Table Of ContentLocal versus global interactions in nonequilibrium transitions: A model of social
dynamics
J. C. Gonz´alez-Avella,1 V. M. Egu´iluz,1 M. G. Cosenza,1,2 K. Klemm,3 J. L. Herrera,4 and M. San Miguel1
1IMEDEA (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
6 2Centro de F´isica Fundamental, Universidad de los Andes, Apartado Postal 26, M´erida 5251, Venezuela
0 3Bioinformatics, Dept. of Computer Science, University of Leipzig, H¨artelstr. 16, 04107 Leipzig, Germany
0 4Centro de F´isica Fundamental, Universidad de los Andes, Apartado Postal 26, Merida 5251, Venezuela
2
(Dated: February 6, 2008)
n
A nonequilibrium system of locally interacting elements in a lattice with an absorbing order-
a
J disorderphasetransition isstudiedundertheeffectofadditionalinteractingfields. Thesefieldsare
shown toproduceinterestingeffectsinthecollectivebehaviorof thissystem. Both for autonomous
6
and external fields, disorder grows in the system when the probability of the elements to interact
1
withthefieldisincreased. Thereexistsathresholdvalueofthisprobabilitybeyondwhichthesystem
] isalwaysdisordered. Thedomainofparametersoftheorderedregimeislargerfornonuniformlocal
h fields than for spatially uniform fields. However, the zero field limit is discontinous. In the limit of
c vanishinglysmallprobabilityofinteraction with thefield,autonomousorexternalfieldsareableto
e ordera system that would fall in a disordered phaseunderlocal interactions of theelements alone.
m
Weconsiderdifferenttypesoffieldswhichareinterpretedasformsofmassmediaactingonasocial
- system in thecontext of Axelrod’s model for cultural dissemination.
t
a
t PACSnumbers: 89.75.Fb,87.23.Ge,05.50.+q
s
.
t
a
I. INTRODUCTION present in that system, such as chaotic synchronization
m
andnewspatialpatterns. However,theclassificationand
-
description of generic effects produced by external fields
d Theemergenceofnontrivialcollectivebehaviorinspa-
n tiotemporal dynamical systems is a central issue in the or global coupling in a nonequilibrium system of locally
o current research on complex systems, as in many physi- interacting units is still an open general question. The
c common wisdom for equilibrium systems is that under
cal,chemical,biological,economicandsocialphenomena.
[
a strong external field, local interactions become negli-
There are a variety of processes occurring in these sys-
1 tems where both spatially local and global interactions gible, and the system orders following the external field.
v Fornonequilibriumnonpotentialdynamics[13]thisisnot
extending all over the system coexist and contribute in
0 necessarilythecase,andnontrivialeffectsmightarisede-
different and competing ways to the collective dynam-
4 pending on the dynamical rules.
