Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 60

dynamics in the evolution of a global parameter called there a magnitude of cellular

disorder. In [62] we refined the above analysis defining a dynamics signature as a

vector of finite size (usually log 2 T , where T is the number of CA iterations) and ( )

we were able to prove that a clustering neural network is able to detect six com-

mon prototypes of these signatures, which may correspond to the same number
of distinct qualitative

behaviors. For any particular dynamic behavior a degree

of membership to one of the prototypes can be easily computed thus providing an

automatic tool for detection of certain types of emergent behaviors. For instance

such tools allow genetic or other types of global optimization algorithms to search
in a huge space of possible cells only for those cells producing a certain type of

emergent behavior. This would be something impossible to do using visual cate-

gorization of global phenomena (as proposed by Wolfram) since a human decision
would be required in each objective function evaluation of the optimization algo-
rithm.
Here we will go further and propose a much simpler and easier to use measure
of complexity. Compared to [62] we eliminate the need to use an adaptive vector
quantization scheme and for each global state of the cellular automata at the ter-
minal state T we evaluate a clustering coefficient “ C(T) ” and a transient length
“ Tr ”. Both can be easily computed from the time series formed of consecutive
clustering coefficients

( . Other measures of complexities were proposed
in the literature (e.g. Wuensche's input entropy [73]). As detailed later it was
found however that the clustering coefficient is computationally less complex than
the input entropy .

C t

t 1 ,.., T

The clustering coefficient is exclusively a feature of the array of cells and it

can be computed for either binary or multi-state cells. The use of a clustering coef-
ficient was suggested by the observation that in a cellular system with a complex
emergent behavior an initially chaotic state pattern evolves towards a sequence of
patterns characterized by a steady state value of the clustering index. The “degree
of clustering” was proposed by several authors as a measure of complexity in
complex networks [63]. Moreover, in biological systems, life can be regarded as
long transient processes from a certain degree of clustering among an array of
cells (the initial or birth state) to another relatively stable clustering degree formed
at the end of the life cycle.
The clustering coefficient may vary between 0 (indicating a “high frequency
order: in the form of a chessboard) to 1 (indicating a “low frequency order”, i.e.
patterns where all cells have their neighbors in the same state). For a random
uniformly distributed array of cells the clustering coefficient is 0.5, thus indicating
a “perfect disorder”. From the examples discussed in Sect. 4.2, several strategies
in locating various behaviors follow:
If after running a CA for T time steps the final clustering coefficient C(T) is
around 0.5 the behavior can be classified as a “chaotic” or Class III one, and its
immediate application may be in building random number generators. Note for in-
stance the example in Fig. 4.7b. It is interesting to observe that in the case of cha-
otic dynamics the time series C(t) has a relatively large variance . As we will see
later, in calculating the transient we will define the admissible variation ' as the