Exponents of Growth - Systematic Design for Emergence in Cellular Nonlinear Networks - page 86

5.4 Influences of the “Small World” Model

As shown previously the model of a cellular system can be simply transformed
into a “small world” model by taking a fraction f of the local connections and re-
routing them to randomly selected distant cells. The random allocation is set
before the CA running and is not changed during the running process. An analysis
over the whole set of 1,024 possible genes from the

2s5” family was performed,
for f = 0.01 . The results are shown as a histogram in Fig. 5.16. A better under-
standing of the “small-worlds” effect is revealed by Table 5.3:

Table 5.3. Distribution of genes among behavioral categories (Small Worlds model)

Category (a) (b) Stable (c) Edge (d) Unstable

near edge

(e) Nested
near edge

(f) Un-
stable

Pas-
sive

near edge

Number of genes 43 185 212 58 0 525

% (Small world) 4 18 21 6 0 51

% (Cellular) 4.2 14.55 31.35 0.98 0.39 48.54

The last row is included for comparison and represents the percentages in the case of the
purely cellular model ( f=0 )

A comparison with results reported in Table 5.2 shows no major influence of
the “small worlds” model on the “passive” genes. The main influence of changing
the purely cellular into a “small world” model is the migration of a fraction from
the “edge” genes into the interesting “stable near edge” or “unstable near edge”
classes. A slight fraction of the “edge” genes also migrates to the “unstable” re-
gime. Indeed, while in the cellular model 321 genes correspond to the “edge” be-
haviors ( U = 1) now their number decreased to 212. Among the difference of 109
genes, 36 migrated to the “stable near edge” category (increasing their number to 185),
44 migrated to the category “unstable near edge” (increasing their number to 58, and

Fig. 5.15. A histogram for the distribution of the exponent of growth U among the set of
1,024 semi-totalistic genes from “2s5” family (“small world” model with f = 0.01). Note
that in comparison to the classic cellular model in Fig. 5.13, now there is a much crisper
distribution into “passive”, “edge” or “unstable” behaviors