ar
X
iv
:0
91
0.
08
66
v1
[
ph
ys
ics
.so
c-p
h]
5
O
ct
20
09
Mass media destabilizes the cultural homogeneous regime in Axelrod’s model
Lucas R. Peres and Jose´ F. Fontanari
Instituto de F´ısica de Sa˜o Carlos, Universidade de Sa˜o Paulo,
Caixa Postal 369, 13560-970 Sa˜o Carlos, Sa˜o Paulo, Brazil
An important feature of Axelrod’s model for culture dissemination or social influence is the
emergence of many multicultural absorbing states, despite the fact that the local rules that specify
the agents interactions are explicitly designed to decrease the cultural differences between agents.
Here we re-examine the problem of introducing an external, global interaction – the mass media
– in the rules of Axelrod’s model: in addition to their nearest-neighbors, each agent has a certain
probability p to interact with a virtual neighbor whose cultural features are fixed from the outset.
Most surprisingly, this apparently homogenizing effect actually increases the cultural diversity of the
population. We show that, contrary to previous claims in the literature, even a vanishingly small
value of p is sufficient to destabilize the homogeneous regime for very large lattice sizes.
PACS numbers: 89.75.Fb, 87.23.Ge, 05.50.+q
I. INTRODUCTION
Why do people have different opinions given that af-
ter repeated interactions some consensus should emerge?
Why are there different cultures given that modern me-
dia has apparently succeeded in transforming the planet
into a global village [1]? These are the issues addressed
by Axelrod’s model for the dissemination of culture or
social influence [2], which is considered the paradigm for
idealized models of collective behavior which seek to re-
duce a collective phenomenon to its functional essence
[3]. In fact, building on just a few simple principles, Ax-
elrod’s model provides highly nontrivial answers to those
questions. In Axelrod’s model, an agent – an individual
or a culturally homogeneous village – is represented by
a string of F cultural features, where each feature can
adopt a certain number q of distinct traits. The interac-
tion between any two agents takes place with probability
proportional to their cultural similarity, i.e., proportional
to the number of traits they have in common. The re-
sult of such interaction is the increase of the similarity
between the two agents, as one of them modifies a previ-
ously distinct trait to match that of its partner.
Notwithstanding the built-in assumption that social
actors have a tendency to become more similar to each
other through local interactions [4, 5], Axelrod’s model
does exhibit global polarization, i.e., a stable multicul-
tural regime [2]. More importantly, however, at least
from the statistical physics perspective, is the fact that
the competition between the disorder of the initial con-
figuration and the ordering bias of the local interactions
produces a nontrivial threshold phenomenon (more pre-
cisely, a nonequilibrium phase transition) which sepa-
rates in the space of parameters of the model the globally
homogeneous from the globally polarized regimes [6, 7].
A feature that sets Axelrod’s model apart from most
lattice models which exhibit nonequilibrium phase tran-
sitions [8, 9] is the fact that all stationary states of the
dynamics are absorbing states, i.e., the dynamics freezes
in the long time regime [6]. This is so because, according
to the rules of Axelrod’s model, two neighboring agents
who do not have any cultural trait in common cannot
interact and the interaction between agents who share
all the cultural traits does not change their cultural fea-
tures. Hence at equilibrium we can safely predict that,
regarding their cultural features, any neighbor of a given
agent is either identical to or completely different from it.
This is a double-edged sword: on the one hand, we can
easily identify the stationary regime, which is a major
problem in the characterization of nonequilibrium phase
transitions; on the other hand, the dynamics can take
an arbitrarily large time to freeze for some parameter
settings and initial conditions [6, 7, 10, 11].
The key ingredient for the existence of a stable globally
polarized state is the rule that prohibits the interaction
between completely different agents (i.e., agents which
do not have a single cultural trait in common). This
was first pointed out by Kennedy [12] who relaxed this
rule and permitted interactions regardless of the simi-
larity between agents. As a result, the system evolved
until all agents became identical, i.e., the only absorb-
ing states were the homogenous ones. (There are qF dis-
tinct absorbing homogenous configurations.) In addition,
Klemm et al. [13] have shown that the introduction of
external noise to the dynamics so that a single trait of an
arbitrarily chosen agent was changed at random ends up
destabilizing the polarized state. Moreover, expansion of
communication modeled by increasing the connectivity
of the lattice [14, 15] or by placing the agents in more
complex networks [16] (e.g., small-world and scale-free
networks) also resulted in cultural homogenization.
