Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 65

The compute_complex Matlab function implements the above concepts and it
can be called from the main CA simulator program at the end of the main loop, as-
suming that the sequence of clustering coefficients is computed as indicated above.

In evaluating the above measures of emergence a certain degree of approxima-
tion has to be accepted. The following are some reasons for this approximation:

x It is unpractical to simulate a cellular system to infinity. Therefore we will de-
fine in advance a stopping moment T . Empirically it was found that in most of
the cases a “steady state” 4 of

is achieved if with typical values

C t T D N

. Therefore instead of a theoretical we use the Clus value esti- D 2 y 10 C f

mated as above.
x As seen in Fig. 4.7b

C may have large fluctuations, impeding on finding the
exact length of the transient. However in practice the exact length of the tran-
sient does not matter too much, and in fact it was found that its dependence on
the number of cells N is more relevant for characterizing the global dynamic
behavior. Therefore, we introduced the admissible variance ' and measure its
estimate as indicated above. This estimate is less accurate for Class III behav-
iors where the fluctuation can be quite large. For a better accuracy of the
transient time Tr, it is also useful to average over several CA runs, each with a
different random initial state. Though, note that improving accuracy comes for
the price of an increase in the computational complexity (proportional with the
number of additional CA runs).

4.4.1 Visualizing Complexity for an Entire Family of Cellular
Systems - Wolfram Classes Revisited

Now we can characterize each individual of a CA family (e.g. one of the 256

members of the 1a3 family) as a point in a plane determined by two of the meas-
ures for emergence defined above. Let us first consider, the plane formed by the
pair of variables

Tr , . Since the above family of CAs was used by Wolfram
[56] to define its four classes, such plots will give a visualization of the class con-
cept when measured as proposed above. For visualization purposes is convenient
to replace Tr with its logarithm

Clus

. The plot for the 1a3 family is depic-
ted in Fig. 4.8. Each point (circle in the graph) is associated to a specific ID, while
its position within the plane is correlated to its associated class (within the four
proposed by Wolfram). Such a visual representation gives now a better perspec-
tive on the relationship between our measures and the 4-class system.

Ltr ln Tr

4 Here by steady state we accept small period oscillations or even chaotic oscillation with
an amplitude within the admissible variance ' defined as above.

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