Exponents of Growth - Systematic Design for Emergence in Cellular Nonlinear Networks - page 78

Fig. 5.4. Time evolution for ID = 1011. It is a typical evolution for edge behaviors. The ex-
ponent of growth in such cases is always U = 1 indicating a delicate balance between stability
(implosion of spatial clusters) and instability (explosion or growth of the spatial clusters)
The longer the transient (i.e. the closer U is to 1) the more complex is the
behavior. The upper bound of U = 4 is empirically determined based on experi-
mental evidence that another category of behaviors (“nested”), to be discussed
later, emerges for 4<U<5.
The unstable near the edge behaviors may have a lot of relevance for biological
modeling. Indeed, self-reproduction starting from an initial pattern (as shown in
experiments of Langton and others [74,78]) is characterized by a slow growing
dynamics, where several novel clusters emerge as “newborns”. Behaviors in this
class may also be related to Echos State Networks (ESN) [79] or liquid state ma-
chines [80,81]. These are recently introduced neural computing paradigms, proved
to have interesting properties for complex signal processing task such as predic-
tion of complex time-series or speech recognition. It is interesting to note that only
a small fraction of the entire population in a CA family belongs to the interesting
“unstable near edge” class, confirming the scarcity of complex behaviors.
Figure 5.5 displays an example from this category (ID = 471).
Fig. 5.5. Time evolution for ID = 471. It is a typical evolution for an unstable near the edge
behavior. In this case U =2.11 indicating a slow growth reminiscent of the self-making or
autopoietic [64] dynamics