs	O
or	O
spline	B-Algorithm
curves	I-Algorithm
,	O
forces	O
may	O
also	O
be	O
placed	O
on	O
the	O
control	O
points	O
of	O
these	O
curves	O
,	O
for	O
instance	O
to	O
improve	O
their	O
angular	O
resolution	O
.	O
</s>
<s>
For	O
forces	O
defined	O
from	O
springs	O
whose	O
ideal	O
length	O
is	O
proportional	O
to	O
the	O
graph-theoretic	O
distance	O
,	O
stress	O
majorization	O
gives	O
a	O
very	O
well-behaved	O
(	O
i.e.	O
,	O
monotonically	O
convergent	B-Algorithm
)	O
and	O
mathematically	O
elegant	O
way	O
to	O
minimize	O
these	O
differences	O
and	O
,	O
hence	O
,	O
find	O
a	O
good	O
layout	O
for	O
the	O
graph	O
.	O
</s>
<s>
Such	O
mechanisms	O
,	O
which	O
are	O
examples	O
of	O
general	O
global	O
optimization	O
methods	O
,	O
include	O
simulated	B-Algorithm
annealing	I-Algorithm
and	O
genetic	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
following	O
are	O
among	O
the	O
most	O
important	O
advantages	O
of	O
force-directed	B-Algorithm
algorithms	I-Algorithm
:	O
</s>
<s>
Flexibility	O
Force-directed	B-Algorithm
algorithms	I-Algorithm
can	O
be	O
easily	O
adapted	O
and	O
extended	O
to	O
fulfill	O
additional	O
aesthetic	O
criteria	O
.	O
</s>
<s>
Simplicity	O
Typical	O
force-directed	B-Algorithm
algorithms	I-Algorithm
are	O
simple	O
and	O
can	O
be	O
implemented	O
in	O
a	O
few	O
lines	O
of	O
code	O
.	O
</s>
<s>
This	O
makes	O
them	O
a	O
preferred	O
choice	O
for	O
dynamic	O
and	O
online	B-Algorithm
graph-drawing	O
systems	O
.	O
</s>
<s>
Strong	O
theoretical	O
foundations	O
While	O
simple	O
ad-hoc	O
force-directed	B-Algorithm
algorithms	I-Algorithm
often	O
appear	O
in	O
the	O
literature	O
and	O
in	O
practice	O
(	O
because	O
they	O
are	O
relatively	O
easy	O
to	O
understand	O
)	O
,	O
more	O
reasoned	O
approaches	O
are	O
starting	O
to	O
gain	O
traction	O
.	O
</s>
<s>
Statisticians	O
have	O
been	O
solving	O
similar	O
problems	O
in	O
multidimensional	O
scaling	O
(	O
MDS	O
)	O
since	O
the	O
1930s	O
,	O
and	O
physicists	O
also	O
have	O
a	O
long	O
history	O
of	O
working	O
with	O
related	O
n-body	B-Algorithm
problems	O
-	O
so	O
extremely	O
mature	O
approaches	O
exist	O
.	O
</s>
<s>
The	O
main	O
disadvantages	O
of	O
force-directed	B-Algorithm
algorithms	I-Algorithm
include	O
the	O
following	O
:	O
</s>
<s>
High	O
running	O
time	O
The	O
typical	O
force-directed	B-Algorithm
algorithms	I-Algorithm
are	O
in	O
general	O
considered	O
to	O
run	O
in	O
cubic	O
time	O
(	O
)	O
,	O
where	O
is	O
the	O
number	O
of	O
nodes	O
of	O
the	O
input	O
graph	O
.	O
</s>
<s>
This	O
is	O
related	O
to	O
the	O
N-body	B-Algorithm
problem	O
in	O
physics	O
.	O
</s>
<s>
Common	O
techniques	O
used	O
by	O
algorithms	O
for	O
determining	O
the	O
layout	O
of	O
large	O
graphs	O
include	O
high-dimensional	O
embedding	O
,	O
multi-layer	O
drawing	O
and	O
other	O
methods	O
related	O
to	O
N-body	B-Algorithm
simulation	I-Algorithm
.	O
</s>
<s>
Force-directed	B-Algorithm
algorithms	I-Algorithm
,	O
when	O
combined	O
with	O
a	O
graph	O
clustering	O
approach	O
,	O
can	O
draw	O
graphs	O
of	O
millions	O
of	O
nodes	O
.	O
</s>
<s>
Poor	O
local	O
minima	O
It	O
is	O
easy	O
to	O
see	O
that	O
force-directed	B-Algorithm
algorithms	I-Algorithm
produce	O
a	O
graph	O
with	O
minimal	O
energy	O
,	O
in	O
particular	O
one	O
whose	O
total	O
energy	O
is	O
only	O
a	O
local	O
minimum	O
.	O
</s>
<s>
Force-directed	B-Algorithm
methods	O
in	O
graph	O
drawing	O
date	O
back	O
to	O
the	O
work	O
of	O
,	O
who	O
showed	O
that	O
polyhedral	O
graphs	O
may	O
be	O
drawn	O
in	O
the	O
plane	O
with	O
all	O
faces	O
convex	O
by	O
fixing	O
the	O
vertices	O
of	O
the	O
outer	O
face	O
of	O
a	O
planar	O
embedding	O
of	O
the	O
graph	O
into	O
convex	O
position	O
,	O
placing	O
a	O
spring-like	O
attractive	O
force	O
on	O
each	O
edge	O
,	O
and	O
letting	O
the	O
system	O
settle	O
into	O
an	O
equilibrium	O
.	O
</s>
<s>
The	O
combination	O
of	O
attractive	O
forces	O
on	O
adjacent	O
vertices	O
,	O
and	O
repulsive	O
forces	O
on	O
all	O
vertices	O
,	O
was	O
first	O
used	O
by	O
;	O
additional	O
pioneering	O
work	O
on	O
this	O
type	O
of	O
force-directed	B-Algorithm
layout	I-Algorithm
was	O
done	O
by	O
.	O
</s>
