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a
b
c
Fig. 5.1. Time evolution of the state of a set of units placed on a circle, subject to periodic
boundary conditions (see Fig. 5.2 for details of the model). Black (gray) squares correspond to
nodes with 0 (1) states. Time goes from top to bottom and unit labels from left to right. All units
process information according to the majority rule (rule 232, see text) and noise intensity is xed.
The three panels dier in the topology; whereas all of them have the same initial circular pattern
they vary in the number of excess connections (short-cuts). They have in common the initial state
and the sequence of random numbers mimicking noise. For a low random connectivity (left) the
initial conguration is rapidly stabilized and patterns keep their shapes, only aected by the noise
from time to time. For larger number of short-cuts, initial patterns develop and can spread over
dierent time scales. Finally, for a larger random connectivity the dynamics becomes dominated
by the external uctuations and correlations are rapidly lost.
where k
i
is the connectivity of unit i | i.e., the number of inputs that unit i receives
| andF
i
is a Boolean function (or rule), that completely species the way that
unit i is processing the information that arrives to it and hence is xed in time.
Note that for a number K of inputs there are 2
2
K
dierent functions: for example,
for K = 3 there are 256 dierent Boolean functions.
Generally, the dynamics in real systems is subject to noise eects. In our case,
the noise is intended to mimic two eects that are always present in living processes:
(i) communication errors due to the intrinsic noise from in vivo conditions, and (ii)
external stimuli aecting the transmission of signals to a unit [Mar et al. (1999)]. We
introduce noise in the model dynamics by assuming that a unit has a probability
of \reading" a random Boolean variable instead of the \true" state of the neighbor.
It is important to note that in our model the noise acts only on the inputs of the
neighbors of a unit. This implies that the state of a unit is not changed because of
noise only.
As an example of this type of dynamics we present in Fig. 5.1 the evolution of
the state of a particular model (to be described in detail in Sect. 5.4). This example
illustrates the great importance of the topology, apart from the dynamical rules of
the units and the intrinsic noise.
