Consensus and ordering in language dynamics

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by  Xavier Castelló

Consensus and ordering in language dynamics
Xavier Castell´o,1 Andrea Baronchelli,2 and Vittorio Loreto3,4
1IFISC, Institut de F´ısica Interdisciplin`aria i Sistemes Complexos (CSIC-UIB),
Campus Universitat Illes Balears E07122 Palma de Mallorca (Spain)
2Departament de F´ısica i Enginyeria Nuclear, Universitat Polit`ecnica de Catalunya,
Campus Nord B4, 08034 Barcelona (Spain)
3Dipartimento di Fisica, “Sapienza” Universit`a di Roma, P.le A. Moro 2, 00185 Roma (Italy)
4Fondazione ISI, Viale S. Severo 65, 10133 Torino (Italy)
(Dated: January 24, 2009)
9 Weconsidertwosocial consensusmodels, theAB-modelandtheNamingGamerestricted totwo
0 conventions,whichdescribeapopulationofinteractingagentsthatcanbeineitheroftwoequivalent
0 states (A or B) or in a third mixed (AB) state. Proposed in the context of language competition
2 andemergence,theABstatewasassociatedwithbilingualismandsynonymyrespectively. Weshow

n that the two models are equivalent in the mean field approximation, though the differences at the
a microscopic level havenon-trivial consequences. To point them out, we investigate an extension of
J these dynamicsin which confidence/trust is considered, focusing on the case of an underlyingfully
4 connected graph, and we show that the consensus-polarization phase transition taking place in the
2 Naming Game is not observed in the AB model. We then consider the interface motion in regular
lattices. Qualitatively, both models show the same behavior: a diffusive interface motion in a one-

] dimensional lattice, and a curvaturedrivendynamics with diffusing stripe-like metastable states in
h
a two-dimensional one. However, in comparison to the Naming Game, the AB-model dynamics is
p
shown toslow down thediffusion of such configurations.
-
c
o PACSnumbers: 64.60.Cn,87.23.Ge
s
.
s
c I. INTRODUCTION not take into account the possibility of homonymy, so
i thatallmeaningsareindependentandonecanworkwith
s
y only one of them, without loss of generality. An exam-
Theformalism,ideasandtoolsfromstatisticalphysics
h ple of such a game is that of a population that has to
and complex systems have successfully been applied to
p reach the consensus on the name (i.e. the form) to as-
different disciplines of science beyond the traditional re-
[ signto anobject (i.e. the meaning)exploiting only local
search lines of physics, ranging from biology, to eco-
1 interactions. However it is clear that the model, origi-
nomics and the social sciences [1]. In particular, there
v nally inspired to robotic experiments [5], is appropriate
hasbeenafruitfuleffortinprovidingmodelsoflanguage
4 to address general situations in which negotiation rules
dynamics, including dynamics of language competition
4 a decision process on a set of conventions (i.e. opinion
8 [2],languageevolution[3,4]andsemioticdynamics[5,6].
dynamics, etc.). The Naming Game has been studied
3 In the field of language competition, the Abrams- in fully connected graphs (i.e. in mean-field or homoge-
.
1 Strogatz model [2] has triggeredthe developmentof sev- neousmixingpopulations)[5,6,17],regularlattices[18],
0 eral models which take into account the competition of smallworldnetworks[19]andcomplexnetworks[20,21].
9 many [7, 8] or few [9, 10] languages. A review of some Thefinalstateofthesystemisalwaysconsensus,butsta-
0 of these models can be found in [11]. Building up upon ble polarized states can be reached introducing a simple
:
v a proposal by Minett and Wang [12, 13], the AB-model confidence/trust parameter [22]. In this paper, we shall
Xi is a model of twonon-excluding options,in which agents focus on the particular case in which only two options
can be in two symmetric states (A or B) and in a third compete within the population [17, 22].
r
mixed (AB) state of coexisting options at the individual
a
level [14]. It has been used to study the competition be-
tweentwosociallyequivalentlanguages,whereABagents
areassociatedtobilingualism. Themodelhasbeenstud-
This paper is structured as follows: in Section II we
iedintwo-dimensionalandsmallworldnetworks[14]and present the microscopic description of the two models
in networks with community structure [15, 16]. The fi-
studied in this paper, the Naming Game restricted to
nal state of the system is always consensus in one of the two conventions and the AB-model. In Section III we
options, A or B.
look at the macroscopic description of the models, while
In semiotic dynamics, the Naming Game [5, 6, 17] de- in Sections IV and V we explore in detail the differences
scribes a population of agents playing pairwise interac- betweenthe twomodels arisingfromthe differentmicro-
tions in order to negotiate conventions, i.e., associations scopicinteractionrules. Finally,wepresentinSectionVI
between forms and meanings, and elucidates the mech- the conclusions as well as a discussion about the impli-
anisms leading to the emergence of a global consensus cations for language competition of the results obtained
among them. For the sake of simplicity the model does in this paper.
