Predicting Emergence from Cell’s Structure - Systematic Design for Emergence in Cellular Nonlinear Networks - page 127

probability ( FP in (7.2)) simpler although not very straightforward.
The cell description as nonlinear function applied to a weighted sum
of inputs is crucial for making these calculations possible. The theory
can be extended to continuous state systems as well. Then, while each
state variable involved can vary within a specified domain it will be
attributed a probability density. Of course the technical difficulties to
compute the cells function of probability densities will be higher but
still the principles of the theory remains the same. Since another cru-
cial feature of this new theory is to rely on next state uncertainties in-
stead of probabilities, as a tool to predict growth or implosion (i.e. by
using a forgetting function like in (7.3)), such a similar function must
be also defined for the case of probability densities. A possible choice
would be:

b b

1 1 1

³ ³

(7.30) u 1 p x pu x dx 1 p x dx

ba

,

k k k

2 2

a a

pu is the uniform density of probability of a signal ranging with the
[a,b] interval and

where

( x )

is the specific density of probability of the cell k. There-

p k x

(maximum uncertainty) if

fore is uniform, for a normal

u 1 p k x u 0 .

k k

(Gaussian) distribution , and becomes 0 (the signal is in a certain state) if p k x

p k is a Dirac's function (as is the case for the probability distribution of signals
that are in a certain single state. The only mathematical problem needed to be ad-
dressed remains an algorithm to compute the output probability density given a
nonlinear description of the cell and the expressions for probability densities of the
inputs.

x

(d) The theory of probabilistic exponents of growth is confirmed by ex-

periments. Particularly, as described within the chapter, for various
measures of complexity computed experimentally it was observed, us-
ing the visualization technique called parametrization , the existence of
a clear functional relationship between the bits expressing the ID and
the underlying complexity measure. Such relationships are now clearly
expressed as (7.13)-(7.27) using the theory of probabilistic exponent
of growth .

(e) Herein we focused only on the exponent of growth as a measure of

complexity and emergence and proved that is possible to compute it
analytically, the result depending solely by the cell structure, the initial
state profile and the neighborhood topology. Previous experiments
have shown that the exponent of growth is a rather synthetic measure
of complexity which may include hints about other measures such as
transient time or clustering coefficients . It might be possible however
to exploit some ideas from this theory to the problem of evaluating
clustering coefficients and transient lengths on a probabilistic (infor-
mation theoretic) base.