Ordering dynamics with two non-excluding options: bilingualism in language competition

Xavier Castelló,Maxi San Miguel,Víctor M Eguíluz

doi:10.1088/1367-2630/8/12/308

Xavier Castelló, Maxi San Miguel + Show 1 more

Open Access

https://doi.org/10.1088/1367-2630/8/12/308

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Abstract

We consider an extension 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. This change in the coarsening mechanism is also shown to originate for a class of 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 characteristic time to reach the absorbing state scales with system size as τ ∼ lnN to be compared with the result τ ∼ N for the voter model in a small world network.

Highlights

The microscopic version [16] of the AS-model for the competition of two equivalent languages is equivalent to the voter model [18]–[23]
We have studied the non-equilibrium transient dynamics of approach to the absorbing state for the AB-model, an extension of the voter model in which the interacting agents can be in either of two equivalent states (A or B) or in a third mixed state (AB)
A domain of agents in the AB state is not stable and the density of AB-agents becomes very small after an initial fast transient, with AB agents placing themselves in the interfaces between single-option domains

Summary

A model with two non-excluding options

We consider a model in which an agent i sits in a node within a network of N individuals and has ki neighbours. Equation (2) gives the probabilities for an agent to move from the AB community towards the A or B communities They are proportional to the local density of agents with the option to be adopted, including those in the AB state (1 − σl = σj + σAB, l, j = A, B; l = j). For a quantitative description of the ordering dynamics towards consensus in the A or B state we use as an order parameter the ensemble average interface density ρ This is defined as the density of links joining nodes in the network which are in different states [20, 22].

Coarsening in a regular lattice

Coarsening in a small world network

Summary and conclusions