ics. Some examples include Turing patterns [1] (with
3
1 slow and fast diffusion), Ginzburg-Landau dynamics [2], Thisproblemis,inparticular,relevantforrecentstud-
0 surface chemical reactions [3], sand dunes (with the mo- iesofsocialphenomenainthegeneralframeworkofcom-
6 tions of wind and of sand) [4], and pattern formation in plex systems. The aim is to understand how collec-
0 some biological systems [5]. Recently, the collective be- tive behaviors arise in social systems. Several mathe-
/
t havior of dynamical elements subject to both local and matical models, many of them based on discrete-time
a
global interactions has been experimentally investigated and discrete-space dynamical systems, have been pro-
m
in arrays of chaotic electrochemical cells [6]. Many of posed to describe a variety of phenomena occurring in
- these systems can be modeled as networks of coupled social dynamics [14, 15, 16, 17, 18, 19, 20, 21, 22]. In
d
n dynamical units with coexisting local and global inter- this context, specially interesting is the lattice model
o actions [7]. Similarly, the phenomena of pattern forma- introduced by Axelrod [23] to investigate the dissemi-
c tion and collective behavior induced by external forcing nation of culture among interacting agents in a society
v: on spatiotemporal systems, such as chemical reactions [22, 24, 25, 26, 27, 28, 29, 30]. The state of an agent in
i [8, 9] or granular media [10], has also been considered. thismodelisdescribedbyasetofindividualculturalfea-
X
The analogy between external forcing and global cou- tures. The local interaction between neighboring agents
r pling in spatiotemporal dynamical systems has recently depends on the cultural similarities that they share and
a
been explored in the framework of coupled map lattice similarityisenhancedasaresultoftheinteraction. From
models [11, 12]. It has been found that, under some the point of view of statistical physics, this model is ap-
circumstances, the collective behavior of an autonomous pealingbecauseitexhibitsanontrivialoutofequilibrium
spatiotemporalsystemwithlocalandglobalinteractions transitionbetweenanorderedphase(ahomogeneouscul-
is equivalent to that of a driven spatiotemporal system ture) and a disordered (multicultural) one, as in other
possessing similar local couplings as in the autonomous well studied lattice systems with phase ordering proper-
system. ties [31]. The additional effect of global coupling in this
The addition of a global interaction to a locally cou- system has been considered as a model of influence of
pledsystemisknowntobeabletoinducephenomenanot mass media [24]. It has also been shown that the addi-
2
tionofexternalinfluences,suchasrandomperturbations mally,wetreatthefieldateachelementiasanadditional
[28] or a fixed field [32], can induce new order-disorder neighbor of i with whom an interaction is possible. The
nonequilibrium transitions in the collective behavior of field is represented as an additional element φ(i) such
Axelrod’smodel. However,aglobalpictureoftheresults that σ =µ in the definition givenbelow of the dy-
φ(i)f if
of the competition between the local interaction among namics. The strength of the field is given by a constant
the agents and the interaction througha globalcoupling parameterB ∈[0,1]that measuresthe probabilityof in-
field or an external field is missing. In this paper we teraction with the field. The system evolves by iterating
address this general question in the specific context of the following steps:
Axelrods model. (1)Selectatrandomanelementionthelattice(called
We deal with states of the elements of the system and active element).
interactingfieldsdescribedbyvectorswhosecomponents (2)Selectthesourceofinteractionj. Withprobability
cantakediscretevalues. Theinteractiondynamicsofthe B set j = φ(i) as an interaction with the field. Other-
elements among themselves and with the fields is based wise,chooseelementj atrandomamongthefournearest
on the similarity between state vectors, defined as the neighbors (the von Neumann neighborhood) of i on the
fraction of components that these vectors have in com- lattice.
mon. We consider interaction fields that originate either (3) Calculate the overlap (number of shared compo-
externally (an external forcing) or from the contribution nents) l(i,j) = PF δ . If 0 < l(i,j) < F, sites i
of a set of elements in the system (an autonomous dy- f=1 σif,σjf
andj interactwithprobabilityl(i,j)/F. Incaseofinter-
namics)suchasglobalorpartialcouplingfunctions. Our
action, choose h randomly such that σ 6= σ and set
ih jh
study allows to compare the effects that driving fields
σ =σ .
ih jh
or autonomous fields of interaction have on the collec-
(4) Update the field M if required (see definitions of
tive properties of systems with this type of nonequilib-
fields below). Resume at (1).
rium dynamics. In the context of social phenomena, our
Step(3)specifiesthebasicruleofanonequilibriumdy-
scheme can be considered as a model for a social system
namicswhichisatthebasisofmostofourresults. Ithas
interactingwithglobalorlocalmassmediathatrepresent
two ingredients: i) A similarity rule for the probability
endogenous cultural influences or information feedback,
of interaction, and ii) a mechanism of convergence to an
as well as a model for a social system subject to an ex-
homogeneous state.
