Evolution toward linguistic coherence in naming game with migrating agents
AArticle
Evolution toward linguistic coherence in naming gamewith migrating agents
Dorota Lipowska ∗ and Adam Lipowski Faculty of Modern Languages and Literature, Adam Mickiewicz University in Pozna ´n, Poland;[email protected] Faculty of Physics, Adam Mickiewicz University in Pozna ´n, Poland; [email protected] * Correspondence: [email protected];† These authors contributed equally to this work.Version February 26, 2021 submitted to Entropy
Abstract:
As an integral part of our culture and way of life, language is intricately related to migrationsof people. To understand whether and how migration shapes language formation processes we examinethe dynamics of the naming game with migrating agents. (i) When all agents may migrate, the dynamicsgenerates an effective surface tension, which drives the coarsening. Such a behaviour is very robustand appears for a wide range of densities of agents and their migration rates. (ii) However, when onlymultilingual agents are allowed to migrate, monolingual islands are typically formed. In such a case,when the migration rate is sufficiently large, the majority of agents acquire a common language, whichspontaneously emerges with no indication of the surface-tension driven coarsening. A relatively slowcoarsening that takes place in a dense static population is very fragile, and most likely, an arbitrarilysmall migration rate can divert the system toward quick formation of monolingual islands. Our workshows that migration influences language formation processes but additional details like density, ormobility of agents are needed to specify more precisely this influence.
Keywords: multi-agent modeling, migration, Naming Game, language formationAlthough there is a multitude of factors that shape our language, including culture, politics, economy,geography or technology, the most important one is the mutual interactions between multiple languageusers. It is thus tempting to examine language formation and its evolution using multi-agent models andstatistical mechanics methodology [1–4]. A valuable insight into language emergence [5], death [6], itsdiversification [7], importance [8], or appearance of grammar or linguistic categories [9] proves that suchan approach is indeed promising.An important factor, which affects various aspects of our life, is migration. This process may mixas well as separate human communities and language formation processes should be thus stronglyinfluenced by it [10,11]. Moreover, some modern trends, related mainly with globalization, most likelyincrease people’s migrations [12]. It would be desirable to examine models that take into account bothlinguistic interactions and migration, and thus gain some understanding of how related are these twoprocesses. A possible candidate is a suitably extended naming game, which proved to be useful in thestudies of various aspects of the emergence of linguistic coherence [13,14]. One of the questions, whichmay be addressed, is how coarsening, which in the naming game is known to be basically similar to theIsing model, is affected by migration. Such a similarity is related to the fact that the dynamics of bothmodels are driven by a surface tension [15], even though in the naming game it is rather an effective surfacetension [16]. Let us notice that recent studies indicate that the dynamics of English dialects evolution is
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Entropy a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ersion February 26, 2021 submitted to Entropy also driven by a certain effective surface tension [17,18]. It would be desirable to check to what extent thesurface-tension driven dynamics of the naming game is robust with respect to migration of agents.An interesting multi-agent model, where migration plays an important role, was proposed sometime ago by Schelling [19]. In his model, an agent is relocated if the number of its neighbours with thesame orientation (an opinion or race) as the agent is below a certain threshold. Numerous versions ofthe Schelling model show that the phase separation (formation of a ghetto) appears to be a very robustfeature of the dynamics [20]. Let us notice that in the Schelling model, only those agents for which certainconditions are satisfied may migrate. Linguistic factors as, for example, acquiring a new language, mightalso influence our ability or willingness to relocate.In the present paper, we examine the naming game with migration and address some of the aboveissues. In particular, we examine whether an effective surface tension persists in such systems. We alsoexamine the implications of a state-dependent migration but, of course, we are aware that its applicabilityto real social systems is limited. Our simulations show that the state-dependent diffusion usually leads tospatial segregation but when sufficiently strong, it can trigger formation of the dominant language in theentire system.