It should be mentioned, however, that other models
of social influence seem to yield a more robust polar-
ized state. For instance, the frequency bias mechanism
[17, 18] for cultural or opinion change assumes that the
number of people holding an opinion is the key factor for
an agent to adopt that opinion, i.e., people have a ten-
dency to espouse cultural traits that are more common in
their social environment. This is then the standard voter
model of statistical physics [19]. Parisi et al. [20] have
replaced the rules of Axelrod’s model by the frequency
bias mechanism (essentially, a majority rule) and found a

2stable polarized state for small lattices. Since similarity
plays no role in the agents’ interactions, the frequency
bias mechanism is naturally robust to noise.
The impression is then that the globally polarized
(multicultural) state is very frail, being disrupted by any
(realistic or not) extension of the original model. In view
of this feeling, it came as a big surprise the finding by
Shibanai et al. [21] that the introduction of a homo-
geneous media effect (i.e., it is the same for all agents)
aiming at influencing the agents’ opinions actually favors
polarization. This finding is at odds with the common-
sense view that mass media, such as newspapers and tele-
vision, are devices that can be effectively used to control
people’s opinions and so homogenize society. Of course,
the effect of media in real personal networks is compli-
cated and seems to follow the so-called ‘two-step flow of
communication’ in which the media affect opinion leaders
first, who then influence the rest of the population [22].
In fact, personal networks seem to serve as a buffer for
the media effect.
Although this counterintuitive effect of the mass media
has been extensively investigated (see, e.g., [23, 24, 25,
26]) there is still no first-principles explanation to it. The
research has focused mostly on the search for a threshold
on the intensity of the media influence such that above
that threshold the population would become polarized
and below it, the population would becomes culturally
homogeneous. In this contribution we show that such
threshold is in fact an artifact of finite lattices: when a
careful analysis of the finite-size effects is carried out we
find that even a vanishingly small media influence is suf-
ficient to destabilize the culturally homogeneous regime.
The rest of this paper is organized as follows. In
Sect. II we describe the original Axelrod’s model, discuss
at some length the basic assumptions of the model and
introduce the effect of an external fixed media [23, 24].
In Sect. III we present an efficient algorithm to simu-
late Axelrod’s model. The simulation results as well as a
discussion of our main results are presented also in that
section. Finally, in Sect. IV we present our concluding
remarks.
II. MODEL
In Axelrod’s model each agent is characterized by a set
of F cultural features which can take on q distinct val-
ues. Hence an agent is represented by a string of symbols,
e.g. 13255 in the case of F = 5 and q = 5. Clearly, for
this parameter setting there are only qF = 3125 different
cultures. The agents are fixed in the sites of a square
lattice of size L × L with periodic boundary conditions
and can interact only with their four nearest neighbors.
The initial configuration is completely random with the
features of each agent given by random integers drawn
uniformly from 1 to q. At each time we pick an agent
at random (this is the target agent) as well as one of its
neighbors. These two agents interact with probability
equal to their cultural similarity, defined as the fraction
of common cultural features. For instance, assuming that
the target agent is described by the string 13255 and its
neighbor by 13425, the interaction occurs with probabil-
ity 3/5. In case the interaction action is not selected,
we pick another target agent at random and repeat the
procedure. An interaction consists of selecting at ran-
dom one of the distinct features, and changing the target
agent’s trait on this feature to the neighbor’s correspond-
ing trait. Returning to our example, if the third feature
is chosen the target agent becomes 13455 and its neigh-
bor remains unchanged. This procedure is repeated until
the system is frozen in an absorbing configuration.
