Exponents of Growth - Systematic Design for Emergence in Cellular Nonlinear Networks - page 87

finally some other 28 “edge” genes in the cellular model migrated to the “unstable”
class increasing their number to 525.
The dramatic increase of “unstable near edge” genes (i.e. those behaviors that
are closest to the behavior of living systems) from 14 to 58 (i.e. almost four times)
gives an argument in the favor of the hypothesis that natural systems are in fact
described better by a “small worlds” rather than a purely cellular model.
As for the “unstable” behaviors, unlike in the case of a pure cellular model,
where some unstable behaviors exhibited a slow growth with half or less than half
of the “speed of light”, in the case of the “small world” model almost all unstable
behaviors migrate to the “fast growing” emergent behaviors (with largest U ).
As a practical consequence, it is expected to have better “pseudo-random” se-
quence generators within the “unstable” behaviors of the small-worlds model.
5.4.1 The “Small Worlds” Model Allows Fine Tuning of the “Edge
of Chaos”
Another interesting fact about the “small worlds” model is the following: Starting
with a gene located in the “edge” category of a cellular model, the f parameter
(representing the fraction of random long-distance connections) can be used to
slightly change the behavior of the resulting “small worlds” system into any of the
behavioral regimes ranging from “stable near edge” to “unstable”.
Indeed let us consider the gene ID = 768 of the “2s5” family (the interesting
“edge” gene producing rectangles around any initial state spatial cluster, in the
case of the pure cellular model). The evolution of the cellular array for three dif-
ferent values of f is shown in Fig. 5.16.
Note that for f = 0.01 the overall behavior is slightly the same as in the case of
a pure cellular model ( f = 0 ). However some small black dots appear out of the
rectangles, as a consequence of sending information to distant cells. Eventually af-
ter about 167 iterations, the evolution enters into a steady state having the same
computational meaning as in the case of the pure cellular model. When the frac-
tion of distant connections is increased beyond a bifurcation value (here f = 0.02)
the spatial clusters continue to expand after 167 iterations, the overall system ex-
hibiting an “unstable” behavior which eventually leads to a final state where all
cells are in the “black” state.
The bifurcation value of f is in a direct relationship with the size of the biggest
spatial cluster in the initial state image. Therefore the above phenomenon may be
applied as an object size detector. Indeed, if the size of the spatial cluster (repre-
senting an object) stands bellow a certain threshold (directly related to f ) the
system will evolve to a steady state containing a square of black pixels surrounding
the object. However, if the size of the object is greater than its critical limit, the