1. Model
In our model, we consider a population of agents placed on a square lattice of linear size N (withperiodic boundary conditions). Each agent has its own inventory, which is a dynamically modified listof words. Initially agents are uniformly distributed on the lattice with the density/probability ρ (doubleoccupancy excluded). The dynamics of our model combines the lattice gas random migration with theso-called minimal version of the naming game [21,22].More specifically, in an elementary step, an agent and one of its (four) neighbouring sites are randomlyselected. With probability d , the agent migrates to the selected neighbouring site provided that the chosensite is empty and some additional conditions (dependent on the number of words in agent’s inventory) aremet. With probability 1 − d and provided that the selected neighbouring site is occupied, the chosen agentbecomes the Speaker, its neighbour becomes the Hearer, and they play the naming game:• Speaker selects a word randomly from its inventory (or invents a new word if its inventory is empty)and transmits it to Hearer. To invent a word all agents have at their disposal M different words andone of them is selected randomly.• If Hearer has the transmitted word in its inventory, the interaction is a success and both playersmaintain only the transmitted word in their inventories.• If Hearer does not have the transmitted word in its inventory, the interaction is a failure and Hearerupdates its inventory by adding this word to it.The unit of time ( t =
1) is defined as ρ N elementary steps, which corresponds to a single (onaverage) update of each agent. Agents may have in their in ventories at most M ≥ M = M > M absorbing states is driven by the effective surface tension [1,23] andanalogous similarities are between the Ising and Potts models [24]. In the following, we will refer to wordscommunicated by agents also as languages.Recently, we have already analysed a naming game model with migration, in which, however, therelocation depended on the language used by an agent [22]. In this model, all agents were allowed tomigrate (albeit with a language-dependent rate) and the main objective of this study was to demonstrate acertain symmetry breaking induced by the difference in mobility. In the present paper, the state-dependentmobility depends on the number of languages known by an agent and not on the particular languageused by the agent. Some other systems with migrating agents but with different ordering dynamics (the ersion February 26, 2021 submitted to Entropy voter model) were also analysed [25]. The emergence of consensus in the population was also examined inthe case of agents moving in a continuous space [26] as well as in some robotic swarms [27]. It would becertainly interesting to replace the local migration of our agents with a possiblity of longer distance steps,as for example in some dynamical models defined on spatial networks [28].Let us also emphasize that migration often refers to the mass movement of people in a certain directionor location, as e.g., in China during the Qing dynasty [29]. To model such phenomena, a random walk ofour agents would have to be considerably modified.
2. Results
Naming game typically evolves toward a linguistic consensus state where all agents have only a singleword in their repositories and thus every communication attempt results in a success. Before reachingsuch a state, monolingual domains are formed and their coarsening, driven by an effective surface tension,leads eventually to the emergence of a linguistic consensus. The surface tension is known to drive thedynamics of many other models as, e.g., the Ising or Potts models [15], and its absence as, e.g., in the votermodel [30], results in much different dynamics. In surface-tension driven dynamics, the correlation lengthcan be related to the total length of domains’ boundaries, which can be easily extracted from the modelconfiguration. In the naming game, a competition between languages, which takes place at domains’boundaries, implies that such interfacial agents are typically bi-(or more)lingual. Their concentration canbe easily measured numerically and, being related to the total length of domains’ boundaries, it determinesthe correlation length in the system.
The case of our main concern is such that only agents with two or more words in their inventorieshave the ability to migrate. However, first we report some results for the case, where all agents are able tomigrate. We made simulations for several values of ρ and d , and we measured the fraction x of agentswith two or more words in their inventories. The results of our simulations are presented in Fig. 1. Most ofour simulations were made for M = M = M = ρ =
1, when the lattice is fully occupied by agents and thus they have no space to migrate, x showsa power law decay x ∼ t − α . From our data we estimate α ≈ ( ) , which agrees with some previousstudies on the naming game [31,32] or related models of opinion formation [23,33]. For ρ < d > x of multilingual agents seems to exhibit nearly the sameasymptotic decay. This is even the case when the lattice is sparsely covered with agents ( ρ = d = t − (and that easily translates into ∼ t increase ofthe correlation length). The decay of x observed in the naming game is very similar and most likely it isrelated to a certain effective surface tension generated in this kind of models [16].Our simulations show that the surface-tension driven dynamics in the naming game is very robustwith respect to the concentration of agents and migration rate. There are some reasons to believe thatsome other factors will not change qualitative features of the dynamics of such systems either. Indeed, wehave recently shown that the surface-tension driven dynamics is restored in the voter model (which isknown to have much different dynamics and no surface tension [30]) with only a small fraction of sitesevolving according to the Ising heat-bath dynamics [34]. It is thus plausible that even in heterogeneoussystems, where some of our naming-game agents would be replaced with agents with dynamics similar tothe voter model, the system would still exhibit surface-tension characteristics. Provided that the naminggame mimics to some extent real linguistic interactions, such a strong robustness could suggest that a ersion February 26, 2021 submitted to Entropy -3-2.5-2-1.5-1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 l og ( x ) log (t) ρ =1.0, M=2 ρ =0.3, d=0.5, M=2 ρ =0.1, d=0.95, M=2 ρ =0.01, d=0.99, M=2 ρ =0.01, d=0.99, M=3 ρ =0.01, d=0.99, M=5-4-3-2-1 0 0 1 2 3 4 5N=500N=1000N=2000N=5000 Figure 1.
