Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 57

clear that a natural d is achieved for neither sparse nor fully connected
networks. A cellular model is thus a good example of a network topology
where a natural connectivity (and consequently degree of separation d ) occurs
by properly choosing the number of cells N and the interconnection topology.
As shown in the previous chapter and implemented in the CA simulator with
Small Worlds effect, by altering a small fraction f of the local connec-

tions within a cellular model, a dramatic change in d can be achieved in the
sense of reducing it (as compared to the pure cellular model, where f=0 )
without changing the number of cells. Therefore, within the cellular model, we
have an instrument to finely tune d without changing the overall architecture.

1

3. Kauffman proposed random Boolean networks as models of biological proc-
esses, showing that emergence (in the sense of behaviors comparable with
some natural phenomena to be modeled) occurs neither for a high connectivity
nor for a very low one but for a certain “optimal” degree of connectivity, i.e. in
a “small worlds” parlance, an optimal d [60].

Since our aim is to search for complex emergent processes in cellular systems,
in the light of the above we are interested to see what are the implications of the
typical structural constraints for the degree of connectivity and therefore for emer-
gence. In a network of N cells, if a cell is connected on average with k other cells
(neighbors, but also distant neighbors for the altered fraction f ), the following rela-
tionship holds:

d

N | . Since in a cellular system k is usually constrained to 5 or
9, the optimal number of cells (in order to achieve a “natural” separation distance
8

k

d , as confirmed by “Small Worlds” models for various natural systems) is
given in Table 4.1:

3

Table 4.1. Number of cells for various neighborhoods and degrees of separations

d 3 d 4 d 6 d 8

k 5 125 (11 x 11) 625 (25 x 25) 15625 (125 x 125) 390625 (625 x 625)

729 (27 x 27) 6561 (81 x 81) 531441 (729 x 729) (6561x 6561)

k 9

The resulting N values correspond to typical cellular systems sizes used in
various models and applications. It turns out that in order to increase the chances
for emergence, a larger number N of cells in needed in the case of nine-cells
neighborhoods than for the cellular models with 5-cells neighborhoods. Therefore
it would be computationally more convenient to investigate first such small
neighborhood models.
Let us now consider the structural constraints of the cell and how they influence
the connectivity of a cell k . From the above discussion we will conclude that
among all possible Boolean cells, the sub-family of semitotalistic cells concentrate
most of the potentially emergent behaviors.
The behavior of a cellular network is defined by the local rule (i.e. the cells ID).
As we already mentioned, there is a huge number of Boolean functions even for