The basic assumption of Axelrod’s model is that sim-
ilarity is a main requisite for social interaction and, as
a result, exchange of opinions. This is the ‘birds of a
feather flock together’ hypothesis which states that in-
dividuals who are similar to each other are more likely
to interact and then become even more similar [5]. Re-
cent empirical evidence in favor of this assumption comes
from the analysis of Web 2.0 social networks [27]. Study
of a population of over 107 people indicates that people
who chat with each other using instant messaging are
more likely to have common interests, as measured by
the similarity of their Web searches, and the more time
they spend talking, the stronger this relationship is. We
note, however, that this assumption is disputed by other
researchers who argue that people are attracted to others
who resemble their ideal, rather than their actual selves
[28].
To introduce the effect of a global media following the
seminal paper by Shibanai et al. [21], we need first to de-
fine a virtual agent whose cultural traits reflect the media
message. In Ref. [21], each cultural feature of the virtual
agent has the trait which is the most numerous in the
population – the consensus opinion. Here we choose to
keep the media message fixed from the outset, so it really
models some alien influence impinging on the population.
Explicitly, we generate the culture vector of the virtual
agent at random and keep it fixed during the dynamics.
Next, we need to specify how the media interact with
the real agents. To do that we introduce a new control
parameter p ∈ [0, 1], which measures the strength of the
media influence. As in the original Axelrod’s model, we
begin by choosing a target agent at random, but now it
can interact with the media with probability p or with
its neighbors with probability 1 − p. Since we have de-
fined the media as a virtual agent, the interaction follows
exactly the same rules as before. The original model is
recovered for p = 0, provided we properly define the halt-
ing criterion of the dynamics, as discussed in the next
section.
III. RESULTS
To simulate efficiently Axelrod’s model we make a list
of the active agents. An active agent is an agent that

3has at least one feature in common and at least one fea-
ture distinct with at least one of its four nearest neigh-
bors. Clearly, since only active agents can change their
cultural features, it is more efficient to select the target
agent randomly from the list of active agents rather than
from the entire lattice. Note that the randomly selected
neighbor of the target agent may not necessarily be an
active agent itself. In the case that the cultural features
of the target agent are modified by the interaction with
its neighbor, we need to re-examine the active/inactive
status of the target agent as well as of all its neighbors
so as to update the list of active agents. The dynamics
is frozen when the list of active agents is empty. This is
the halting criterion we mentioned in the last section.
The important point in this halting criterion is that
the virtual agent does not enter the procedure to deter-
mine whether a real agent is active or not; otherwise the
dynamics would not freeze. Actually, there are only two
situations where the dynamics could freeze in the case the
virtual agent is used in that procedure: in the uniform
regime where all agents become identical to the virtual
agent, and in a two-domains regime where one domain
is identical to the virtual agent and the other is com-
pletely opposed (there are (q − 1)F distinct realizations
of this possibility). However, since the dynamics does not
lead in general to these situations, it gets stuck in a trite
position in which changes occurs due to the interaction
with the virtual agent only. Although it has never been
explicitly pointed out, this must have been the halting
criterion used in previous analyses of the effect of media
in Axelrod’s model [21, 23, 24, 25, 26].
In order to explore fully the dependence of the frozen
configuration on the lattice size, in this contribution we
restrict our analysis to the parameter setting F = q = 5.
A feature that sets our results apart from those reported
previously in the literature is that our data points rep-
resent averages over at least 103 independent runs for
lattices of linear size up to L = 3000. (For comparison,
we note that the results of Refs. [23, 24] are derived from
simulations of lattices with L = 40 and 50 independent
runs.) This requires a substantial computational effort,
especially in the regime where the number of cultures
decreases with the lattice size since then the time for
absorption can be as large as 106 × L2. In the figures
presented in the following, the error bars are smaller or
at most equal to the symbol sizes.
