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In particular, we considered dierent distributions of number of incoming and
outgoing connections for networks with exponential distribution of local links with
dierent means. Allowing the number of incoming connections to uctuate also
changes the distribution of number of outgoing connections, which becomes a Pois-
son distribution. The change in the increased number of local connections leads to
no signicant change in the results. The reason is that the complex dynamics are
generated at the boundary between domains: when one allows some units to have
more local connections, these units still have the same number of units to the left
and to the right, so the existence of a single long-range connection is enough to
destabilize the boundary.
Additionally, we considered networks with power-law distributions either of in-
coming or outgoing links. Taking the one-dimensional ring of Fig. 5.2 again as
initial setup, we add the additional connections according to the preferential at-
tachment rule for outgoing units and incoming units [Albert and Barabasi (2002)].
The former case gives rise to a network with a broad distribution of outgoing links
while the latter gives rise to a network with broad distribution of incoming links. As
one might expect, a power law distribution of outgoing links leads to no signicant
change in the phase-space describing the dynamic behaviors since it only makes the
network a small-world more eciently than random long-range connections [Cohen
and Havlin (2003)]. In contrast, a power law distribution of incoming links does
lead to a change in the phase-space of dynamical behaviors. The reason may be
that since information travels only one way on the connections, the fact that some
units are receiving so many of the long-distance connections will make it harder for
the system to reach the small-world regime. Interestingly, this asymmetrical distri-
bution of incoming/outgoing links has been observed in genetic regulatory networks
for species across four dierent kingdoms [Balleza et al. (2008)].
The second class of change we considered is the way in which the units imple-
ment the rules. We looked at this from two dierent perspectives: one consists in
restricting the majority rule to need more than the simple majority and the second
in allowing a subset of units to operate according to another rule, which accounts
for the eect of \errors" in the units implementation.
The rst attempt was to construct a more restrictive majority rule. To do this
we considered a clear majority rule by demanding the majority (half plus one) plus
one to adopt the state. In this way we have a more restrictive majority rule, that is,
a unit would remain in its initial state unless a clear majority operates. We perform
simulations to evaluate the correlation exponent for systems with a fraction of units
operating according either to a clear majority or simple majority rule. The use of a
clear majority rule forces the units to remain in its state for more extended periods
before switching to the opposite state. The phase space shows the same regimes as
in Fig 5.5 (left) but a more extended region corresponding to Brownian dynamics is
observed, due to the fact that the dynamics is more persistent in the present case.
We also investigated the eect of allowing the co-existence in the system of
distinct Boolean rules. To this end, we rst explored systems composed of units
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