Spatial Evolution of Human Dialects

The geographical pattern of human dialects is a result of history. Here, we formulate a simple spatial model of language change which shows that the final result of this historical evolution may, to some extent, be predictable. The model shows that the boundaries of language dialect regions are controlled by a length minimizing effect analogous to surface tension, mediated by variations in population density which can induce curvature, and by the shape of coastline or similar borders. The predictability of dialect regions arises because these effects will drive many complex, randomized early states toward one of a smaller number of stable final configurations. The model is able to reproduce observations and predictions of dialectologists. These include dialect continua, isogloss bundling, fanning, the wavelike spread of dialect features from cities, and the impact of human movement on the number of dialects that an area can support. The model also provides an analytical form for Séguy’s curve giving the relationship between geographical and linguistic distance, and a generalization of the curve to account for the presence of a population center. A simple modification allows us to analytically characterize the variation of language use by age in an area undergoing linguistic change.

Popular Summary

A “dialect” is defined as a set of linguistic idiosyncrasies shared by speakers of a particular geographical region, and language surveys can be used to define dialect regions and measure dialect differences. Here, we claim that regional dialects are driven by individuals’ tendency to conform. We show that the evolution of isoglosses—geographic boundaries between dialects—is driven by two opposing forces: An effect analogous to surface tension renders isoglosses smoother over time, but variations in population density create curvature. These two effects, encapsulated in a single equation drawing on ideas from statistical mechanics, allow us to predict the spatial distribution of dialects.

Using population density census data from the United Kingdom, we predict the most likely pattern of English dialects assuming that people living within roughly 10 to 20 kilometers of one another come into linguistic contact. We consider these interactions between individual speakers and model the geographical distribution of dialects throughout the United Kingdom. We find that many isoglosses follow similar patterns, which results in “isogloss bundling” despite initial differences in language use. Our model also reproduces several observed phenomena, including the expansion of urban dialects—such as those in the London area—and the relation between linguistic distance and geographical distance, which is of long-standing interest to dialectologists.

Our model is deliberately simple in its assumptions yet still yields predictive power. It lays a theoretical foundation to investigate patterns in other countries and in other cultural or natural processes that involve copying, such as technology, art, and possibly evolution.

Figure 1

Dashed line shows the function $p_1(\mathbf{m})\equiv f_1$, defined by Eq. (4) for the $V=2$ model with conformity number $\beta =2$. Dotted line shows the neutral version $p_1(\mathbf{m})=m_1$ (when $\beta =1$) for comparison. Solid line shows the function $p_1(\mathbf{m})-m_1$ in the case $\beta =2$, giving the time derivative of the memory in the absence of spatial variation. Note that the dashed line shows that when speakers’ memories contain a majority of variant 1, they use this variant in a greater proportion than they recall it being used. This leads to progressively greater levels of conformity: more speakers using variant 1.

Figure 2

Evolution of the $V=3$ model from randomized initial condition with $\sigma =15\mathrm{km}$ and $\beta =1.1$ at times $t\in \{1,2,4,8,16,32\}$, where one time unit corresponds to one memory length. Colors indicate which variant is most common at each position. Numerical solution implemented in c++ on grid with 2-km spacing [63] (GB is $\approx 1000\mathrm{km}$ north to south). Each grid point initialized with randomly selected variant.

Figure 3

The surface tension effect at domain boundaries. Blue dots represent speakers and black circles give an approximate representation of interaction ranges. In the red shaded parts of these interaction ranges, variant $A$ is more common, and in the yellow shaded parts, variant $B$ is more common.

Figure 4

Behavior of an isogloss surrounding a densely populated area. The blue dot represents a speaker on the isogloss between variants $A$ and $B$. Other speakers are shown as black dots. The circle around our speaker represents their typical range of interaction. Within red shaded part of this interaction range, variant $A$ is more common, and in the yellow shaded part, variant $B$ is more common. Because of variation in population density, they hear more of variant $B$ (dashed lines indicate interactions) despite the fact that a greater area (red shaded) of their interaction range lies within the domain of variant $A$.