2
II. THE MODELS both, A and B (using both languages, bilingual agent).
An agent changes its state with a probability which
We presenthere the two models consideredin this pa- depends on the fraction of agents in the other states,
per: thegeneralizedNamingGamerestrictedtotwocon- nj (j=A, B, AB). The transition probabilities are the
ventions[6,22] andthe AB-model[14], extendedin such following[24]:
a way that confidence/trust is considered. In both mod-
1 1
els, we consider a set of N interacting agents embedded pA→AB = nB, pB→AB = nA (3)
2 2
in a network. At each time step, and starting from a
1 1
given initial condition, we select randomly an agent and pAB→A = β(1 nB), pAB→B = β(1 nA) (4)
2 − 2 −
weupdateitsstateaccordingtothedynamicalrulescor-
responding to each model. An agent changes from the A or B state towards the
Inthe NamingGame[6,17],anagentis endowedwith AB state (equation (3)), with a probability proportional
an internal inventory in which it can store an a priori to the agents in the opposite option. The probabil-
unlimited number of conventions. Initially, all invento- ity that an AB agent moves towards the A or B state
ries are empty. At each time step, a pair of neighboring (equation (4)) is proportional to the density of agents
agentsis chosenrandomly,one playingas “speaker”,the sharing that option, including those in the AB state
other as “hearer”,and negotiateaccordingto the follow- (1 n = n +n ; i,j = A,B, i = j). In this paper,
i j AB
− 6
ing rules: weextendthe originalmodelinanalogyto the extension
proposed for the Naming Game in [22]: an agent aban-
the speaker selects randomly one of its conven- donsanoptionorlanguageaccordingto the dynamicsof
•
tions and conveys it to the hearer (if the inven- theAB-model(changesfromABtoAorB)withaprob-
tory is empty, a new convention is invented by the abilityβ,whilewithaprobability1 β nothinghappens.
speaker); In the context oflanguage competit−ion, the parameter β
canbeinterpretedasaninertiatostopusingalanguage,
ifthehearer’sinventorycontainssuchaconvention,
• andatthe same time, as areinforcementofthe status of
the two agents update their inventories so as to
being bilingual, which was not taken into account in the
keeponlytheconventioninvolvedintheinteraction
original model (recoveredby setting β =1).
(success);
In both models, a unit of time is defined as N it-
otherwise, the hearer adds the convention to those erations, so that at every unit of time each agent has
• already stored in its inventory (failure). been updated on average once. To describe the dynam-
icsofthesystemweuseasanorderparameterthe inter-
Here we are interested in the particular case in which face density ρ,definedasthe fractionoflinks connecting
a population deals with only two competing conventions nodes in different states. When the system approaches
(say A or B) [17]. We therefore assignto each agentone consensus, domains grow in size, and the interface den-
of the two conventions at the beginning of the process, sity decreases. Zero interface density indicates that an
preventinginthiswayfurtherinvention(thatcanhappen absorbing state, consensus, has been reached. We also
only when the speaker’s inventory is empty). Moreover, use the averageinterface density, ρ , where indicates
h i h·i
we adopt the generalized Naming Game scheme [22], in averageoverrealizationsofthestochasticdynamicsstart-
whichaconfidence/trustparameterβ determinestheup- ing from different random initial conditions.
