Coherence thresholds in models of language change and evolution: The effects of noise, dynamics, and network of interactions

A simple model of language evolution proposed by Komarova, Niyogi, and Nowak is characterized by a payoff in communicative function and by an error in learning that measure the accuracy in language acquisition. The time scale for language change is generational, and the model’s equations in the mean-field approximation are a particular case of the replicator-mutator equations of evolutionary dynamics. In well-mixed populations, this model exhibits a critical coherence threshold; i.e., a minimal accuracy in the learning process is required to maintain linguistic coherence. In this work, we analyze in detail the effects of different fitness-based dynamics driving linguistic coherence and of the network of interactions on the nature of the coherence threshold by performing numerical simulations and theoretical analyses of three different models of language change in finite populations with two types of structure: fully connected networks and regular random graphs. We find that although the threshold of the original replicator-mutator evolutionary model is robust with respect to the structure of the network of contacts, the coherence threshold of related fitness-driven models may be strongly affected by this feature.

Figure 1

(Color online) Symbols: mean fraction of speakers of language 1 $\left(x\right)$ during 5000 generations for different values of $u$ and a population $N={10}^4$ (simulation results in a fully connected network for an initial condition $x=1$). Circles: replicator-mutator dynamics. Triangles: fitness-driven Glauber dynamics. Squares: fitness-driven voter dynamics. Lines: fixed points $x^{\ast }$ from (13, 20, 25) for $x\ge \frac{1}{2}$.

Figure 2

Symbols: mean fraction of speakers of language 1 as a function of $u$ for the replicator-mutator dynamics from simulations (10 000 generations for an initial condition $x=1$) in a random regular lattice with degree $k$. Solid symbols: $k=4$. Open symbols: $k=20$. Circles: $N={10}^3$. Squares: $N={10}^4$. Lines: fixed points $x^{\ast }$ from (9, 16) for $x\ge \frac{1}{2}$. Solid line: $k=20$. Dot-dashed line: $k=4$.

Figure 3

(Color online) The same as in Fig. 2, but for the fitness-driven Voter dynamics. The fixed points $x^{\ast }$ were calculated using (9, 22, 18) for $x\ge \frac{1}{2}$.

Figure 4

(Color online) The same as in Fig. 2, but for the fitness-driven Glauber voter dynamics. The fixed points $x^{\ast }$ were calculated using (9, 26) for $x\ge \frac{1}{2}$.