Utterance selection model of language change

Baxter, G. J.; Blythe, Richard A.; Croft, William; McKane, Alan J.

doi:10.1103/physreve.73.046118

Cited by 111 publications

(212 citation statements)

References 33 publications

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“…where we have used (14) and where u i = f i (x) is the transformation (10). Use of Taylor's theorem then gives…”

Section: B Coordinate System In Which the Kolmogorov Equation Is Sepamentioning

confidence: 99%

“…Note that this equation has been derived using the new variables-just changing variables using (10) in the original Kolmogorov equation (8) will not give (12), but will instead simply give an equation for P (x, t|x 0 , 0) with x i replaced by u i j<i (1 − u j ). The new conditional pdf in (12), denoted by P(u, t|u 0 , 0), is related to P (x, t|x 0 , 0) through the Jacobian of the transformation to the new variables:…”

Section: Change Of Variablesmentioning

confidence: 99%

“…As a first step in the mathematical formulation of this model, we developed a "drift model" of language change [10], which turns out to be identical to the diffusion models of population genetics discussed above. Particularly relevant in this application to language change is the situation where the number of variants (the linguistic counterpart to alleles) may be large.…”

Section: Introductionmentioning

confidence: 99%

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Exact solution of the multi-allelic diffusion model

Baxter

Blythe

McKane

2007

Mathematical Biosciences

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We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results-which were for the cases of two and three alleles only. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.

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“…where we have used (14) and where u i = f i (x) is the transformation (10). Use of Taylor's theorem then gives…”

Section: B Coordinate System In Which the Kolmogorov Equation Is Sepamentioning

confidence: 99%

Section: Change Of Variablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact solution of the multi-allelic diffusion model

Baxter

Blythe

McKane

2007

Mathematical Biosciences

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“…The voter model has been extensively studied on lattices [1] and, in recent years, on complex networks [2][3][4][5][6] and is closely related to models of language evolution [7], ecological dynamics [8], opinion dynamics [9], and epidemic spread [10]. The voter model defines a dynamical process where nodes are each assigned one of two states, +1 or −1.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Voter Model on Complex Networks

Schneider-Mizell

Sander

2009

J Stat Phys

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We study a generalization of the voter model on complex networks, focusing on the scaling of mean exit time. Previous work has defined the voter model in terms of an initially chosen node and a randomly chosen neighbor, which makes it difficult to disentangle the effects of the stochastic process itself relative to the network structure. We introduce a process with two steps, one that selects a pair of interacting nodes and one that determines the direction of interaction as a function of the degrees of the two nodes and a parameter α which sets the likelihood of the higher degree node giving its state. Traditional voter model behavior can be recovered within the model. We find that on a complete bipartite network, the traditional voter model is the fastest process. On a random network with power law degree distribution, we observe two regimes. For modest values of α, exit time is dominated by diffusive drift of the system state, but as the high nodes become more influential, the exit time becomes becomes dominated by frustration effects. For certain selection processes and parameters values, an intermediate regime occurs where exit occurs after exponential mixing.

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“…Neutral diffusion-like or copying process models have been applied in a very broad range of fields, from social phenomena [1] and language change [2,3], to ecology [4,5,6] and population genetics [7] among many more. In all such models, different alternative items -species, opinions or language variants for example -are copied between neighbouring sites until one finally dominates the whole system.…”

Section: Introductionmentioning

confidence: 99%

A voter model with time dependent flip rates

Baxter¹

2011

J. Stat. Mech.

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Abstract. We introduce time variation in the flip-rates of the Voter Model. This type of generalisation is relevant to models of ageing in language change, allowing the representation of changes in speakers' learning rates over their lifetime. and may be applied to any other similar model in which interaction rates at the microscopic level change with time. The mean time taken to reach consensus varies in a nontrivial way with the rate of change of the flip-rates, varying between bounds given by the mean consensus times for for static homogeneous (the original Voter Model) and static heterogeneous flip-rates. By considering the mean time between interactions for each agent, we derive excellent estimates of the mean consensus times and exit probabilities for any time scale of flip-rate variation. The scaling of consensus times with population size on complex networks is correctly predicted, and is as would be expected for the ordinary voter model. Heterogeneity in the initial distribution of opinions has a strong effect, considerably reducing the mean time to consensus, while increasing the probability of survival of the opinion which initially occupies the most slowly changing agents. The mean times to reach consensus for different states are very different. An opinion originally held by the fastest changing agents has a smaller chance to succeed, and takes much longer to do so than an evenly distributed opinion.

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Utterance selection model of language change

Cited by 111 publications

References 33 publications

Exact solution of the multi-allelic diffusion model

Exact solution of the multi-allelic diffusion model

A Generalized Voter Model on Complex Networks

A voter model with time dependent flip rates

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