date rule following a success: with probability β the
usual dynamics takes place, while with the complemen-
tary probability 1 β nothing happens. The usual case III. MACROSCOPIC DESCRIPTION
−
is thus recoveredfor β =1. For brevity we shall refer to
this setting (generalizedNaming Game restricted to two In Section II we have presented the microscopic de-
conventions) as the 2c-Naming Game. In this simplified scriptionofthe2c-NamingGameandtheAB-model,i.e.,
case,itis easyto seethatthe transitionprobabilitiesare the set of local interactions among the agents. In order
the following [22]: to have a macroscopic description of the dynamical evo-
lution of the system as a whole we derive the mean-field
1 1
pA→AB =nB+ nAB, pB→AB =nA+ nAB (1) equationsforthefractionofagentsineachstate. Forthe
2 2
2c-Naming Game one has [22]:
3β 3β
pAB→A = 2 nA+βnAB, pAB→B = 2 nB+βnAB(2) dnA = n n +βn2 + 3β−1n n (5)
dt − A B AB 2 A AB
where n (j=A, B,AB) arethe fractionofagentsstoring
j dn 3β 1
in their inventory the conventions A, B or both A and B = n n +βn2 + − n n (6)
dt − A B AB 2 B AB
B, respectively.
Forthe AB-model[14], anagentcanbe inthreepossi- and n =1 n n .
AB A B
− −
ble states: A, choosing option A (using language A), B, The stability analysis showed that there exist three
choosing option B (using language B) or AB, choosing fixed points [22]: (1) n = 1,n = 0,n = 0; (2)
A B AB
3
n = 0,n = 1,n = 0 and (3) n = b(β),n =
A B AB A B
b(β),n = 1 2b(β) with b(β) = 1+5β−√1+10β+17β2 100
AB − 4β
(andb(0)=0). Anon-equilibriumphasetransitionoccurs
for a critical value β = 1/3. For β > 1/3 consensus is
c c
stable. For βc < 1/3 a change of stability gives place to 10-1
a stationary coexistence of n =n and a finite density
A B
ofundecidedagentsnAB,fluctuatingaroundtheaverage <ρ>
values b(β) and 1 2b(β). In the Naming Game with
invention,infact,th−eoneobservedatβc =1/3isthefirst 10-2
of a series of transitions yielding the asymptotic survival
of a diverging (in the thermodynamic limit) number of
conventions as β 0 [22].
For the AB-mo→del one has: 10-1300 101 102 103 104 105
time
dn 1
A = ( n n +βn2 +βn n ) (7)
dt 2 − A B AB A AB FIG. 1: AB-model: time evolution of the average inter-
dn 1 face density, hρi, in a fully connected network of N =
dtB = 2(−nAnB+βn2AB +βnBnAB) (8) 10000 agents for different values of β. From left to right:
β = 1.0,0.2,0.05,0.01,0.002,0.0005,0.0001,0.0. Averaged
and n =1 n n . over1000 runs.
AB A B
− −
Thestabilityanalysisshowsthatthereexistthreefixed
points: (1) n = 1,n = 0,n = 0; (2) n = 0,n =
A B AB A B
1,n = 0 and (3) n = f(β),n = f(β),n = 1
AB A B AB
−
3β−√β(β+4) 104
2f(β) with f(β)= .
2(2β−1)
Notice that in both models, at β = 0 the third fixed tconv103
point becomes a stable absorbingstate in which the sys- 102
tem reaches consensus in the AB-state.
Surprisingly,thetwooriginalmodels(β =1)areequiv- 10110-3 10-2 10-1 100
β
alent in the mean-field approximation. There is just a 70
different time scale coming from the prefactor 1/2in the 60
AprBox-mimoadteilon(eqisuaetxiaocntsi(n7)thaendth(e8r)m).odTyhneammiecanli-mfieitl,daanpd- tconv50
40
validforlargesystemsinfullyconnectednetworks. How-
30
ever,thetwomodelsdifferattheirlocalinteractions(see 102 103 104
N
equations(1-4)forβ =1). Toexploretheeffectsofthese
differences at the microscopic level, in Section IV we in-
FIG. 2: AB-model. Top: scaling of the time of convergence
vestigate,in a complete graph,the role of the parameter with β for N = 10000: tconv ∼ β−1. Averaged over 200-800
β as described in equations (5-8), while in Section V we runsdependingonthevalueofβ. Bottom: scalingofthetime
focus on the interface motion in regular lattices for the of convergence with system size N for β =1: tconv ∼ln(N).
original case β =1. Averaged over 10000 runs.
