Language change in a multiple group society

The processes leading to change in languages are manifold. In order to reduce ambiguity in the transmission of information, agreement on a set of conventions for recurring problems is favored. In addition to that, speakers tend to use particular linguistic variants associated with the social groups they identify with. The influence of other groups propagating across the speech community as new variant forms sustains the competition between linguistic variants. With the utterance selection model, an evolutionary description of language change, Baxter et al. [Phys. Rev. E 73, 046118 (2006)] have provided a mathematical formulation of the interactions inside a group of speakers, exploring the mechanisms that lead to or inhibit the fixation of linguistic variants. In this paper, we take the utterance selection model one step further by describing a speech community consisting of multiple interacting groups. Tuning the interaction strength between groups allows us to gain deeper understanding about the way in which linguistic variants propagate and how their distribution depends on the group partitioning. Both for the group size and the number of groups we find scaling behaviors with two asymptotic regimes. If groups are strongly connected, the dynamics is that of the standard utterance selection model, whereas if their coupling is weak, the magnitude of the latter along with the system size governs the way consensus is reached. Furthermore, we find that a high influence of the interlocutor on a speaker's utterances can act as a counterweight to group segregation.

Figure 1

Scaling plot of the time to consensus as a function of group size $N$ and group affinity $f$ for two coupled groups. There are two asymptotic regimes, for strong and weak coupling, respectively. The boundary is marked by the intersection $N_{\times }(1-f)$ of the curves fitting the power laws. The other parameter values are $T=1,\lambda =0.01,h=0.01$. Inset: The same curves before rescaling.

Figure 2

The dependence of the boundary between the asymptotic limits of the scaling law, $(1-f)N_{\times }$, on the magnitude of change in vocabulary $\lambda$ is linear for $\lambda <1/h$. For large values of $\lambda$, the function $\eta$ becomes independent of this parameter. Other parameter values are $N=2$ and $T=1$.

Figure 3

For very small values of the relative influence of the interlocutor, $h$, the boundary $(1-f)N_{\times }$ does not depend on this parameter. In the limit of large $h$, the boundary becomes proportional to $1/h$. The values of the other parameters are $N=2$ and $T=1$.

Figure 4

Typical time trajectories of the consensus measure $x_0$ of two groups with (a) weak ($f=0.9999$), (b) intermediate ($f=0.998$), and (c) strong ($f=0.9$) coupling. Other parameter values: $T=1$, $\lambda =0.01$, $h=0.01$.

Figure 5

(a) The trajectory from Fig. 4c, for strong group coupling, here in ($x_{0,1}$,$x_{0,2}$) coordinates. The dynamics is that of a one-dimensional random walk with absorbing boundaries. (b) The trajectory for intermediate group coupling from Fig. 4b. It is the dynamics of a two-dimensional random walk in a rectangular area with reflecting boundaries and two absorbing points.

Figure 6

A typical coarse-grained trajectory of the consensus measure for two groups ($x_{0,1}$ and $x_{0,2}$, respectively), as well as the consensus measure throughout the system, $\langle x_0\rangle$, for $N_G=256$ groups with group affinity $f=0.5$ (other parameter values: $N=2$, $T=1$, $\lambda =0.01$, $h=0.01$). The points represent averages over 1000 simulation runs. For larger numbers $N$ of speakers in a group, the trajectory looks similar.

Figure 7

A coarse-grained trajectory of the consensus measures $x_{0,1}$, $x_{0,2}$, and $\langle x_0\rangle$ in a system of $N_G=256$ groups with group affinity $f=0.99$ (values of the other parameters: $N=2$, $T=1$, $\lambda =0.01$, $h=0.01$). The points are averaged over 1000 simulation runs.

Figure 8

Scaling plot of the time to consensus for a well-mixed system with different number of groups and fixed group size. Inset: the horizontal scaling factor obtained by shifting the curves in order to obtain the master curve. For $N_G<32$, the scaling is different from the one for large $N_G$ due to finite-size effects. Other parameters: $N=4$, $T=1$, $\lambda =0.01$, $h=0.01$.

Figure 9

Scaling plot of the time to consensus for a system of groups on a square lattice. Inside the groups the configuration is well-mixed. Inset: horizontal scaling factor, obtained by shifting the simulation data curves so that they all fall onto the master curve. Other parameters: $N=2$, $T=1$, $\lambda =0.01$, $h=0.01$.