Figure 5

Behavior of isoglosses at an indentation in coastline or other boundary (political or naturally occurring). Dashed isogloss is unstable and will evolve toward the solid isogloss which emerges perpendicular from the coast. We assume that both isoglosses are effectively anchored to a feature some distance away, opposite the boundary shown. Speakers are shown as blue dots, and colors have the same meanings as in Fig. 3.

Figure 6

Superposition of the isoglosses at $t=50$ produced by 20 solutions of the $V=2$ model with $\beta =1.1$, each with different randomized initial conditions. For the left-hand map, $\sigma =5\mathrm{km}$, and for the right-hand map, $\sigma =10\mathrm{km}$ (see video in Supplemental Material [38]). Background shading indicates population density with brightest orange corresponding to 7200 inhabitants per ${\mathrm{km}}^2$.

Figure 7

Left map: Future England dialect boundaries predicted by Trudgill [4]. Right map: Future dialect boundaries predicted using $k$-medoids cluster analysis of 20 synthetic binary linguistic variables when $\sigma =10\mathrm{km}$ and $\beta =1.1$ at $t=150$. Levenshtein distance (or “edit distance”) [9] used as distance metric. Colors, determined by Hungarian method, show mapping between dialect areas. Black dotted line shows north-south isogloss.

Figure 8

Left map: Future England dialect boundaries predicted by Trudgill [4]. Right map: Voronoi tessellation with the same number of cells as Trudgill’s prediction. Colors determined by Hungarian algorithm.

Figure 9

Results of $k$-medoids cluster analysis of 20 synthetic binary linguistic variables when $\sigma =10\mathrm{km}$ and $\beta =1.1$ at $t=150$. Edit distance [9] used as distance metric. $k$ values from left to right are $k=16$, 22, 30.

Figure 10

Evolution of isoglosses in a $400\times 200$ system with two opposing boundary indentations and unit background population density, together with a collection of cities contributing additional population densities $\rho (r)=({\rho }_1-1)\exp \{-r^2/(2R^2)\}$, where $r$ is distance from city center, $R=10$, and ${\rho }_1=4$ (measuring ratio of peak city density to background). Parameter values $\sigma =4$, $\beta =2$. Evolution times $t=10$, 50, 300, 1660. See video in Supplemental Material for full animation [38].

Figure 11

Evolution of isoglosses in a $400\times 200$ rectangular system with uniform population density, starting from randomized initial conditions. We show a superposition of 10 such solutions. Notice that all isoglosses join the two long sides of the system. Parameter values $\sigma =4$, $\beta =2$. Evolution times $t=30$, 90, 270, 810.

Figure 12

Evolution of a circular isogloss (initial radius 30) in a $200\times 200$ system with unit background population density, together with a central city contributing additional density $\rho (r)=({\rho }_1-1)\exp \{-r^2/(2R^2)\}$, where $r$ is distance from city center, $R=40$, and ${\rho }_1=21$. Parameter values $\sigma =4$, $\beta =2$. Evolution times $t\in \{10,20,30,\dots ,300\}$.

Figure 13

Séguy’s curve showing linguistic distance ($l$) versus geographical distance ($r$). Dashed line shows Eq. (19) in the case $\sigma =4$, $\beta =2$. Open and closed dots show simulated linguistic distances using same parameter values in a $1000\times 1000$ system at times $t=80$, 160. Note that linguistic distance depends only on the combination $r{t}^{-1/2}$, so curves evaluated at different times collapse onto one another.

Figure 14

Radial cross sections (along the line $y=0$) through numerical approximations to fundamental solutions of the modified OJK equation (21) with population distribution Eq. (20) at time $t=300$. Here, $r=\sqrt{x^2+y^2}=x$. Parameter values are $\beta {\sigma }^2=1$ and $R=10$. Initial conditions are ${\mathbf{r}}_0=(1,0),(10,0),(20,0)$ (solid, dashed, dotted lines, respectively).