IV. PHASE TRANSITION for a small fraction of agents (less than 1% on average
for N = 10000). This fraction decreases as N increases,
Here we consider the extension of the AB-model pre- andcompleteconsensusintheAB-stateisreachedinthe
sented above in a fully connected network, with the aim thermodynamic limit (see stability analysis in the pre-
to explore a possible non-equilibrium phase transitionin vious section). For β = 0, pAB→A = pAB→B = 0, so
β similar to the one found in the 2c-Naming Game. In the only possible evolution is that A and B agents move
Figure1weshowthetimeevolutionoftheaverageinter- towards the AB state. At the last stage of the dynam-
face density, ρ , for different values of the parameter β. ics, when nA and nB approachto zero, as soonas one of
h i
For large values of β, ρ reachesa plateau followedby a the two single-option densities, ni, vanishes, nj remains
h i
finitesizefluctuationthatdrivesthesystemtoanabsorb- constant (i,j =A,B,i=j) because pj→AB ni, giving
ingstate. However,forβ <0.01weobservethatafter ρ rise to the small fractio6n of agents in the sta∼te j present
reaches the plateau, it inc∼reases again, reaching a mahxii- in the frozen state. The time to convergence scales with
mumvalueafterwhichafinitesizefluctuationdrivesthe beta as tconv β−1 (Figure 2-top), as observed for the
systemtoconsensus. Infinitesystemsandforβ =0,the 2c-Naming Ga∼me for β >βc (tconv (β βc)−1) [22].
∼ −
systemreachesaconstantvalueof ρ ,afrozenstatecor- Contrary to the phase transition described in Sec-
h i
responding to almost consensus in the AB-state, except tionIIIobtainedinthe2c-NamingGame[22],thereisno
4
phase transition in the AB-model: at β = 0, the system
reaches trivially a frozen state (dominance of the AB- 100
n
state, with complete consensus in the thermodynamic AB
limit); while for β > 0 the final absorbing state is, as ρ
usual, consensus in the A or B option. Even though
the two original models are equivalent in the mean-field 10-1 nB
ahpavpiroorxsimwhateinonth(ecapsaeraβm=et1e)r,βwiesotabkseernvienttwooacdcioffuenret.ntTbheis- densities nA
can be shown formally by looking at the time evolution
of the magnetization, m n n . For the 2c-Naming 10-2
A B
≡ −
Game and the AB-model, we have respectively:
dm 3β 1
dt = 2− nABm (9) 10-1300 101 102 103
time
dm 1
= βn m (10)
AB
dt 2 FIG. 3: AB-model. Density of agents in state A, nA (dot-
In equation (9), we can observe the origin of the non- ted gray), in state B, nB (dotted black), in state AB, nAB
(dashed);andinterface density,ρ(solid line) fora typicalre-
equilibrium phase transition described in Section III for
alization of the AB-model. In the plateau, nA ∼ nB ≃ 0.1,
the2c-NamingGame. Thetimederivativeofthemagne-
while the majority of agents are in the AB-state. Fully con-
tization, dm, vanishes at the criticalpoint β =1/3. For
dt c nected network, β=0.01 and N =10000 agents.
β > 1/3, sign(dm) = sign(m), and therefore m 1:
c dt | | →
thesystemisdriventoanabsorbingstateofconsensusin
the A or B option. For β <1/3, sign(dm)= sign(m) realization of the dynamics and a given small β. Be-
c dt −
and m 0, giving rise to stationary coexistence of cause of the inertia of the AB agents to move away from
| | →
the three phases, with n = n and a finite density of their state (small β), at the beginning we observe an in-
A B
AB agents. For the AB-model, instead, we can see in crease of n together with the corresponding decrease
AB
equation (10) that for β > 0, sign(dm) = sign(m) so ofn , n andρ. Then, ρandthe threedensities reacha
dt A B
that consensus in the A or B option is always reached. plateau. Most of the agents are in the AB state, while a
The time derivative of the magnetization, dm, vanishes competition between options A and B takes place, with
dt
at β =0, where the dynamics gets stuck in anabsorbing n n < n . This metastable state lasts longer as
A B AB
≃
state corresponding to consensus in the AB state in the weincreasethe systemsize. Atacertainpoint,however,
thermodynamic limit. Therefore, the phase transition a system size fluctuation drives the density of one of the
observed in the 2c-Naming Game is not observed. two states (A in the figure) to zero, while the other (B
The reason of the difference shown above has to be in the figure) starts gaining ground. Since there are less
found in the differences that these models have at the andlessagentsinthestatebecomingextinct,andagents
microscopic level. The fact that in the AB-model A and having one option do feel only the presence of agents in
BagentsdonotfeeltheinfluenceofABagentsisthekey the opposite state, agents in the dominant state become
pointwhichexplainsthedifferentnatureofthetransition more and more stable, until, when the other state disap-
for this model. For the case β = 1, in the 2c-Naming pears, they become totally stable. During this process,
Game the second term in equation (1) (influence of AB the interface density increases as n decreases. The
AB
agents on the A or B agents) and the first term in equa- peak of ρ corresponds to the point where n =n (be-
AB i
tion(2)(influenceoftheAorBagentsontheABagents) ing i the state which takes over the whole system, B in
combine in such a way that the mean-field equations are the figure). When one of the states has vanished (A in
equivalent to the ones in the AB-model. However, when the figure), the AB agents slowly move towards the re-
β <1thecombinationofthesetermsoriginatethephase maining state (B in the figure) and the system reaches
transition from an absorbing final state towards a dy- consensus.
