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
We consider two social consensus models, the AB-model and the Naming Game restricted to two conventions, which describe a population of interacting agents that can be in either of two equivalent states (A or B) or in a third mixed (AB) state. Proposed in the context of language competition and emergence, the AB state was associated with bilingualism and synonymy respectively. We show that the two models are equivalent in the mean field approximation, though the differences at the microscopic level have non-trivial consequences. To point them out, we investigate an extension of these dynamics in which confidence/trust is considered, focusing on the case of an underlying fully connected graph, and we show that the consensus-polarization phase transition taking place in the Naming Game is not observed in the AB model. We then consider the interface motion in regular lattices. Qualitatively, both models show the same behavior: a diffusive interface motion in a onedimensional lattice, and a curvature driven dynamics with diffusing stripe-like metastable states in a two-dimensional one. However, in comparison to the Naming Game, the AB-model dynamics is shown to slow down the diffusion of such configurations.
We address the role of community structure of an interaction network in ordering dynamics, as well as associated forms of metastability. We consider the voter and AB model dynamics in a network model which mimics social interactions. The AB model includes an intermediate state between the two excluding options of the voter model. For the voter model we find dynamical metastable disordered states with a characteristic mean lifetime. However, for the AB dynamics we find a power law distribution of the lifetime of metastable states, so that the mean lifetime is not representative of the dynamics. These trapped metastable states, which can order at all time scales, originate in the mesoscopic network structure. * These authors contributed equally to this work.
We search for conditions under which a characteristic time scale for ordering dynamics toward either of two absorbing states in a finite complex network of interactions does not exist. With this aim, we study random networks and networks with mesoscale community structure built up from randomly connected cliques. We find that large heterogeneity at the mesoscale level of the network appears to be a sufficient mechanism for the absence of a characteristic time for the dynamics. Such heterogeneity results in dynamical metastable states that survive at any time scale.
Motivated by the idea that some characteristics are specific to the relations between individuals and not to the individuals themselves, we study a prototype model for the dynamics of the states of the links in a fixed network of interacting units. Each link in the network can be in one of two equivalent states. A majority link-dynamics rule is implemented, so that in each dynamical step the state of a randomly chosen link is updated to the state of the majority of neighboring links. Nodes can be characterized by a link heterogeneity index, giving a measure of the likelihood of a node to have a link in one of the two states. We consider this link-dynamics model in fully connected networks, square lattices, and Erdös-Renyi random networks. In each case we find and characterize a number of nontrivial asymptotic configurations, as well as some of the mechanisms leading to them and the time evolution of the link heterogeneity index distribution. For a fully connected network and random networks there is a broad distribution of possible asymptotic configurations. Most asymptotic configurations that result from link dynamics have no counterpart under traditional node dynamics in the same topologies.
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