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real population, which is supposedly composed by a mixture of conformist and
non-conformist people who do not change easily their attitude.
In Sec. 2 we introduce a class of probabilistic cellular automata characterized
by the size 2
r
+1 of the neighborhood, a majority threshold
q
, a coupling constant
J
and an external field
H
.
We are interested in the two extreme cases: people living on a one-dimensional
lattice, interacting only with their nearest neighbors (
r
= 1) and people inter-
acting with a mean-field opinion.
2
In Sec. 3 we present the simplest case where each individual interacts with his
two nearest neighbors (
r
= 1), the mean field phase diagram and the one found
from numerical experiments. For this simple case, we find a complex behavior
which includes first and second order phase transitions, a chaotic region and
the presence of two universality classes [6]. In Sec. 4 we discuss the mean field
behavior of the model for arbitrary neighborhoods and majority thresholds when
the external field is zero and the coupling constant is negative (non-conformist
society). The phase diagram of the model exhibits a large region of coherent
temporal oscillations of the whole populations, either chaotic or regular. These
oscillations are still present in the lattice version with a su
9
cient large fraction
of long-range connections among individuals, due to the small-world effect [7].
2 The Model
We denote by
x
i
the opinion assumed by individual
i
at time
t
. We shall limit
to two opinions, denoted by
−
1 and 1 as usual in spin models. The system is
composed by
L
individuals arranged on a one-dimensional lattice. All operations
on spatial indices are assumed to be modulo
L
(periodic boundary conditions).
The time is considered discontinuous (corresponding, for instance, to election
events). The state of the whole system at time
t
is given by
x
t
=(
x
0
,...,x
t
L−
1
)
with
x
i
∈{−
1
,
1
}
;
The individual opinion is formed according to a local community “pressure”
and a global influence. In order to avoid a tie, we assume that the local commu-
nity is formed by 2
r
+1 individuals, counting on equal ground the opinion of the
individual himself at previous time. The average opinion of the local community
around site
i
at time
t
is denoted by
m
i
=
j
=
−r
x
i
+
j
.
The control parameters are the probabilities
p
s
of choosing opinion 1 at time
t
+1 if this opinion is shared by
s
people in the local community, i.e. if the local
“field” is
m
=2
s −
2
r −
1.
Let
J
be a parameter controlling the influence of the local field in the opinion
formation process and
H
be the external social pressure. The probability
p
s
are
given by
p
s
=
p
(
m
+2
r
+1)
/
2
∝
exp(
H
+
Jm
)
.
2
Related models in one and two dimensions have been studied in Refs [4,5].