namical stationary state of coexistence, as found in [22].
In Figure 2-bottom, we observe for the AB-model
the scaling of t with the system size for β = 1: V. INTERFACE DYNAMICS: 1-D AND 2-D
conv
t ln(N),indicatingthatt increasesslowlywith LATTICES
conv conv
∼
the system size. As expected, this compares properly
to the same scaling already obtained for the 2c-Naming We study and compare here the interface dynamics
Game in [17], since the two models are equivalent in the in regular lattices with periodic boundary conditions for
mean field for β =1. the two original models (β = 1). The 2c-Naming Game
To understand the time evolution of the average in- has been shown to exhibit a diffusive interface motion
terface density ρ shown in Figure 1 for β < 0.01, we in a one-dimensional lattice, with a diffusion coefficient
show in Figure h3 ithe time evolution of ρ and∼the densi- D = 401/1816 0.221 [18]. We therefore focus on the
≃
tiesofagentsineachstate,n ,n andn ,foratypical AB-model, and, to analyze the interface dynamics in a
A B AB
5
one-dimensionallatticewithN agents,weconsiderasin-
gle interface between two linear clusters of agents. In 0.2
each of the clusters, all the agents are in the same state.
0.15 t=10 t=102 t=103
We consider a cluster of agents in the state A on the- x,t) 0.1
left and another cluster of agents in the B state on the P(
right. We call C an interface of m agents in state C 0.05
m
(for clarity, here C labels an AB agent). Due to the dy- 0
-40-20 0 20 40 -40-20 0 20 40 -40-20 0 20 40
namical rules, the only two possible interface widths are x x x
200
C , corresponding to a two directly connected clusters
0 150
w··idAtAhAoBneB,B ·A·,AoAr CC1B,BcoBrres.poIntdiinsgsttroaiagnhtifnotrewrfaarcdetoof 2x>(t)100
· · · · <
compute the probabilityp =1/2N that a C interface 50
0,1 0
becomes a C1 in a single time step. Otherwise, it stays 00 200 400 600 800 1000
in C0. In the same way, p1,0 =1/2N. We are now inter- t
ested in determining the stationary probabilities of the
Markov chain defined by the transition matrix FIG. 4: AB-model: evolution of the position of an interface
in a one-dimensional regular lattice. Top: time evolution of
thedistribution P(x,t). Bottom: time evolution of the mean
1 1 1 square displacement hx2(t)i = 2Dexpt. The value Dexp =
M =(cid:18) −12N 1 2N1 (cid:19) (11) 0.06205obtainedfromthefittingisinperfectagreementwith
2N − 2N thetheoretical prediction D=1/16=0.0625.
inwhichthebasisis C ,C . Thestationaryprobability
0 1
vector, P = P ,P{ is co}mputed by imposing P(t +
0 1
1) P(t) = 0{, i.e., (}MT I)P = 0. We obtain P =
0
− −
1/2,P =1/2. Since the interface has a bounded width,
1
we assume that it can be modeled as a point-like object
localized at position x = (x + x )/2, where x is the
l r l
position of the rightmost site of cluster A, and x the
r
leftmost site of cluster B. An interaction Cm → Cm′
correspondstoasetofpossiblemovementsforthecentral
position x. We denote by W(x x δ) the transition
→ ±
probabilitythataninterfacecenteredinxmovestotothe
positionx δ. Theonlypossibletransitionsare: W(x
x 1)= 1±P + 1 P . Usingtheresultsobtainedfort→he
±2 4N 0 4N 1
stationaryprobabilityvectorwegetW(x x 1)= 1 .
