Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 62

(a) (b)

(c) (d)

Fig. 4.7. Evolution of cells and clustering coefficients for several types of behaviors: ( a )
Class II with very long transients; ( b ) Class III; ( c ) Class IV; ( d ) Class I (using Wolfram's
taxonomy). Note that the clustering coefficient (plotted on vertical axis in the lower graphs)
converges towards a “steady state” in the sense of a constant variance. This variance is lar-
ger for chaotic dynamics and is considered for the last 20% of the iterations, to detect the
transient phenomena much accurately

element is the absolute value of the corresponding element in the input matrix. For
instance, in the case of one-dimensional arrays with N cells, the above formula be-
comes:

ª º

N

x x 2 x

1

i 1 i 1 i

, (4.4)

C

¦

1

« »

N 2

i 1

¬ ¼

> @

where .

X x ,... x ,.. x

1 i N

It is interesting to note that the clustering coefficient defined as above is based
on a diffusive filter applied to the matrix of cell states. It is also related to a quan-
tity, the current entering a multiport-cell in a Reaction-Difussion CNN , essential
for the local activity theory [8].
Before computing the transient length, it is necessary to have the whole time-
sequence ^` T

computed. Let us now consider the following two Matlab

C t

t 1 ,..