Predicting Emergence from Cell’s Structure - Systematic Design for Emergence in Cellular Nonlinear Networks - page 122

(c) Case 3: (MSB = 1, LSB = 0). In this case, the stable quiescent state is

always the one from the initial state. Therefore since we assumed “0”
initial quiescent states, this case collapses to case 1, discussed above.

(d) Case 4: (MSB = 1, LSB = 1). In this case, the stable quiescent state is

always “1”. Therefore, even if we assumed an initial state with “0”
quiescent states, after a first iteration the quiescent state remains “1”.
Consequently the computations of ue and ua will consider this situa-
tion as follows:

The average uncertainty of the expansion area
The one-dimensional case (four cells in the expansion area, see Fig. 7.5.):

1

1 1

(7.23) ue u 12. u

2

The two-dimensional case (12 cells in the expansion area, according to Fig. 7.6.
all have the same uncertainty):

1.

(7.24) ue u

The average uncertainty index ua
The one-dimensional case (five cells in the active area, see Fig. 7.5.):

1

ª º

ua 23 u

1

u 4

1

u 5.

(7.25)

¬ ¼

5

The two-dimensional case (nine cells in the active area, according to Fig. 7.6):

1

ª º

(7.26) ua 43 u

1

u 4

1

u 5.

¬ ¼

9

7.5 Exponents of Growth and Their Significance

For any of the above cases, a unique scalar parameter probabilistic exponent of
growth with a similar significance to U defined in Chap. 5 is finally computed U

p

as follows:

-
®

ue 1, i f ue ! 0,

(7.27)

U

p

ua , if ue 0.

¯

The above formula can be interpreted as follows: If ue is 0, then surely there
will be no initial state such that an expansion dynamics takes place. This guaran-
tees that only “imploding” phenomena may take place. If the uncertainty within