→ ±2 4N
We are now able to write the master equation for the
probability P(x,t) to find the interface in position x at
time t. In the limit of continuous time and space:
∂P(x,t)
P(x,t+1) P(x,t) δt , (12)
− ≈ ∂t FIG.5: InitialconditionswithonehalfofthelatticeintheA
2
∂P(x,t) state, and the other half in the B state. N = 64 Top: AB-
P(x+δx,t) P(x,t)+δx + (13) model. Stripe-like metastable state. t=0, 100, 200 from left
≈ ∂x
toright. Bottom: 2c-NamingGame: t=0,50,100fromleftto
1 ∂2P(x,t)
+ (δx)2 right. Snapshotsareselectedtakingintoaccountthedifferent
2 ∂x2 time scale coming from the prefactor 1/2 in the meanfield
equations (7) and (8).
In this limit, the master equation reads:
∂P(x,t) D∂2P(x,t)
= (14) Thus,theAB-modelandthe2c-NamingGamedisplay
∂t N ∂x2
the same diffusive interface motion in one-dimensional
where the diffusion coefficient is D = 1/16 = 0.0625 (in lattices, but they differ in about one order of magnitude
the appropriate dimensional units (δx)2/δt). These an- in the diffusion coefficient, indicating that in the AB-
alytical results are confirmed by numerical simulations. model interfaces diffuse muchslower. It can also be seen
In Figure 4 we show the time evolution of P(x,t), which that the growth of the typical size of the clusters is ζ
displays a clear diffusive behavior. The mean-square tα, with α 0.5, leading to the well known coarsenin∼g
≃
distance follows a diffusion law x2 = 2D t, with process found also in SFKI models [23] (not shown).
exp
h i
D = 0.06205 being the diffusion coefficient obtained In two-dimensional lattices, on the other hand, it
exp
numerically. has been shown that starting from random initially dis-
6
VI. DISCUSSION AND CONCLUSION
0.1
Wehaveanalyzedandcomparedthe2c-NamingGame
2c-Naming Game
AB-model and the AB-model, originally defined in the context of
language emergence and competition, respectively. We
0.01 have shown that although these two models are equiva-
p(t) lent in mean-field, their microscopic differences give rise
to different behaviors. In particular, we have focused on
(1)theextensionofthemodelsbyintroducingtheparam-
0.001
eterβ,describingtheinertiaoftheagentstoabandonan
acquired option, and (2) the interface dynamics in one
and two-dimensional lattices.
0.0001
0 1×105 2×105 As for the extension of the models incorporating the
t parameter β, even though the originalmodels are equiv-
alent in the mean field approximation for β = 1, an
FIG. 6: Probability distribution for the time to reach con-
important difference concerns the existence of a phase
sensus, starting with stripe-like configurations. Black: 2c-
Naming Game, τNG ≃ 1.2×104; gray: AB-model, τAB ≃ transition. While the 2c-Naming Game features a non-
6.3×104. Averages are over 5000 runs. equilibrium phase transition between consensus and sta-
tionary coexistence of the three phases present in the
system, in the AB-model such a transition does not ex-
ist, the model featuring a trivial frozen state for β = 0
tributed options among the agents, both models present (dominance of the AB-state, with complete consensus in
a coarsening ζ tα, α 0.5, with a curvature driven the thermodynamic limit), and the usual consensus in
∼ ≃
interfacedynamics [14, 18]andAB-agentsplacingthem- the A or B state, for 0<β 1.
selvesattheinterfacesbetweensingle-optiondomains. In As for the interface dyna≤mics, we have shown in one-
Figure 5 we show snapshots comparing the two dynam- dimensionallatticesthattheAB-modelhasadiffusivein-
ics, starting from initial conditions where we have half terfacemotionanalogousto the onealreadyfoundinthe
of the lattice in state A, and the other half in state B. 2c-Naming Game, but with a diffusion coefficient nearly
Given that the interface dynamics is curvature driven, one order of magnitude smaller. In two-dimensional lat-
flat boundaries are very stable. In both models these tices, we have studied the time evolution of stripe-like
stripe-like configurations give rise to metastable states, configurations,which are metastable in both models but
already found in [14] for the AB-model: dynamical evo- have a larger life time in the AB-model. Both results
lution of boundaries close to flat interfaces but with in- indicatethatincomparisontothe2c-NamingGame,the
terfacial noise present. These configurations evolve by AB-modelinterfacedynamicsslowsdownthediffusionof
diffusion of the two walls (average interface density fluc- these configurations.
