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Fig. 5.5. Phase space for signaling networks with units obeying uniformly rules 232, 19
and 1. In the phase spaces, we use the color scheme shown in the bar to indicate the value
of the scaling exponent characterizing the auto-correlations in the dynamics of the system by
means of the DFA method [Peng et al. (1995)]. The exponent is systematically estimated for
time-scales 40 < n < 4000. We show for 6161 pairs of values of k
e
and the noise in the
communication between the units comprising the network. For all simulations, we follow the time
evolution of systems comprising 4,096 units for a transient period lasting 8,192 time steps, and
then record the time evolution of the system for an additional 10,000 time steps. The rules shown
in this gure display dierent types of noisy uctuations depending on the values of and k
e
. For
all three rules, the system generates 1=f-noise for a broad range of noise intensities.
Concerning (i) we note that the topology considered so far takes as initial setup,
before adding the extra long-range connections, a one-dimensional ring of nearest-
neighbor links. One of the rst questions one asks itself is whether this particular
setup is relevant in the development of the complexity of the signals discussed
in the previous section. We have tackled this problem, from two dierent points:
a) considering next-nearest neighbors connections, and b) connectivity distributions
other that the homogeneous ones considered so far [Diaz-Guilera et al. (2007)].
In the rst case, we considered as initial setup a one-dimensional array in which
each unit i has bidirectional connections to its nearest neighbors (1) and to
its next-nearest neighbors (2). We noticed that the inclusion of two additional
neighbors before we add the extra connections leads to a similar structure of the
phase space and the three dierent regimes are still observed. It is noted, however,
that more noise intensity is needed to obtain the regimes observed in the original
system with only nearest neighbors.
In the second case, we note that the network topologies considered so far span
the cases of ordered one-dimensional lattices, small-world networks, and random
graphs [Amaral et al. (2000)]. However, all networks considered are comprised of
units with approximately the same degree, i.e., the same number of connections.
To investigate the role of the distribution of number of connections, we also studied
networks which span the range of empirically observed degree distribution: a delta-
distribution, an exponential distribution, and a power law distribution.