ternal cultural influence. Our results indicate that the
Before considering the effects of the field M, let us re-
usual equilibrium notion that the application of a field
view the originalmodel without field (B =0). In any fi-
shouldenhanceorderinasystemdoesnotholdhere. On
nitenetworkthedynamicssettlesintoanabsorbingstate,
the contrary,disorderbuilds-upby increasingthe proba-
characterized by either l(i,j) = 0 or l(i,j) = F, for all
bility of interaction of the elements with the field. This
pairsofneighbors(i,j). Homogeneous(”monocultural”)
occurs independently of the nature (either external or
statescorrespondtol(i,j)=F,∀i,j,andobviouslythere
autonomous)of the field of interactionadded to the sys-
are qF possible configurations of this state. Inhomoge-
tem. Moreover,we find that a spatially nonuniformfield
neous(”multicultural”)statesconsistoftwoormoreho-
of interaction may actually produce less disorder in the
mogeneousdomainsinterconnectedbyelementswithzero
system than a uniform field.
overlapandthereforewithfrozendynamics. Adomainis
The model, including the description of three types of
a set of contiguous sites with identical state vectors. It
interactionfieldsbeingconsidered,ispresentedinSec.II.
has been shown that the system reaches ordered, homo-
In Sec. III, the effects of the fields in the ordered phase
geneousstatesforq <q anddisordered,inhomogeneous
c
of the system are shown, while Sec. IV analyzes these
statesforq >q ,whereq isacriticalvaluethatdepends
c c
effects in the disordered phase. Section V contains a
onF [25,26, 27, 28,29]. This order-disordernonequilib-
global picture and interpretation of our results.
riumtransitionisofsecondorderinone-dimensionalsys-
tems and of first order in two-dimensional systems [30].
It has also been shown that the inhomogeneous configu-
II. THE MODEL rationsarenotstable: singlefeatureperturbationsacting
ontheseconfigurationsunfreezethedynamics. Underre-
The system consists of N elements as the sites of a peated action of these perturbations the system reaches
square lattice. The state c of element i is defined as a an homogeneous state [28].
i
vector of F components σ = (σ ,σ ,...,σ ). In Ax- To characterize the transition from an homogeneous
i i1 i2 iF
elrod’s model, the F components of c correspondto the state to a disordered state, we consider as an order pa-
i
culturalfeatures describing theF-dimensionalculture of rameter the average fraction of cultural domains g =
element i. Each component σ can take any of the q hN i/N. Here N is the number of domains formed in
if g g
valuesinthe set{0,1,...,q−1}(calledculturaltraitsin the final state of the system for a given realization of
Axelrod’s model). As an initial condition, each element initial conditions. Figure 1 shows the quantity g as a
is randomly and independently assigned one of the qF function of the number of options per component q, for
state vectors with uniform probability. We introduce a F = 5, when no field acts on the system (B = 0). For
vector field M with components (µ ,µ ,...,µ ). For- values of q < q ≈ 25, the system always reaches a ho-
i1 i2 iF c
3
1 0.08
0.8
0.06
N
>/ 0.6
max g0.04
S
<
g, 0.4
0.02
0.2
0 0
0 10 20 30 40 50 0 0.02 0.04 0.06 0.08 0.1
q q
FIG. 2: Order parameter g as a function of the coupling
FIG.1: Orderparametersg(circles)andhSmaxi/N (squares)
strength B of an external (squares), global (circles) and lo-
as a function of q,in theabsence of a field B=0.
cal (triangles) field. Parameter value q=10<qc.
mogeneous state characterized by values g → 0. On the
quent value present in component f of the state vectors
other hand, for values of q > q , the system settles into
c oftheelementsbelongingtothevonNeumannneighbor-
a disordered state, for which hN i ≫ 1. Another pre-
g hood of element i. If there are two or more maximally
viously used order parameter [25, 27], the average size
abundant values of component f one of these is chosen
of the largest domain size, hS i/N, is also shown in
max at random with equal probability. The local field can
Fig. 1 for comparison. In this case, the ordered phase
be interpreted as local mass media conveying the “local
corresponds to hS i/N = 1, while complete disorder
max cultural trend” of its neighborhood to each element in a
is given by hS i/N →0. Unless otherwise stated, our
max social system.