tuating around a fixed value) until they meet and the
It is interesting to discuss the implications of our re-
systemisdriventoanabsorbingstate. IntheAB-model,
sultsontheAB-modelinthe contextofdynamicsoflan-
also when starting from options randomly distributed
guage competition. In the original AB-model (β = 1),
through the lattice, 1/3 of the realizations end up in
the density of bilingual individuals remains small during
such stripe-like metastable states [14].We checked that
the language competition process (around 20%), and in
the same turns out to be true also for the 2c-Naming
theendbilingualindividualsdisappeartogetherwiththe
Game. In the usual Naming Game with invention, on
language facing extinction. When introducing the pa-
the other hand, stripes are better avoided since in that
rameter β, interpreted here as a sort of inertia to stop
casethetwoconventionstateisusuallyreachedwhenone
using a language, and, at the same time, as a reinforce-
cluster is already considerably larger than the other.
mentofthestatusofbeingbilingual,weobservedthefol-
We show in Figure 6 the distribution of survivaltimes lowing. Whenβ issmallenough,bilingualagentsrapidly
for the two models, i.e., the time needed for a stripe-like becomethemajority,whilethetwomonolingualcommu-
configuration to reach an absorbing state. The distri- nitiescompetebetweeneachotherandhavesimilarsizes.
bution displays an exponential tail, p(t) e−t/τ with a The smaller the parameter β, the larger the bilingual
∼
characteristictimeτ. ThecharacteristictimefortheAB- community is at this point. After this stage of coexis-
model is however larger than the one for the 2c-Naming tence, a symmetry breaking takes place and one of the
Game(τ >τ ),confirmingthattheAB-modelinter- two monolingual communities starts to grow, while the
AB NG
face dynamics slows down the diffusion of configurations other looses ground. When this monolingual community
suchasstripesintwodimensionallattices,orwallsinone faces extinction, the languagespokenin that community
dimensional lattices. Notice that in both cases, the dif- survivesinthebilingualcommunityuntilthiscommunity
ferences found are beyond the trivial different time scale also disappears.
correspondingtotheprefactor1/2inthemeanfieldequa- Inother words,contraryto the originalmodel, the ex-
tions for the AB-model (equations (7) and (8)). tinction of a language takes place in two steps. At first,
7
the agents who speak just that language disappear, but to understand the implications of different complex so-
the language does not, as it is still spoken in the society cial networks (small world effect, community structure,
by the bilingual agents. Then bilingual agents disap- etc) for the extension of the model presented here, it is
pear, too, which leads to the extinction of the language. worthinvestigatingin future this extensionin topologies
WithinthelimitedframeworkoftheAB-model,inwhich of increasing complexity.
there does not exist any political measure enhancing the
prestige of an endangered language, these results come
tosupporttheideathat,insocietieswithtwolanguages,
the disappearance of a monolingual community using a
Acknowledgments
language as its only way of communication could rep-
resent the first step in the extinction of that language.
The other language could indeed become eventually the X.Castell´oacknowledgesfinancialsupportfromaphD
onlyspokenone,asthebilingualagentswouldeventually fellowship of the Govern de les Illes Balears (Spain). A.
end up using only the languagespoken by the remaining BaronchelliacknowledgessupportfromtheDURSI,Gen-
monolingual community. eralitat de Catalunya (Spain) and from Spanish MEC
The dynamics of the original Naming Game as well (FEDER) through project No: FIS2007-66485-C02-01.
as that of the AB-model are strongly affected by the This research has been party supported by the TAGora
underlying interaction network, as it has been shown projectfundedbytheFutureandEmergingTechnologies
in [14, 15, 16, 18, 19, 20, 21, 22] (times to consensus, program(IST-FET) of the European Commissionunder
apparition of trapped metastable states, etc). In order the European Union RD contract IST-034721.
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