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Fig. 1.
Mean-field phase diagram for the density
c
coded as gray levels ranging from
white (
c
= 0) to black (
c
= 1). The dashed upper-left region denotes the coexistence
phase, in which both states
c
= 0 and
c
= 1 are stable, and the final state depends on
the initial density (first-order transition).
Rule T23 is a strict majority rule, whose evolution starting from a random
configuration leads to the formation of frozen patches of zeros and ones in a
few time steps. A small variation of the probabilities induces fluctuations in the
position of the patches. Since the patches disappear when the boundaries collide,
the system is equivalent to a model of annihilating random walks, which, in one
dimensions, evolves towards one of the two absorbing states according to the
asymmetries in the probabilities or in the initial configuration.
Rule T13 on the other hand is a “chaotic” one,
4
leading to irregular pattern
for almost all initial conditions (except for the absorbing states). These patterns
are microscopically very sensitive to perturbations, but very robust for what
concerns global quantities (magnetization).
The role of frustrations (the di
9
culty of choosing a stable opinion) is eviden-
tiated by the following considerations. Rule T23 corresponds to a ferromagnetic
Ising model at zero temperature, so an individual can simply align with the local
majority with no frustrations. On the contrary, rule T13 is unable to converge
towards an absorbing state (always present), because these states are unstable:
a single individual disagreeing with the global opinion triggers a flip in all local
community to which he/she belongs to. It is possible to quantify these concepts
by defining stability parameters similar to Lyapunov exponents [10].
We start by studying the mean-field approximation for the generic case, with
p
1
and
p
2
different from zero or one.
Let
c
and
c
denote the density of opinion 1 at times
t
and
t
+1 respectively.
We have
c
=3
p
1
c
(1
− c
)
2
+3
p
2
c
2
(1
− c
)+
c
3
.
4
Called rule 150 in Ref. [9]