For our purposes, the frozen configuration can be char-
acterized by the ratio between the number of clusters (or
cultural domains) S and the lattice area L2. A clus-
ter is simply a bounded region of uniform culture. In
the case of diasporas [14], the two or more cultural do-
mains (which are characterized by the same culture) are
counted separately. We note that since S is bounded by
L2 we have g ≡ S/L2 ≤ 1. In the uniform regime we have
S = 1 and so g = 1/L2. Figure 1 exhibits this measure
as function of the strength of the media influence p for
different lattice sizes. The suitability of the measure g is
demonstrated by the fact that the data converge to well-
0
0.002
0.004
0.006
0.008
0 0.02 0.04 0.06 0.08 0.1
g
p
FIG. 1: Ratio g between the number of cultural domains and
the lattice area as function of the strength of the media influ-
ence for L = 50 () and 200 (△). The solid line is the result
of the extrapolation of the data to the limit L2 → ∞. The
parameters are F = 5 and q = 5.
defined values (solid line in Fig. 1) as the lattice size is
increased. In other words, S increases with L2 for p not
too small. Indeed, from Fig. 1 it seems that the measure
g vanishes for small p which would indicate the existence
of a minimum strength value pc, above which the uniform
regime is destabilized [23, 24, 25, 26]. Visual inspection
of the data shown in Fig. 1 yields pc ≈ 0.03, which agrees
with the estimate of Ref. [24] (see their Figure 3).
A more careful analysis reveals a different story, how-
ever, as shown in Fig. 2. In fact, consider the data
for p = 0.01, which is well below our initial estimate,
pc ≈ 0.03. An analysis of lattices of sizes up to L = 600
indicates a clear tendency of convergence towards the
uniform regime (i.e., g = 1/L2 fits the data almost per-
fectly in that range of L), but this trend changes com-
pletely when lattices of sizes greater than L = 1000 are
considered. In this case, rather than vanishing as 1/L2,
g tends to a nonzero value when L → ∞. To verify
whether this finding holds true for all values of p we need
first estimate the value of g = g (p) for infinite lattices
and nonzero p and then try to figure out the dependence
of the extrapolated value of g on p in the limit p → 0. As
suggested by Fig. 2, direct simulations using small val-
ues of the parameter p would require very large lattice
sizes in order to produce significative deviations from the
uniform regime.
Figure 3 illustrates the procedure used to obtain the
measure g in the limit L →∞. The key point is the use
of the fitting function g (p) = Bp+Ap/L2 which describes
the data very well for L > 500: the statistical error in
the estimate of Bp = limL→∞ g (p) is less than 2% for
all values of p considered here. The solid curve shown
in Fig. 1 was obtained following this procedure. Finally,
Fig. 4 presents the dependence of Bp on p. For small p

4 1e-008
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.1
1
1 10 100 1000
g
L
FIG. 2: Logarithmic plot of the ratio g as function of
the linear size L of the lattice for (top to bottom) p =
0.05, 0.04, 0.03, 0.02 and 0.01. The solid straight line is 1/L2,
which corresponds to the value of g in the uniform regime.
The parameters are F = 5 and q = 5.
1e-007
1e-006
1e-005
0.0001
0 0.002 0.004 0.006 0.008
g
1/L
FIG. 3: Ratio g between the number of cultural domains and
the lattice area as function of the reciprocal of the lattice size
1/L for p = 0.02 (△) and 0.01 (◦). The solid lines are the
fittings g = Bp + Ap/L2 and the dashed line is the function
1/L2. The parameters are F = 5 and q = 5.
the data is fitted very well by the equation
Bp = lim
L→∞
g (p) = (260± 29) p4.38±0.03 (1)
as indicated in the figure. The large value of the power
of p may explain why the numerical simulations yielded a
nonzero value for pc: for small p it is virtually impossible
to distinguish the result of Eq. (1) from zero.
In sum, Axelrod’s model does not exhibit a phase tran-
sition for p > 0: the only stable regime for infinite lattice
sizes is the polarized one. Strictly, this conclusion is valid
for a single setting of the control parameters, namely,
1e-008
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.01 0.1
B p
p
FIG. 4: The ratio between the number of cultural domains
and the lattice area for L →∞ obtained through the extrap-
olation procedure shown in Fig. 3 as function of the strength
p of the media influence. The straight line is the fitting given
by Eq. (1). The parameters are F = 5 and q = 5.
F = q = 5, but we see no reason why it should not hold
for other values of these parameters as well.
