Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 49

Let first observe that there is a transient length ( Tr ), to be defined systematically
later in Sect. 4.4, that differs among simulations. It can be measured as the number
of iterations until the equilibrium CA state (all cells “white”) is reached. While the
CA in Fig. 4.1a) (ID = 855) requires only a few iterations to reach the equilibrium,
the longest transient (apparently associated with the most complex behavior from
this class) is observed for the CA in Fig. 4.1c) (ID = 852), which needs more than
130 iterations to reach an equilibrium. The CA with ID = 852 in Fig. 4.1c) exhibits
two species of gliders (traveling configurations of bits) emerging from a complex
interaction process after some 20 iterations. One of these gliders “travels” down-
wards and the other “rightwards” but since the CA boundaries are connected it
eventually collides with the first glider producing an annihilation effect around it-
eration 110. The existence of such gliders is a clear sign of emergent complexity.
Metaphorically speaking gliders may be regarded as stable formations emerging
from a low level structure (that of the coupled cells) much like stable words may
emerge from a finite alphabet of letters. Moreover, gliders do interact, eventually
producing higher effects much like words can interact and form phrases in a sen-
tence. In our particular case the two gliders annihilate each other but in general,
one may assume that the same CA using a different initial state may produce quite
different behaviors. This is just an example of ambiguity in Wolfram's definition,
showing that a measurable numerical parameter such as Tr would be useful for a
finer and more precise characterization of emergence in cellular systems.
In terms of the equilibrium reached, as we will see later in Sect. 4.4, a numeri-
cal measure (called a clustering coefficient ) tells us precisely what kind of stable
(or unstable pattern) is reached after the transient. For Class I behaviors this meas-
ure is always at its maximum value, i.e. C = 1.
Another interesting difference among the above five examples, and somehow in
relation with the transient time, is given by the “speed” of a hypothetical wave,
which propagates from the block of 11 initially random bits. In Fig. 4.1d the speed
is slower while in Fig. 4.1e the speed is the greatest, both cases indicating an “ex-
ploding” kind of phenomena (after the first iterations more than 11 cells are en-
gaged in forming a coherent spatial pattern). Figure 4.1c (glider producing CA)
also displays an explosion with a “speed” similar to that of the CA in Fig. 4.1d). In
fact at a closer look, the CA in Fig. 4.1d) also produces two gliders (one moving
to the right and one to the left) which eventually collide around iteration 90 (with
less collateral effects) into a stable equilibrium of the entire CA. For the CA in
Fig. 4.1a there is a “shrinking” or “implosion” effect, i.e. after the first iterations
the number of cells engaged in forming a coherent spatial pattern decreases, while
in Fig. 4.1b there is a delicate balance between some initial growth followed sud-
denly by an “implosion” after iteration 30.
These observations on “growing” and “imploding” behaviors led to the notion
of an exponent of growth ( U ), as detailed in Chap. 5. Imploding effects correspond
to U< 1 while growing or exploding effects correspond to U>1 . Edge phenomena
(such as the delicate balance in Fig. 4.1b) correspond to values of U close to 1.
From the examples above it turns out that complex behaviors are expected when