numerical results throughout the paper are based on av-
Case (i) corresponds to a driven spatiotemporal dy-
eragesover50realizationsforsystemsofsizeN =40×40,
namical system. On the other hand, cases (ii) and (iii)
and F =5.
can be regarded as autonomous spatiotemporal dynam-
Let us now consider the case where the elements on
ical systems. In particular, a system subject to a global
the lattice have a non-zero probability to interact with
field correspondstoanetworkofdynamicalelementspos-
the field (B >0). We distinguish three types of fields.
sessingboth localandglobalinteractions. Both the con-
(i) The external field is spatially uniform and con- stantexternal field and the global field are uniform. The
stant in time. Initially for each component f, a value local field isspatiallynon-uniform;itdependsonthesite
ǫf ∈ {1,...,q} is drawn at random and µif = ǫf is set i. In the context of cultural models, systems subject to
for all elements i and all components f. It corresponds either local or global fields describe social systems with
to a constant, external driving field acting uniformly on endogenous cultural influences, while the case of the ex-
the system. A constant external field can be interpreted ternal field represents and external cultural influence.
as a specific cultural state (such as advertising or propa- The strength of the coupling to the interaction field is
ganda) being imposed by controlled mass media on all controlled by the parameter B. We shall assume that B
the elements of a social system [32]. isuniform,i.e.,thefieldreachesalltheelementswiththe
(ii) The global field is spatially uniform and may vary same probability. In the cultural dynamics analogy, the
in time. Here µif is assigned the most abundant value parameter B can be interpreted as the probability that
exhibited by the f-th component of all the state vectors themassmediavectorhastoattractthe attentionofthe
in the system. If the maximally abundant value is not agentsin the socialsystem. The parameterB represents
unique, one of the possibilities is chosen at random with enhancingfactorsofthemassmediainfluencethatcanbe
equal probability. This type of field is a global coupling varied, such as its amplitude, frequency, attractiveness,
functionofalltheelementsinthesystem. Itprovidesthe etc.
sameglobalinformationfeedbacktoeachelementatany
given time but its components may change as the sys-
tem evolves. In the context of cultural models [24], this III. EFFECTS OF AN INTERACTING FIELD
fieldmay representa globalmassmedia influence shared FOR q<qc
identicallybyallthe agentsandwhichcontainsthe most
predominant trait in each cultural feature present in a In the absence of any interactionfield, the system set-
society (a “global cultural trend”). tles into one of the possible qF homogeneous states for
(iii) The local field, is spatially non-uniform and non- q < q (see Fig. 1). Figure 2 shows the order parameter
c
constant. Each component µ is assigned the most fre- g as a function of the coupling strength B for the three
if
4
0.05 0.8
0.04
0.6
0.03
B
c g 0.4
0.02
0.2
0.01
0
5 10 15 20 25 0
0 0.3 0.6 0.9
q B
FIG. 3: Threshold values Bc for q <qc corresponding to the FIG. 4: Order parameter g as a function of the coupling
differentfields. Eachlineseparatestheregionoforder(above strength B of an external (squares), global (circles) and lo-
the line) from the region of disorder (below the line) for an cal (triangles) field. The horizontal dashed line indicates the
external (squares), global (circles), and local (triangles) field. valueof g at B =0. Parameter value q=30.
of the probability B.