Figure 15

Continuous line line shows radial cross section (along the line $y=0$) through numerical solution of the modified OJK equation (21) with population distribution Eq. (20) at time $t=700$, with initial condition ${\mathbf{r}}_0=(0,0)$. Here, $r=\sqrt{x^2+y^2}=x$. Parameter values are $\beta {\sigma }^2=1$ and $R=10$. Data points show asymptotic analytical solution $G(|r|,t;0)$ [Eq. (30)], with $t=700$, time offset $t_0=-4.4$, and $A=0.0109$ (found by maximum likelihood).

Figure 16

Dashed lines show theoretical shape of Séguy’s curve Eq. (32) centered at peak of population density $\rho =e^{-r/R}$ with $R=20$. Curve computed using Eq. (32) when $\beta =1.4$, $\sigma =5$, times are $t=10$, 20, 30 with offset $t_0=-5.77$ (maximum likelihood estimate). Simulation points give equivalent correlations in the full model computed from 100 independent simulations in a $400\times 400$ system.

Figure 17

Dashed lines show theoretical shape of Séguy’s curve centered at peak of population density $\rho =e^{-r/R}$ with $R=20$, and with time evolution accelerated by factor of 1.25. Curve computed using Eq. (32) when $\beta =1.1$, $\sigma =5$, and times are $t_0+1.25t$, where $t=10$, 20, 30 and $t_0=4.43$. Linear scaling of time determined by maximum likelihood fit of simulation to analytical prediction. Simulation points give equivalent correlations in the full model computed from 100 independent simulations in a $400\times 400$ system at times $t=10$, 20, 30.

Figure 18

Isogloss evolution in a $400\times 400$ system with $V=2$, $\beta =1.1$, $\sigma =5$ at $t=10$, 15, 25, 35 with $\rho =e^{-r/R}$, where $R=20$ and $r=0$ corresponds to the center of the system. Plot is a superposition of 20 simulations with different initial conditions. Central peak repels isoglosses. See video in Supplemental Material for full animation [38].

Figure 19

Schematic diagram of isoglosses (dashed lines) for three language areas or “island nations.” Yellow and ochre circles represent cities. Nation $A$ supports dialect areas formed by coastal boundary shape and repulsion from cities. Nation $B$ largely exhibits a continuum of variation apart from two somewhat indistinct city dialects. Nation $C$ has a single city dialect, but without this city (or if the city were not sufficiently densely populated) it would have no linguistic variation due to its entirely convex boundary and evenly distributed population.

Figure 20

Sequence of radial cross sections of the frequency of a linguistic variable whose initial domain is concentric with a city. Snapshots taken at times $t=10,30,50,\dots$ starting from initial radius $r=55$. Model parameters $\sigma =5$, $\beta =1.1$. In this example, the city has radius $R_c=50$ and peak density ${\rho }_1=3$. Vertical line gives theoretical stable radius $R_2=125.3$ computed from Eq. (41). Unstable radius [Eq. (40)] is $R_1=47.2$. Red dashed line gives the theoretical frequency gradient in transition region [Eq. (36)].

Figure 21

Thick dashed line shows theoretical frequency of new variant for age distribution Eq. (b18)—using linear approximations Eq. (b15) and (b16) for $v(a)$ and $\mathrm{\Lambda }(a)$, and assuming $\mathrm{\Lambda }(0)=0$—when $a_0=10$, $A=90$, $\sigma =5$, $\epsilon =0.1$ at time $t=A/2=45$ (chosen so that oldest speaker at $x=0$ has just switched variable). For these parameter values $\overline{a}=35.23$. Vertical dashed line is drawn at location of youngest adopter of new variable (giving width of transition region). Open circles give frequency values using an age distribution discretized into two-year bins, computed by numerically evolving the full model.

Figure 22

Expected frequency Eq. (b20) with which new variant is used by speakers of different ages from a random sample. Mean and variance of speaker locations relative to left boundary $x_0$ of transition region are $h=\frac{1}{2}\mathrm{\Delta }(0,A)$ and ${\omega }^2=1$. Model parameter values are $a_0=10$, $A=90$, $\sigma =5$, $\epsilon =0.1$. For these parameter values, $\overline{a}=35.23$.