IV. CONCLUSION
In this contribution we have revisited an important
extension of Axelrod’s model in which, in addition to
the local interactions between agents, there is a global
element – the media – that influences the agents’ opin-
ions or cultural traits [21]. In stark contrast to the com-
mon sense opinion that the media effect is to homogenize
the society we find, in agreement with previous studies
[21, 23, 24, 25, 26], that the media actually promotes po-
larization or the diversity of opinions. However, we have
shown that this effect is so powerful that a vanishingly
small influence strength p is sufficient to destabilize the
cultural homogenous state for very large lattices. This
finding calls for a re-examination of the claim, which is
based on the analysis of small lattices, that there exists
a threshold value pc below which the homogeneous state
is stable. At present we have no idea of why the me-
dia promotes polarization rather than the expected ho-
mogenization. An analysis of the distribution of sizes of
the cultural domains as well as of the distance between
domains may provide some cue on this counterintuitive
effect. Work in this line is under way.
Acknowledgments
This research was supported by CNPq and FAPESP,
Project No. 04/06156-3.

5[1] M. McLuhan, Understanding Media: The Extensions of
Man (Signet Books, New York, 1966).
[2] R. Axelrod, J. Conflict Res. 41, 203 (1997).
[3] R. L. Goldstone and M. A. Janssen, Trends Cog. Sci. 9,
424 (2005).
[4] B. Latane´, American Psychologist 36, 343 (1981).
[5] S. Moscovici, Handbook of Social Psychology 2, 347
(1985).
[6] C. Castellano, M. Marsili and A. Vespignani, Phys. Rev.
Lett. 85, 3536 (2000).
[7] L. A. Barbosa and J. F. Fontanari, Theor. Biosc.
DOI:10.1007/s12064-009-0066-z (2009)
[8] J. Marro and R. Dickman, Nonequilibrium Phase Tran-
sitions in Lattice Models (Cambridge University Press,
Cambridge, UK, 1999).
[9] H. Hinrichsen, Adv. Phys. 49, 815 (2000).
[10] D. Vilone, A. Vespignani and C. Castellano, Europ. Phys.
J. B 30, 399 (2002).
[11] F. Vazquez and S. Redner, EPL 78, 18002 (2007).
[12] J. Kennedy, J. Conflict Res. 42, 56 (1998).
[13] K. Klemm, V. M. Egu´ıluz, R. Toral, M. San Miguel,
Phys. Rev. E 67, 045101R (2003).
[14] J. M. Greig, Conflict Res. 46, 225 (2002).
[15] K. Klemm, V. M. Egu´ıluz, R. Toral, M. San Miguel,
Physica A 327, 1 (2003).
[16] K. Klemm, V. M. Egu´ıluz, R. Toral, M. San Miguel,
Phys. Rev. E 67, 026120 (2003).
[17] R. Boyd and P. J. Richerson, Culture and the evolution-
ary process (University of Chicago Press, Chicago, 1985).
[18] A. Nowak, J. Szamrej and B. Latane´, Psychological Re-
view 97, 362 (1990).
[19] T. M. Ligget, Interacting Particle Systems (Springer,
New York, 1985).
[20] D. Parisi, F. Cecconi, F. Natale, J. Conflict Res. 47, 163
(2003).
[21] Y. Shibanai, S. Yasuno, I. Ishiguro, J. Conflict Res. 45,
80 (2001).
[22] P. Lazarsfeld, B. Berelson and H. Gaudet, The people’s
choice (Columbia University Press, New York, 1948).
[23] J. C. Gonza´lez-Avella, M. G. Cosenza and K. Tucci,
Phys. Rev. E 72, 065102R (2005)
[24] J. C. Gonza´lez-Avella, M. Egu´ıluz, M. G. Cosenza, K.
Klemm, J. L. Herrera and M. San Miguel, Phys. Rev. E
73, 046119 (2006).
[25] K. I. Mazzitello, J. Candia and V. Dossetti, Intern. J.
Mod. Phys. C 18, 1475 (2007).
[26] J. Candia and K. I. Mazzitello, J. Stat. Mech. P07007
(2008).
[27] P. Singla P and M. Richardson, Proceedings of the 17th
International World Wide Web Conference (ACM Press,
Toronto, 2008) pp. 655–664.
[28] C. G. Wetzel and C. A. Insko, J. Exp. Soc. Psych. 18,
253 (1985).