Alan J. McKane

2012

Fast fixation with a generic network structure

Physical Review E 86(3):031142, 2012

We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model), and opinion dynamics ...MORE ⇓

We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model), and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics while holding the degree distribution constant. In particular, we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to. We further show that there is a complex interplay between degree distribution and asymmetry when they may covary, and that the results are robust to correlations in the network or the initial condition.

2009

Language Variation and Change 21(2):257-296, 2009

Trudgill (2004) proposed that the emergence of New Zealand English, and of isolated new dialects generally, is purely deterministic. It can be explained solely in terms of the frequency of occurrence of particular variants and the frequency of interactions between different ...MORE ⇓

Trudgill (2004) proposed that the emergence of New Zealand English, and of isolated new dialects generally, is purely deterministic. It can be explained solely in terms of the frequency of occurrence of particular variants and the frequency of interactions between different speakers in the society. Trudgill's theory is closely related to usage-based models of language, in which frequency plays a role in the representation of linguistic knowledge and in language change. Trudgill's theory also corresponds to a neutral evolution model Of language change. We use a mathematical model based on Croft's usage-based evolutionary framework for language change (Baxter, Blythe, Croft, \& McKane, 2006), and investigate whether Trudgill's theory is a plausible model of the emergence of new dialects. The results of our modeling indicate that determinism cannot be a sufficient mechanism for the emergence of a new dialect. Our approach illustrates the utility of mathematical modeling of theories and of empirical data for the study of language change.

2008

Physical Review Letters 101(25):258701, 2008

We investigate a set of stochastic models of biodiversity, population genetics, language evolution and opinion dynamics on a network within a common framework. Each node has a state, 0 < xi < 1, with interactions specified by strengths mij. For any set of mij we derive an ...MORE ⇓

We investigate a set of stochastic models of biodiversity, population genetics, language evolution and opinion dynamics on a network within a common framework. Each node has a state, 0 < xi < 1, with interactions specified by strengths mij. For any set of mij we derive an approximate expression for the mean time to reach fixation or consensus (all xi=0 or 1). Remarkably in a case relevant to language change this time is independent of the network structure.

2006

Physical Review E 73:046118, 2006

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of ...MORE ⇓

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of population genetics, but there are significant differences. The continuous time formulation of the model is expressed in terms of a Fokker-Planck equation. This equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight. Whilst the stationary properties of this system have much in common with the single-speaker case, time-dependent properties are richer. In the particular case where linguistic forms can become extinct, we find that the presence of many speakers causes a two-stage relaxation, the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations.