Using spatial patterns of English folk speech to infer the universality class of linguistic copying

Both linguistic and genetic evolution involve copying and mutation of variants. The simplest copying process assumes that variants are reproduced at a rate equal to their current frequency, exemplified by Kimura's stepping stone model of neutral evolution, and the voter model. In this case, spatial patterns are driven by noise. In the linguistic context, an alternative possibility is that speakers preferentially select variants which are already popular, yielding patterns driven by surface tension, exemplified by the Ising model. In this paper, we model language change using a spatial network of speakers, inspired by the Hopfield neural network. The model's universality class—Voter or Ising—is determined by speakers' learning function. We view maps generated by the Survey of English Dialects as samples from our network. Maximum likelihood analysis, and comparison of spatial auto-correlations between real and simulated maps, indicates that the underlying copying processes is more likely to belong to the conformity-driven Ising class.

Figure 1

Grid used for simulating English language features. Each light blue dot is the center of a 10 km $\times$ 10 km grid square. Each red star is an SED survey location. There are 1329 grid squares, and 310 survey locations.

Figure 2

Interface shape when $\sigma =1$ and $\beta \in \{2.5,3,4\}$ (blue, black, red). Dashed line shows gradient of interface for $\beta =4$, and red vertical lines show width of interface $\omega (\sigma ,\beta )$ in this case.

Figure 3

Spatial distribution of the two state probability mass function $\mathbf{v}\left(\mathbf{r}\right)$ over a $1000\text{km}\times 1000\text{km}$ toroidal system with $a=10\mathrm{km}$. Parameter values $\sigma =5\mathrm{km}$, $N={10}^4$, $\beta =2.5$, and ${\tau }_m=2,{\tau }_s=1$. Evolution shown after 25 time steps starting from randomized initial conditions. Red and blue correspond to the two possible variants.

Figure 4

Interface crossing count, traveling between ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ in a binary linguistic system. If an even number of interfaces are crossed, then the speakers at ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ will match with high probability.

Figure 5

Blue curves show conformity driven matching probability functions given by (27) for the five $\lambda$ values given in Table 1. Red curves show matching probabilities in the case of proportional copying given by (49) for five $(b,c)$ values, also given in Table 1.

Figure 6

External sates of proportional copying $q=2$ state system with ${\tau }_m=2,{\tau }_s=1$. The simulated population per cell and interaction range are $N_{\text{sim}}=5$ and ${\sigma }_{\text{sim}}=3.16\mathrm{km}$ giving an effective interaction range of ${\sigma }_{\text{eff}}\approx 70\mathrm{m}$ when $N={10}^4$ and $a=10\mathrm{km}$. Histogram shows distribution of external states.

Figure 7

External sates of proportional copying $q=2$ state system with ${\tau }_m=2,{\tau }_s=1$. The simulated population per cell and interaction range are $N_{\text{sim}}=50$ and ${\sigma }_{\text{sim}}=3.16\mathrm{km}$ giving an effective interaction range of ${\sigma }_{\text{eff}}\approx 220\mathrm{m}$ when $N={10}^4$ and $a=10\mathrm{km}$. Histogram shows distribution of external states.

Figure 8

Dots show equal time snapshots of the simulated correlation function $m\left(\mathbf{r}\right)$ calculated from 500 realizations of proportional copying $q=2$ state model with ${\tau }_m=2,{\tau }_s=1$, simulated on England starting from randomized, spatially uncorrelated initial conditions. Snapshots of the ensemble of systems were taken at a (quadratically increasing) sequence of 8 times in the interval $T\in [1,2\times {10}^4]$. The simulated population per cell and interaction range are $N_{\text{sim}}=5$ and ${\sigma }_{\text{sim}}=3.16\mathrm{km}$ giving an effective interaction range of ${\sigma }_{\text{eff}}\approx 70\mathrm{m}$ when $N={10}^4$ and $a=10\mathrm{km}$. Red curves show least squares fits to $max\left(m\right(\mathbf{r}),1/2)$ where $m\left(\mathbf{r}\right)$ defined by (49) and $\epsilon =10\mathrm{km}$.

Figure 9

Relationships between the parameters of the matching probability (49), with $\epsilon =10\mathrm{km}$, obtained by fitting to snapshots at times in the interval $T\in [1,2\times {10}^4]$ of the simulated correlation function $m\left(\mathbf{r}\right)$ calculated from 500 realizations of the proportional copying model. In every case we have ${\tau }_m=2,{\tau }_s=1,{\sigma }_{\text{sim}}=3.16\mathrm{km}$, $N={10}^4$ and $a=10\mathrm{km}$. The three curves correspond to different simulated cell populations $N_{\text{sim}}\in \{5,10,50\}$. Solid lines show kernel density regression on simulated data with bandwidth $h=0.05$.

Figure 10

A third variant embedded within a clearly defined domain.

Figure 11

A third variant not well-embedded within a clearly defined domain.

Figure 12

SED maps of the twenty variables with highest Moran $I$. The title of each map gives the formal linguistic description of the variable, and the Moran $I$ value using six nearest neighbors.

Figure 13

SED maps of the twenty variables with lowest Moran $I$. The titles give the formal linguistic description of each variable, and the Moran $I$ value calculated using six nearest neighbors.

Figure 14

Distributions of $I$ values (six nearest neighbors) from direct simulation on England using $q=2,{\tau }_m=2,{\tau }_s=1$. In the proportional copying case, we have ${\sigma }_{\text{sim}}=3.16\mathrm{km}$, and $N_{\text{sim}}\in \{5,10,50\}$ giving effective interaction ranges ${\sigma }_{\text{eff}}\in \{70\text{m},100\text{m},220\text{m}\}$ when the true cell population is $N={10}^4$. In the conformity driven model we use ${\sigma }_{\text{sim}}=5\mathrm{km}$, $N={10}^4$, and $\beta \in \{2.5,3,3.5\}$. Vertical line indicates where proportional/conformity model cases are more common.

Figure 15

Distributions of $I$ values (six nearest neighbors) from the SED data. Vertical line indices where proportional/conformity model cases are more common in direct simulations. 82% of cases have $I>0.3.$

Figure 16

Samples from Markov graphical models calibrated proportional copying and conformity driven matching curves in Fig. 5.

Figure 17

Distributions of $I$ values (six nearest neighbors) from the SED data, and from Markov random field models calibrated to matching functions described by the parameters in Table 1 and plotted in Fig. 5.

Figure 18

(a) Each curve shows the average logarithmic probability (proportional to logarithmic likelihood) of 100 samples from a single conformity driven model, calibrated to one of the five possible matching curves (see Table 1 for parameter values). Vertical dashed lines show $\lambda$ values for each model, with curves/dashed lines of matching color corresponding to the same $\lambda$ value. The maximum value of each curve occurs at the $\lambda$ value used to generate its samples, as expected. Plot (b) as for plot (a), but using the proportional copying models with parameter $b$ used to specify model [see Table 1 for $(b,c)$ values].

Figure 19

Relative rate of 'tis and it's in the Google Books corpus by date.