types of fields. When the probability B is small enough, Note thatthe regionofhomogeneousorderedstates in
the system still reaches in its evolution a homogeneous the (B,q) space in Fig.3 is largerfor the local field than
state (g → 0) under the action of any of these fields. for the external and the global fields. A nonuniform field
In the case of an external field, the homogeneous state provides different influences on the agents, while the in-
reached by the system is equal to the field vector [32]. teractionwithuniformfieldsissharedbyalltheelements
Thus, forsmallvalues of B, a constantexternal field im- in the system. The local field (spatially nonuniform) is
posesitsstateoverallthe elementsinthe system,asone less efficient than uniform fields in promotingthe forma-
may expect. With a global or with a local field, however, tion of multiple domains, and therefore order is main-
for small B the system can reach any of the possible qF tained for a larger range of values of B when interacting
homogeneousstates, depending onthe initial conditions. with a local field.
Regardless of the type of field, there is a transition at a
thresholdvalueoftheprobabilityB fromahomogeneous
c
statetoadisorderedstatecharacterizedbyanincreasing IV. EFFECTS OF AN INTERACTING FIELD
number of domains as B is increased. Thus, we find the FOR q>qc
counterintuitive result that, above some threshold value
of the probability of interaction, a field induces disorder
When there are no additional interacting fields (B =
in a situationin whichthe systemwould order(homoge-
0), the system always freezes into disordered states for
neous state) under the effect alone of local interactions
q >q . Figure 4 shows the order parameter g as a func-
c
among the elements.
tionoftheprobabilityBforthethreetypesoffields. The
The threshold values of the probability B for each effect of a field for q > q depends on the magnitude of
c c
type of field, obtained by a regression fitting [32], are B. In the three cases we see that for B → 0, g drops to
plotted as a function ofq in the phase diagramofFig. 3. valuesbelowthereferencelinecorrespondingtoitsvalue
The threshold value B for each field decreases with in- whenB =0. Thus,thelimitB →0doesnotrecoverthe
c
creasing q for q < q . The value B = 0 for the three behavior of the model with only local nearest-neighbor
c c
fields is reached at q = q ≈ 25, corresponding to the interactions. The fact that for B → 0 the interaction
c
critical value in absence of interaction fields observed in with a field increases the degree of order in the system
Fig. 1. For each case, the threshold curve B versus q is related to the non-stablenature of the inhomogeneous
c
in Fig. 3 separates the region of disorder from the re- states in Axelrod’s model. When the probability of in-
gion where homogeneous states occur on the space of teraction B is very small, the action of a field can be
parameters (B,q). For B >B , the interaction with the seen as a sufficient perturbation that allows the system
c
field dominates overthe local interactions among the in- toescapefromtheinhomogeneousstateswithfrozendy-
dividual elements in the system. Consequently, elements namics. The role of a field in this situation is similar
whose states exhibit a greater overlap with the state of to that of noise applied to the system, in the limit of
the fieldhavemoreprobabilitytoconvergetothat state. vanishingly small noise rate [28].
This process contributes to the differentiation of states The drop in the value of g as B → 0 from the refer-
between neighboring elements and to the formation of ence value (B =0) that takes place for the local field in
multiple domains in the system for large enough values Fig. 4 is more pronounced than the corresponding drops
5
1 0.55
0.5
g g 0.45
0.4
0.3 0.35
0.1 1 10-5 10-4 10-3 10-2 10-1
B B
FIG. 5: Scaling of the order parameter g with the coupling FIG.6: FinitesizeeffectsatsmallvaluesofthestrengthB of
strengthtotheglobalfield B. Theslopeofthefittingstraight a global field. Orderparameter g as a function of B is shown
line is β =0.13±0.01. Parameter valueq=30>qc. for system sizes N = 202, 302, 402, 502, 702 (from top to
bottom). Parameter value q=30.
for uniform fields. This can be understood in terms of a
greaterefficiency of a nonuniformfield as a perturbation
the system is driven to full order for B →0 in the limit
that allows the system to escape from a frozen inhomo-
of infinite size by any of the interacting fields considered
geneous configuration. Increasing the value of B results,
here.
