General three-state model with biased population replacement: Analytical solution and application to language dynamics

Empirical evidence shows that the rate of irregular usage of English verbs exhibits discontinuity as a function of their frequency: the most frequent verbs tend to be totally irregular. We aim to qualitatively understand the origin of this feature by studying simple agent-based models of language dynamics, where each agent adopts an inflectional state for a verb and may change it upon interaction with other agents. At the same time, agents are replaced at some rate by new agents adopting the regular form. In models with only two inflectional states (regular and irregular), we observe that either all verbs regularize irrespective of their frequency, or a continuous transition occurs between a low-frequency state, where the lemma becomes fully regular, and a high-frequency one, where both forms coexist. Introducing a third (mixed) state, wherein agents may use either form, we find that a third, qualitatively different behavior may emerge, namely, a discontinuous transition in frequency. We introduce and solve analytically a very general class of three-state models that allows us to fully understand these behaviors in a unified framework. Realistic sets of interaction rules, including the well-known naming game (NG) model, result in a discontinuous transition, in agreement with recent empirical findings. We also point out that the distinction between speaker and hearer in the interaction has no effect on the collective behavior. The results for the general three-state model, although discussed in terms of language dynamics, are widely applicable.

Figure 1

Plot of $I$, the fraction of times a verb is used in the irregular past tense form as a function of the inverse of its frequency of usage $\nu$, for 2840 English verbs in the CoHA corpus for the decade 1980–1989. The color (shade) of each symbol represents the average value of $I$ across 16 decades between 1830 and 1989. (Adapted from Ref. [9].)

Figure 2

Phase diagram of the NG model as a function of $n=r/\nu$. The black line indicates the stationary solution ${\rho }_I^{\left(1\right)}$, while green (light gray) and red (dark gray) curves indicate solutions ${\rho }_I^{\left(2\right)}$ (stable) and ${\rho }_I^{\left(3\right)}$ (unstable), respectively. Symbols are the stationary values of the fraction of irregulars ${\rho }_I$ and regulars ${\rho }_R$ from numerical simulations with different initial conditions: from top to bottom ${\rho }_I(t=0)=0.8,0.5$, and $0.3$. The three panels show the dependence of the stationary state on the initial condition.

Figure 3

Phase-space (${\rho }_I,{\rho }_R$) for the NG model with biased population replacement illustrating fixed points, separatrix, and phase flows (represented by blue dashed lines) for (left) $n=0$ and (right) $n=0.025$. The green squares are the attractive stable solution (1) and (2), while the red circles are the solution (3) that correspond to a saddle point with an attractive and a repulsive direction. Only the fraction of the physical phase space with ${\rho }_I+{\rho }_R<1$ is represented.

Figure 4

Stationary value of the fraction of irregulars ${\rho }_I$, regulars ${\rho }_R$, and mixed ${\rho }_M$, as a function of $n$, for the rules of model CT. Results of numerical simulations (symbols) are compared with the analytical predictions (solid lines). The rounding of the transition in simulation results is due to finite-size effects.

Figure 5

Plot of the two conic sections ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ for ${\gamma }_3>{\gamma }_2$ (top) and for ${\gamma }_3<{\gamma }_2$ (bottom), illustrating the intersections above [red (gray) curves] and below (black curves) the critical $n_c$. The triangle is the region of physical interest. The circles denote the intersections other than the fully regular solution.

Figure 6

Phase flow in the $(x,y)$ space for discontinuous (top) and continuous (bottom) transition both below (left) and above (right) the transition. The triangle is the region of physical interest. The thick curves are ${\mathcal{C}}_1$ (solid) and ${\mathcal{C}}_2$ (dashed). Symbols indicate stationary solutions. The red circles are saddle points with an attractive and a repulsive direction. The squares are attractive stable solutions. Thin solid lines are flow lines converging to the stable solution denoted by symbols with the same color (shade).