Exponents of Growth - Systematic Design for Emergence in Cellular Nonlinear Networks - page 74

5
Exponents of Growth
In the previous chapter several scalar complexity measures (the clustering coefficient ,
its variance and the transient length ) were introduced to characterize a given cellular
automata system from the point of view of its behavior and likelihood of emergent
phenomena. These were the first steps in establishing a set of methods called
design for emergence.
All these emergence measures have in common their determination based on
the temporal evolution of the clustering coefficient .
In the next, another point of view is considered to define a novel complexity
measure, also proved to be useful in the process of designing for emergence. This
point of view has to do with the answer to the following question:
Given an initial state pattern with a random distribution of states in a localized
spatial region (a spatial cluster of active cells) what happens during the evolution
of the CA: Will this pattern expand, much like fire in an combustible environment
or will the pattern implode until it will eventually vanish?
An in depth qualitative answer is possible through the introduction of a novel
measure called an exponent of growth (U). We will see that by properly character-
izing the “growth” of a spatial localized pattern it is possible to locate the “edge of
chaos” as a complex dynamic regime at the boundary between order (here re-
garded as an “implosion” of the initial state cluster) and disorder (here regarded as
a rapid “explosion” of the initial state cluster.
Two-dimensional and semi-totalistic CA models are considered herein to give
examples, but the same method for determining exponents of growth may be ap-
plied with minor modifications to any other type of cellular computing system.
5.1 Exponents of Growth and Their Relationship
to Various Emergent Behaviors
Let us consider an array of cells where all cells are in a quiescent state except a
small square in the middle where cells are randomly assigned different values. The
same random initial state will be used for all cellular automata. We will exemplify
here for the binary case where each cell may have only two values.
Before explaining how a measure based on “growth” can be evaluated, let us
consider one special cell, the one with ID = 768 from the “2s5” family. It is a
linearly separable cell with an interesting property when used in a CA.