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f (as it might be intuitively expected, since the introduction of many long-range
connections interferes with the establishment of local domains largely independent
from the configurations which take place far away). However, at smaller k values the
transient duration displays a maximum for an intermediate value of k, and then
declines, as shown in fig. 4.
5 Conclusions
Further work is needed to explore the effects of topological perturbations in the case
where different transition functions are used, like e.g. in boolean models of genetic
circuits [3,4,6], as well as in the case of different updating methods, like synchronous
updating.
However, the results reported here already demonstrate that the introduction of
long range connections (without changing the number of connections of every node)
can have a profound effect on the dynamics. In the case of a small world network, it
has also been shown that major dynamical features are affected at a fairly small
fraction of redirected links. While the literature on this subject is still limited, by
taking into account other results (see e.g. [14] and [18]) one is led to guess that the
influence of topological modifications of this kind on the dynamics is a generic
property.
In the particular case considered here, it has been observed that these changes lead,
with respect to the regular case, to a decrease in the number of attractors that are
reached, which can be observed even at small f values. Qualitatively, one sees that a
"cloud" of attractors that correspond to minor perturbations of the major, uniform
attractors disappear. This is a very nice property that holds in this case, and the
possibility of finding similar features in different dynamical rules has yet to be
examined.
References
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Serra, R., Villani, M., Salvemini, A.: Continuous genetic networks. Parallel Computing
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(1970)
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Kauffman, S.A.: The origins of order. Oxford University Press (1993)
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Serra, R., Zanarini, G.: Complex systems and cognitive properties. Springer-Verlag (1990)
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