in all three types of fields, in an enhancement of the de-
gree of disorder in the system, but the local field always
keeps the amount of disorder, as measured by g, below
the value obtained for B = 0. Thus a local field has V. SUMMARY AND CONCLUSIONS
a greater ordering effect than both the global and the
external fields for q >q . Wehaveanalyzedanonequilibriumlatticemodeloflo-
c
Thebehavioroftheorderparametergforlargervalues callyinteractingelementsandsubjecttoadditionalinter-
of B can be described by the scaling relation g ∼ Bβ, actingfields. Thestatevariablesaredescribedbyvectors
wheretheexponentβ dependsonthevalueofq. Figure5 whose components take discrete values. We haveconsid-
showsalog-logplotofgasafunctionofB,forthecaseof ered the cases of a constant external field, a global field,
a global field, verifyingthis relation. This result suggests anda local field. The interactiondynamics,basedonthe
that g should drop to zero as B → 0. The partial drops similarityoroverlapbetweenvectorstates,producessev-
observed in Fig. 4 seem to be due to finite size effects eralnontrivialeffectsinthecollectivebehaviorofthissys-
for B → 0. A detailed investigation of such finite size tem. Namely,wefindtwomaineffectsthatcontradictin-
effectsisreportedinFig.6forthecaseoftheglobal field. tuitionbasedonthe effectofinteractingfields inequilib-
It is seen that, for very small values of B, the values of rium systems where the dynamics minimizes a potential
g decrease as the system size N increases. However, for function. First, we find that an interacting field might
valuesofB >∼10−2,thevariationofthesizeofthesystem disorder the system: For parameter values for which the
does not affect g significantly. systemordersduetothelocalinteractionamongthe ele-
Figure 7 displays the dependence of g on the size of ments, there is a thresholdvalue B ofthe probabilityof
c
the system N when B → 0 for the three interaction interactionwith afield. ForB >B the systembecomes
c
fields being considered. For each size N, a value of g disordered. This happens because there is a competition
associated with each field was calculated by averaging between the consequences of the similarity rule applied
over the plateau values shown in Fig. 6 in the interval to the local interactions among elements, and applied to
B ∈ [10−5,10−3]. The mean values of g obtained when the interaction with the field. This leads to the forma-
B =0arealsoshownforreference. Theorderparameter tion of domains and to a disordered system. A second
g decreases for the three fields as the size of the system effect is that, for parametervalues for which the dynam-
increases; in the limit N → ∞ the values of g tend to ics based on the local interaction among the elements
zero and the system becomes homogeneous in the three leads to a frozen disordered configuration, very weak in-
cases. ForsmallvaluesofB,thesystemsubjecttothelo- teracting fields are able to order the system. However,
cal field exhibitsthe greatestsensitivitytoanincreaseof increasing the strength of interaction with the field pro-
the system size, while the effect of the constant external ducesgrowingdisorderinthesystem. ThelimitB →0is
field islessdependentonsystemsize. Theorderingeffect discontinuous and the ordering effect for B <<1 occurs
of the interaction with a field as B → 0 becomes more because the interaction with the field acts as a pertur-
evident for a local (nonuniform) field. But, in any case, bation on the non stable disordered configurations with
6
1 1
0.8
N 0.6
>/
g 0.1 max
S
< 0.4
0.2
0
0.01 10 20 30 40 50
1000 q
N
FIG. 8: Influence of the interacting field on the nonequi-
FIG. 7: Mean value of the order parameter g as a function
librium order-disorder transition as described by the order
of the system size N without field (B = 0, solid circles),
andwithanexternal (squares),global (circles) andlocal field parameter hSmaxi/N. Results are shown for B = 0 (solid
squares), a global (B = 10−5 (empty squares), B = 0.3 (cir-
(triangles). Parameter value q=30.
cles)) and a local (B = 10−5 (triangles)) field. Parameter
valueF =3.
