Exponents of Growth - Systematic Design for Emergence in Cellular Nonlinear Networks - page 75

Starting from a localized random initial state which will be next referred as a
spatial cluster , the CA employing this gene evolves towards a steady state which
develops a square with all black cells. This rectangle marks the smallest possible
boundary enclosing all cells of the spatial cluster that are different from the quies-
cent state (e.g. black cells vs. white quiescent states in a graphical representation,
as seen in Fig. 5.1). This is thus a convenient way to determine the area of the
spatial cluster A as the area of the steady state rectangle. Although there are many
other possibilities to calculate the area of a spatial cluster, the CA-based method
using gene 768 has the advantage that it can use the same CA simulator as for the
CA under investigation (e.g. the ca_sim.m introduced in Chap. 3).
Fig. 5.1. Consecutive iterations of a cellular automata employing the semi-totalistic gene
768. Several spatial clusters evolve towards squares with an area easy to be determined.
The smaller the area of an initial state spatial cluster the lower is the number of iterations
needed to reach the steady state rectangle
The idea to compute an exponent of growth “ U ” is straightforward. We always
start with the same initial state spatial cluster of a given area. In the next examples
we will use 11×11 spatial clusters evolving within a CA array of 77×77 cells.
Therefore the reference initial state area of the spatial cluster is A(0)=121 . In prac-
tice one may choose different other values without changing the significance of
the method. Of course, there is a tradeoff between high speeds demanding less
cells in the CA array and a good accuracy in determining U which demands large
CA arrays.
The CA with a specified ID gene is run from the initial state with an area A(0)
of the spatial cluster for a reasonable number T of iterations. What happens is ei-
ther an implosion (shrinking) or an explosion (growth) of the initial spatial cluster.
In case of explosion, the “speed” of the process is limited by what is called “the