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a
b
c
0
200
400
600
800
1000
Time (a.u.)
Fig. 5.3. We show S(t) for a system with N = 4096 units, = 0:1,F
i
= 232, and k
e
= 0:90,
k
e
= 0:45, k
e
= 0:15 (from top to bottom corresponding to the three panels of Fig. 5.1). The three
values of k
e
lead to quite dierent dynamics of the system. (a) For a large number of random links,
k
e
= 0:90, the dynamics are very poorly correlated. (b) For an intermediate value of k
e
, long-range
correlations emerge and the power spectrum displays a power-law behavior,S(f)/1=f
, with
1. (c) For a small number of random links, the time correlations display trivial long-range
correlations such as found for Brownian noise.
In Fig. 5.1 we show the evolution of single nodes (it displays the specic state of
each unit) in time. There we can see how a system with a low number of shortcuts
(left) evolves quasi-deterministically since the majority rule (232) is quite stable in
a ring, even with the eect of external noise. On the other hand, when the number
of shortcuts is very large the system becomes quite disordered and information
about the initial state is rapidly lost. Interestingly from a biological point of view,
we notice that for an intermediate number there is a tradeo between these two
eects, since a clear mixing of robustness and exibility is displayed.
A more global picture is given by the state of the system, as dened by Eq. (5.2).
This is what we plot in Fig. 5.3. Here we can notice again the dierent behaviors
already shown in the previous gure. A low number of shortcuts makes the system
to keep trapped in some of the attractors of the dynamics whereas a large number
of extra connections makes the system to wander around not being able to stabilize
in any of the deterministic congurations. Finally, for an intermediate value the
system is able to jump in a reasonable time from one attractor to the other showing
eects that are important at all time scales.
