We consider a modification of the voter model in which a set of interacting elements (agents) can be in either of two equivalent states (A or B) or in a third additional mixed AB state. The model is motivated by studies of language competition dynamics, where the AB state is associated with bilingualism. We study the ordering process and associated interface and coarsening dynamics in regular lattices and small world networks. Agents in the AB state define the interfaces, changing the interfacial noise driven coarsening of the voter model to curvature driven coarsening. We argue that this change in the coarsening mechanism is generic for perturbations of the voter model dynamics. When interaction is through a small world network the AB agents restore coarsening, eliminating the metastable states of the voter model. The time to reach the absorbing state scales with system size as τ ∼ ln N to be compared with the result τ ∼ N for the voter model in a small world network.

We investigate the dynamics of two agent based models of language competition. In the first model, each individual can be in one of two possible states, either using language X or language Y , while the second model incorporates a third state XY , representing individuals that use both languages (bilinguals). We analyze the models on complex networks and two-dimensional square lattices by analytical and numerical methods, and show that they exhibit a transition from one-language dominance to language coexistence. We find that the coexistence of languages is more difficult to maintain in the Bilinguals model, where the presence of bilinguals in use facilitates the ultimate dominance of one of the two languages. A stability analysis reveals that the coexistence is more unlikely to happen in poorly-connected than in fully connected networks, and that the dominance of only one language is enhanced as the connectivity decreases. This dominance effect is even stronger in a two-dimensional space, where domain coarsening tends to drive the system towards language consensus.

Abstract:The differential equations of Abrams and Strogatz for the competition between two languages are compared with agent-based Monte Carlo simulations for fully connected networks as well as for lattices in one, two and three dimensions, with up to 10 9 agents.Keywords: Monte Carlo, language competition Many computer studies of the competition between different languages, triggered by Abrams and Strogatz [1], have appeared mostly in physics journals using differential equations (mean field approximation [2, 3, 4, 5]) or agent-based simulations for many [6,7,8,9] or few [10, 11] languages. A longer review is given in [12], a shorter one in [13]. We check in this note to what extent the results of the mean field approximation are confirmed by agent-based simulations with many individuals. We do not talk here about the learning of languages [14,15].The Abrams-Strogatz differential equation for the competition of a language Y with higher social status 1 − s against another language X with lower social status s iswhere a ≃ 1.3 [1] and 0 < s ≤ 1/2. Here x is the fraction in the population speaking language X with lower social status s while the fraction 1−x speaks language Y. As initial condition we consider the situation in which both languages have the same number of speakers, x(t = 0) = 1/2. Figure 1 shows exponential decay for a = 1.31 as well as for the simpler linear case a = 1. For s = 1/2 the symmetric situation x = 1/2 is a stationary solution 1