Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 68

Fig. 4.10. Projection of the “1s5” CA family within the ( Tr,Clus,Var ) plane. Observe that
complex behaviors (gliders) are associated with either long transients or large variances.
Note also that Tr and Var are orthogonal, i.e. there are little CAs having both large vari-
ances and transient times

Such a list contains the following IDs: 378, 634, 645, 650 but it can be ex-
panded if a larger range for Clus is accepted. All these cells are associated with
large values of the local structural complexities m as seen from Table 3.3.
The following “sieve” reveals the complex behaviors, with long transients
( Tr >200):

>>ix=find(Tab(:,2)>200),

returning the following list of IDs: 21, 212, 294, 475, 507, 684, 851, 866, 870.
Note the presence of ID = 684, already described in detail in Fig. 4.4. Also when
checked with Table 3.3. it is found that local structural complexities

^

are m 4

lower than in the case of cells leading to “chaotic” behaviors, but still large.
A closer look to Fig. 4.10 reveals that complex behaviors are rather seldom,
most of the CA individuals (IDs) being located near the axis Tr=0 with low vari-
ance and low transient lengths and with a wide range of clustering coefficients.
Indeed, as many authors already reported, complex emergent behaviors in cellular
automata are rare, and consequently difficult to locate. For mid values of the clus-
tering coefficients, there is a large concentration of IDs with 0.1< Var <0.3 all cor-
responding to “random” number generators. There is a relatively large number of
CAs having this property, and in fact applications in cryptography were among the
first considered for cellular nonlinear networks. When the neighborhood changes,
a slight change in the aspect of the profile is observed in Fig. 4.11, where the cells
come now from the two-dimensional instead of the two-dimensional grid topology.