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7.3 Parametrizations as Tools to Locate Similar Behaviors
From the above definition of a parametrization it follows that for a given family of
cells with a gene ID defined by n bits there are many possible parametrizations, as
many as the arrangements of bits used to define the line and column indexes.
Thus, the following question arises: What is a “good parametrization”?
In the following we will assume that a good parametrization is one with less
changes between adjacent elements
P , . Therefore a convenient measure of the
overall parametrization quality may be defined as:
j
n
1
1
n
2
1
Q
¦¦
P
P
P
P
4
P
(7.1)
i
1
j
i
1
j
i
,
j
1
i
,
j
1
i
,
j
i
2
j
2
becoming 0 if there is no change at all (best possible parametrization) and is larger
for “bad” parametrizations. Therefore one may easily write a program to arrange a
given set of emergence measures for an entire CA family into various parametriza-
tions and compare the resulting overall qualities such that in the end the pa-
rametrization with smallest Q value will be selected as the best choice.
Less “change” in the parametrization means that it would be easier to observe a
relationship between cells' structure and the behavioral characteristic and there-
fore it gives a smoother way to a consistent theory relating cells structure to the CA
emergent behavior. The right side of Fig. 7.2 is an example of a “good” parametri-
zation as compared to a “bad” one depicted on the left side. It is clearly much eas-
ier to infer a simple relationship between cells ID and the emergent behaviors (e.g.
white pixels representing large values of the clustering coefficient) from the
“good” parametrization of the right of Fig. 7.2.
2s5 Column=[8 7 5 1 9] Line=[3 4 6 10 2] P i , j =Clus
2s5 Column=[4 7 2 1 6] Line=[3 8 9 10 5] P i , j =Clus
5
5
10
10
15
15
20
20
25
25
30
30
5
10
15
20
25
30
5
10
15
20
25
30
Fig. 7.2. Two different parametrizations (the ID bits used to create the column and line in-
dexes are indicated above each picture). The best parametrization ( Q = 0.21) is the one on
the right while the left one is among the worse possible ( Q = 0.43). In this case the measure
of emergence considered to be parametrized was
P i , . White pixels correspond to
values of the clustering coefficient close to 1 while black pixels correspond to 0-valued
clustering coefficients
Clus
 
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