The time dependence of the fraction x of bilingual agents for each set of parameters averaged over100 independent runs. For ρ ≥ N = ρ = N = ρ = d = M = N ≥ x is seen, which we attribute to the formation of stripe-like structures [32]. For increasing N ,the influence of such stripes on the fraction x seems to diminish and a power-law decay sets in. surface-tension driven dynamics could operate in (real) processes responsible for the evolution of naturallanguages, which certainly complies with some recent analysis indicating that an effective surface tensionseems to shape patterns of dialect changes [17,18]. Agents with two or more words in their inventories, whose fraction is denoted by x , are typicallylocated at the interface of monolingual domains and thus may be considered as multilingual. Since theability (or willingness, or need) to migrate is not necessarily homogenous in the population, we wouldlike to examine the case when only such multilingual agents may migrate.Such a state-dependent migration resembles the Schelling model of ghetto formation, where an agentis relocated if the number of its neighbours of the same (as the agent) orientation is too small [19,20]. Letus notice that in the Schelling model the orientation of an agent is fixed during the evolution of the model,which is not the case in our model. In such an analogy, our model could be considered as driven by anonconservative dynamics. Simulations of such a model reveal that both the density ρ and migration rate d influence the dynamics of the model and its final state. We made simulations in the low- ( ρ = ersion February 26, 2021 submitted to Entropy d=0.8 d=0.95
Figure 2.
The spatial distribution of agents and languages they use after t = for d = t = for d = N =
200 and ρ = high-density ( ρ = ρ = ρ = M = d . Calculation of the average sizeof such clusters S suggests that S may diverge at the value d = d c close to but smaller than d = d c < d <
1. In particular, for d = x drops to 0 but at a time scale that shows a pronouncedsize dependence (Fig. 4). For d = d = d = d c is very close to 1. Although the size ofthe surviving languages (Fig. 3) for both N =
200 and 300 seems to diverge at nearly the same value d = d c = ( ) , we cannot exclude that this is actually the finite size effect and, in the thermodynamiclimit, d c =
1. However, the true thermodynamic limit is perhaps not that important in the linguisitccontexts and the dynamics that we observe might be relevant and interesting also as a finite size effect. ersion February 26, 2021 submitted to
Entropy S d N=200N=300 1.5 2 2.5 3 3.5 4 -2.4 -2 -1.6 -1.2 -0.8 l og ( S ) log (d c -d) Figure 3.
The average size of the surviving languages as a function of the mobility d . The inset presentsthe same data on a logarithmic scale. The least-square fit shows that numerical data follow a power-lawdivergence S ∼ ( d c − d ) − γ with d = d c = ( ) and γ = ( ) (dashed line). ersion February 26, 2021 submitted to Entropy
0 50000 100000 150000 200000 x td=0.999, N=200d=0.999, N=300d=0.999, N=500 0 0.2 0.4 0.6 0.8 110 d=0.8d=0.95 d=0.999 Figure 4.
The time dependence of the fraction x of mobile agents calculated for d = ρ = N =
200 (continuous lines), 300 (dashed),and 500 (dotted). The presented results are averages over 30 independent runs. For d = x takes place on a short time scale and is nearly size independent. The inset shows the same databut with a logarithmic time axis. ersion February 26, 2021 submitted to Entropy t=5000 t=60000t=85000 t=150000
Figure 5.
Snapshots of spatial distribution of agents and languages they use for d = ρ = N = t = ersion February 26, 2021 submitted to Entropy ρ = ρ = ρ = d results in the increase ofthe size of islands, contrary to the ρ = d . However, configurations presented in Fig. 6 are obtained for small values of migration rate d . Forlarger d (close to d = ρ = ρ = N = M = x (Fig. 7). In the absence ofmigration ( d = x but some bending of our data suggests thatthe estimated decay ∼ t − may not be truly asymptotic. A slower decay, perhaps logarithmically slow,would be actually consistent with the coarsening of the Ising model on diluted lattices [35]. Leaving asidethe correct asymptotic form of the decay of x , we would like to point out that the d = ρ = ρ c ≈ ρ < ρ c ), our agents are located in finite clusters, which rather quicklybecome monolingual due to the naming game.For d >
0, we observe that x shows a rapid (probably faster than the power-law) decay (Fig. 7).We associate such a decay with the formation of monolingual islands (Fig. 6). What is in our opinionsurprising is that even a very small migration rate d is sufficient to bring the system to such a multi-islandconfiguration. Indeed, even for d = − the data seem to veer off the d = d decreasing,this deviation takes place at an increasing time scale, which is related to the formation of islands of anincreasing size.The overall behaviour of our model for the state-dependent migration is presented in the phasediagram in Fig.8.
