Language competition in a population of migrating agents

Influencing various aspects of human activity, migration is associated also with language formation. To examine the mutual interaction of these processes, we study a Naming Game with migrating agents. The dynamics of the model leads to formation of low-mobility clusters, which turns out to break the symmetry of the model: although the Naming Game remains symmetric, low-mobility languages are favored. High-mobility languages are gradually eliminated from the system, and the dynamics of language formation considerably slows down. Our model is too simple to explain in detail language competition of migrating human communities, but it certainly shows that languages of settlers are favored over nomadic ones.

Figure 1

Illustration of the Naming Game dynamics. In the case of success—when the word selected by Speaker (dog) is known to Hearer—they both retain only the transmitted word in their inventories. Failure occurs when Hearer does not know the transmitted word, which is then added by Hearer to its inventory.

Figure 2

The average lifetime $\tau$ of language B whose users are initially in a square of size $M$, surrounded by language A users ($L=300$). The straight and dotted lines have slopes corresponding to $\tau \approx M^2$ and $\tau \approx M^{1.1}$, respectively. The data for $\rho <1$ and language-independent mobility ($d_A=d_B=d=1$) also seem to obey such scaling. A different behavior can be seen for language-specific mobility: the language B is either quickly extinct ($d_B>d_A$) or is relatively persistent ($d_B<d_A$). Numerical results are averages over 100 independent samples.

Figure 3

The initial configuration ($L=200,\rho =0.2$) includes in the center a square of size $M=30$ with B-speaking agents (red/gray), surrounded by A-speaking agents (green/light gray). B-speakers are less mobile ($d_B=0.1$) than A-speakers ($d_A=1.0$). The figure presents a configuration after $t=5\times {10}^4$ steps. Agents that have both A and B in their inventories are marked in blue (dark gray).

Figure 4

The time evolution of the density ${\rho }_B$ of language B users. In the inital state, the lattice ($L=500$) is divided into two halfs containing only A- or B-speakers, respectively ($\rho =0.7$). The vertical interface thus created moves in the direction which depends on the difference in mobilities of languages. Only for equal mobilities ($d_A=d_B=0.5$), the interface remains immobile. The results are averages over 100 independent runs.

Figure 5

(Left) Due to the symmetry of the Naming Game, the A-B pair with equal probabilities ends up in the A-A or B-B state. When language A is more mobile, the pair is likely to drift apart, while the B-B pair is more stable. (Right) Apparently, the A-B symmetry is broken for more complex interactions. Even if A happens to graft his language onto a cluster, it is likely to drift apart. Then a single A user, surrounded by three B neighbors, is likely to be converted back to B. Thus less mobile clusters are relatively immune to encounters with more mobile agents and the dynamics (effectively) favors less mobile languages.

Figure 6

Probability that consensus will be reached on language A ($d_A=1$) as a function of the number of agents $n$. Simulations were made for a small lattice with $L=6$. For each $n$ we made ${10}^6$ runs with a random distribution of agents (initially, languages A and B were also assigned randomly). Only for $n=2$ (pair interactions) and $n=36$ (no migration), the dynamics of the model remains symmetric and the probability to end up with language A is the same as with language B.

Figure 7

The configuration obtained after $t=5\times {10}^2$ (top), $3\times {10}^3$ (middle) and $t={10}^4$ (bottom) MC steps ($L=200, \rho =0.1$). Low-mobility languages form more dense clusters, surrounded by depleted zones, and gradually grow at the expense of more mobile neighbors. The color bars apply only to the mobility panels.

Figure 8

The time dependence of the average mobility $\langle d\rangle$; simulation for $L={10}^3$ with random initial conditions and averaged over 100 independent runs. The dash-dotted line corresponds to the decay $\langle d\rangle \sim t^{-0.9}$.

Figure 9

The time dependence of the number of users of the most common language $l_M$; simulation for $L={10}^3$ with random initial conditions and averaged over 100 independent runs. The dashed line corresponds to the increase $l_M\sim t^{0.8}$.