frozendynamicsappearingforB =0. Inthisregard,the
field behavessimilarlyto a randomfluctuationacting on
behavior of the system displays analogous phenomenol-
the system, which always induces order for small values
ogyforthethreetypesoffieldsconsidered,althoughthey
of the noise rate [28].
have different nature. At the local level, they act in the
These results are summarized in Fig. 8 which shows,
same manner, as a “fifth” effective neighbor whose spe-
for different values of B, the behavior of the order pa-
cific source becomes irrelevant. In particular, both uni-
rameter hS i/N previously considered in Fig. 1. For
max form fields, the global coupling and the external field,
small values of B, the interaction with a field can en-
produce very similar behavior of the system. Recently,
hance order in the system: for q < q interaction with a
c ithasbeenfoundthat, undersomecircumstances,anet-
field preserves homogeneity, while for q > q it causes a
c work of locally coupled dynamical elements subject to
drop in the degree of disorder in the system. In an ef-
either global interactions or to a uniform external drive
fective way the nonequilibrium order-disorder transition
exhibits the same collective behavior [11, 12]. The re-
is shifted to larger values of q when B is non-zero but
sults from the present nonequilibrium lattice model sug-
verysmall. Forlargervalues ofB the transitionshifts to
gest that collective behaviors emerging in autonomous
smaller values of q and the system is always disordered
and in driven spatiotemporal systems can be equivalent
inthelimitingcaseB →1. Thislimitingbehaviorisuse-
in a more general context.
fultounderstandthedifferenceswithordinarydynamics
leading to thermal equilibrium in which a strong field InthecontextofAxelrod’smodelforthedissemination
would order the system. In our nonequilibrium case, the of culture [23] the interacting fields that we have consid-
similarity rule of the dynamics excludes the interaction ered can be interpreted as different kinds of mass media
ofthefieldwithelementswithzerooverlapwiththefield. influences acting on a socialsystem. In this context, our
Since the local interaction among the elements is negli- results suggest that both, an externally controlled mass
gible in this limit, there is no mechanism left to change media or mass media that reflect the predominant cul-
situations of zero overlap and the system remains disor- tural trends of the environment, have similar collective
dered. We have calculated, for the three types of field effects on a social system. We found the surprising re-
considered, the corresponding boundary in the space of sult that, when the probability of interacting with the
parameters (B,q) that separates the orderedphase from massmediaissufficientlylarge,massmediaactuallycon-
the disordered phase. In the case of a constant exter- tribute to cultural diversity in a social system, indepen-
nal field, the ordered state in this phase diagram always dently of the nature of the media. Mass media is only
converges to the state prescribed by the constant field efficient in producing culturalhomogeneity in conditions
vector. The nonuniformlocal field hasagreaterordering ofweakbroadcastofamessage,sothatlocalinteractions
effect than the uniform (global and constant external) among individuals can be still effective in constructing
fields in the regime q >q . The range of values of B for some culturaloverlapwith the mass media message. Lo-
c
which the system is ordered for q < q is also larger for calmass mediaappear tobe moreeffective inpromoting
c
the nonuniform local field. uniformity in comparison to global, uniform broadcasts.
In spite of the differences mentioned between uniform Future extensions of this work shouldinclude the con-
andnonuniformfields,itisremarkablethatthecollective sideration of noise and complex networks of interaction.
7
Acknowledgments andJ.L.H.acknowledgesupportfromC.D.C.H.T.,Uni-
versidad de Los Andes (Venezuela) under grant No. C-
J.C. G-A.,V.M.E.andM.SMacknowledgefinancial 1285-04-05-A. K. K. acknowledges support from DFG
supportfromMEC (Spain)throughprojectsCONOCE2 Bioinformatics Initiative BIZ-6/1-2 and from Deutscher
(FIS2004-00953) and FIS2004-05073-C04-03. M. G. C. Akademischer Austausch Dienst (DAAD).
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Minority Games: Interacting Agents in Financial Mar-