3. Conclusions
In summary, we examined how an ordering dynamics of the naming game is affected by migration. Inthe version where all agents are allowed to migrate, the coarsening of our model indicates the presence ofan effective surface-tension. Such a behaviour is very robust with respect to the concentration of agents ortheir migration rate. Recently, we have shown that an effective surface tension appears in a heterogenousvoter model with a small fraction of agents operating with the Ising heat-bath dynamics [34]. Since thenaming game shares some similarity with the Ising model, one may thus hope that an effective surfacetension should be a generic feature of language formation models, which would be resilient againstdilution, migration or dynamical heterogeneities. This is very much in accord with some recent analysis,where the surface tension was shown to shape the English dialects evolution [17,18]. It was also suggested[18] that the diffusion of language users may reduce the surface tension and shift the dynamics towardvoter-like. Our simulations do not support such a behaviour, at least within the scope of our model.When only multilingual agents were allowed to migrate, we observed formation of monolingualislands. Such a state-dependent migration dynamics resembles that of the Schelling model and islands maybe considered as analogues of ghettos that are typically formed in this model. Similarly to the Schellingmodel, the formation of islands is a robust feature of the dynamics and it takes place for small as well aslarge concentration of agents. Let us notice that, unlike the Schelling model, our agents might change their ersion February 26, 2021 submitted to
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Figure 6.
The final spatial distributions of agents and languages they use for ρ = N = M = d = d = d = ersion February 26, 2021 submitted to Entropy
11 of 14 -3.5-3 -2.5-2-1.5-1-0.5 0
0 1 2 3 4 5 6 l og ( x ) log (t) d=0d=0.1d=0.01d=0.001d=0.0001 Figure 7.
The time dependence of the fraction x of bilingual agents calculated for ρ = N = d . The solid straight line has a slope corresponding to x ∼ t − and somebending of our data indicates that the asymptotic decay for d = Figure 8.
The ( ρ , d ) phase diagram as inferred from our simulations. In the largest portion of the phasediagram the final state is quickly reached and is made of finite-size monolingual islands. Only for d = ρ > ρ c we expect a power-law coarsening and for ρ < t . For d very close to 1 we expect the regime where the dominant language isformed via a spontaneous fluctuation ( SF, the dashed line is not in scale). ersion February 26, 2021 submitted to Entropy
12 of 14 language and with this respect they are driven by nonconservative dynamics. Our simulations suggestthat when state-dependent migration rate is sufficiently large, a certain language becomes dominant andspreads over the majority of agents. However, the transition toward such a linguistic coherence is not asurface-tension driven coarsening but rather a spontaneous fluctuation, similar perhaps to the transitionin the voter model. The predicted migration-induced reduction of the surface tension [18] would thustake place but only with migration of multilingual agents. When the concentration of agents is abovea site percolation threshold and the state-dependent migration is absent, agents form an infinite clusterand the naming game dynamics induces the coarsening albeit slower than on an undiluted lattice. Such acoarsening appears to be very fragile with respect to the state-dependent migration and most likely, anarbitrarily small migration directs the dynamics toward formation of monolingual islands.Migration is an important factor that should be taken into account in studying language formationmodels as well as some other agreement dynamics systems. It would be certainly desirable to developalternative approaches that would allow for at least qualitative understanding of our results, which arebased only on numerical simulations. Field-theory techniques based on the Fokker-Planck equation wereused to analyse a related class of models, the so-called voter model with intermediate states [33], and itwould be interesting to develop a similar approach in the context of our models. However, taking intoaccount migration of our particles is likely to result in a more complex field-theory description. It wouldbe also interesting to examine whether an effective surface tension appears also in reinforcement learningsystems with migration [40] or in heterogeneous systems, where agents evolve with different kinds ofdynamics. Elucidation of the role of the state-dependent migration in formation of a dominant languageor in a high fragility of slow coarsening on a diluted lattice would be also desirable.
Author Contributions: conceptualization, D.L. ; methodology, A.L.; software, A.L. and D.L.; validation, A.L. and D.L.;investigation, A.L. and D.L..; writing–review and editing, A.L. and D.L.; visualization, A.L. All authors have read andagreed to the published version of the manuscript.”, please turn to the CRediT taxonomy for the term explanation.Authorship must be limited to those who have contributed substantially to the work reported.
Conflicts of Interest:
The authors declare no conflict of interest.
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