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variation of
C(t)
for the last 20% steps of the run. To precisely locate the transient
we will look back in the past to detect the transient when the value of C(t) goes
out of the boundaries imposed by
'
. This procedure makes possible to avoid false
results in terms of
transition time
, particularly for the case of chaotic behaviors.
Let us also define
C
E
. The closest this value is to 0, the better is the
quality of the random number generator. Experimental results indicate that
if
T
0
!E
, the
transient length
should be additionally considered in order to have
a better description of the emergent behavior.
Large values of
Tr
clearly indicates the possibility of gliders and other com-
plex phenomena, with many potential applications. Two such examples are given
in Fig. 4.7a, c. In the first, cell ID = 133 from the
1s5
family is used and the final
clustering coefficient
C(T)
has a value far from pure disorder (note the variance
which is always smaller when
E
is large). The transient length in this case is 173,
for a cellular system with 31 cells. In the following we will accept that a transient
is large if it is larger than the number of cells in the CA. This indicates that in or-
der to detect large transients we shall run the CA for a large number of iterations.
In practice we found convenient to choose
0
05
. In the second case (Fig. 4.7c),
the cell ID = 372 from
1s5
gives a behavior with a moderate to low transient of 92
(comparable with
N=100
in this case), a signature for less complexity. Indeed, the
T
10
N
final clustering coefficient is now closer to the 0.5 indicating “pure disorder” and
less interacting structures (gliders) are observed. As
and approaches 0, the
E
0
04
variance of the clustering coefficient is larger.
Finally, there are many low complexity behaviors characterized by a short
transient (smaller or much smaller in comparison to
N
) towards as near “pure or-
der” state, characterized by a large
deviation from disorder
E
. Such an example,
plotted in Fig. 4.7d) is characterized by
C(T)=1
, after a short transient of
.
Tr
15
Note again the small admissible variance
'
, as
E
reaches its maximum.
We need now some systematic procedures to compute the above parameters:
At a given time moment
t
the global state of a cellular automata is
^ `
^
x
. For a given array
X
of
cells the following formula is used to compute the clustering coefficient
C
:
. It is assumed that each cell output
X
t
x
t
,
0
i
,
j
j
ª
º
§
·
abs
©
k
X
¦
X
¹
«
»
l
«
l
»
(4.3)
C
mean
1
k
«
»
¬
¼
Y
,
1
is a neighborhood formed by the immediately adjacent cells (excluding diago-
nal cells and the central cell),
k
is the number of cells in the neighborhood
1
and
X
is the array of cells outputs relative to the center cell according to neighbor in-
dex
l
. The
abs
where
mean
Y
is the average value of the whole set of elements of matrix
operator applied to a matrix returns another matrix where each
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