Language Dynamics in the Framework of Complex Networks: A Case Study on Self-Organization of the Consonant Inventories

Abstract

In this chapter, we present a statistical mechanical model of language acquisition and change at a mesoscopic level, and validate our model for the sound systems of the languages across the world. We show that the emergence of the linguistic diversity that exists across the consonant inventories of some of the major language families of the world can be explained through a complex network based growth model, which has only a single tunable parameter that is meant to introduce a small amount of randomness in the otherwise preferential attachment based growth process. The experiments with this model parameter indicates that the choice of consonants among the languages within a family are far more preferential than it is across the families. Furthermore, our observations indicate that this parameter might bear a correlation with the period of existence of the language families under investigation. These findings lead us to argue that preferential attachment seems to be an appropriate high level abstraction for language acquisition and change.

Appendix: Derivation of the Analytical Solution

We shall solve the model for μ = 1 and then generalize for the case μ > 1. Please refer to Sects. 5 and 6 for explanation of the model and the notations used.

1.1 Solution for μ = 1

Since μ = 1, at each time step a node in the V _L partition essentially brings a single incoming edge as it enters the system. The evolution of p _k, t can be expressed as

$${p}_{k,t+1} = (1 -\widetilde{ P}(k,t)){p}_{k,t} +\widetilde{ P}(k - 1,t){p}_{k-1,t}$$

(6)

where $\widetilde{P}(k,t)$ refers to the probability that the incoming edge lands on a consonant node of degree k at time t. $\widetilde{P}(k,t)$ can be easily derived for μ = 1 using together the Eqs. 2 and 3 and takes the form

$$\widetilde{P}(k,t) = \left \{\begin{array}{ccc} \frac{\gamma k+1} {\gamma t+N} & \mbox{ for} &0 \leq k \leq t \\ 0 &\mbox{ otherwise}& \\ \end{array} \right.$$

(7)

for t > 0 while for t = 0, $\widetilde{P}(k,t) = \frac{1} {N}{\delta }_{k,0}$.

Equation 6 can be explained as follows. The probability of finding a consonant node with degree k at time t + 1 decreases due to those nodes, which have a degree k at time t and receive an edge at time t + 1 therefore acquiring degree k + 1, i.e., $\widetilde{P}(k,t){p}_{k,t}$. Similarly, this probability increases due to those nodes that at time t have degree k − 1 and receive an edge at time t + 1 to have a degree k, i.e., $\widetilde{P}(k - 1,t){p}_{k-1,t}$. Hence, the net increase in the value of p _{k, t + 1} can be expressed by the Eq. 6.

In order to have an exact analytical solution of the Eq. 6 we express it as a product of matrices

$${ \mathbf{p}}_{t+1} ={ \mathbf{M}}_{t}{\mathbf{p}}_{t} = \big{[}\prod \limits _{\tau =0}^{t}{\mathbf{M}}_{ \tau }\big{]}{\mathbf{p}}_{0}$$

(8)

where p _t denotes the degree distribution at time t and is defined as ${\mathbf{p}}_{t} = {[{p}_{0,t}\ {p}_{1,t}\ {p}_{2,t}\ \ldots ]}^{T}$ (T stands for the standard transpose notation for a matrix), p ₀ is the initial condition expressed as p ₀ = [1 0 0 …]^T and M _τ is the evolution matrix at time τ which is defined as

$$\mathbf{{M}_{\tau }} = \left (\begin{array}{c c c c c} 1 -\widetilde{ P}(0,\tau )& 0 & 0 &0&\ldots \\ \widetilde{P}(0,\tau ) &1 -\widetilde{ P}(1,\tau )& 0 &0&\ldots \\ 0 & \widetilde{P}(1,\tau ) &1 -\widetilde{ P}(2,\tau )&0&\ldots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \right )$$

(9)

Let us further define a matrix H _t as follows.

$${ \mathbf{H}}_{0} ={ \mathbf{M}}_{0}$$

(10)

$${ \mathbf{H}}_{t} ={ \mathbf{M}}_{t}{\mathbf{H}}_{t-1} = \big{[}\prod \limits _{\tau =0}^{t}{\mathbf{M}}_{ \tau }\big{]}$$

(11)

Thus we have,

$${ \mathbf{p}}_{t+1} ={ \mathbf{H}}_{t}{\mathbf{p}}_{0}$$

(12)

Since our initial condition (i.e., p ₀) is a matrix of zeros at all positions except the first row therefore, all the relevant information about the degree distribution of the consonant nodes is encoded by the first column of the matrix H _t . The (k + 1)th element of this column essentially corresponds to p _k, t . Let the entry corresponding to the ith row and the jth column of H _t and M _t be denoted by ${h}_{i,j}^{t}$ and ${m}_{i,j}^{t}$ respectively. On successive expansion of H _t using the recursive definition provided in Eq. 11, we get (see Fig. 7 for an example)

$${h}_{i,j}^{t} = {m}_{ i,i-1}^{t}{h}_{ i-1,j}^{t-1} + {m}_{ i,i}^{t}{h}_{ i,j}^{t-1}$$

(13)

or,

$$\begin{array}{rcl}{ h}_{i,j}^{t}\! =\! ({m}_{ i,i-1}^{t}{m}_{ i-1,i-2}^{t-1}){h}_{ i-2,j}^{t-2} + ({m}_{ i,i-1}^{t}{m}_{ i-1,i-1}^{t-1} + {m}_{ i,i}^{t}{m}_{ i,i-1}^{t-1}){h}_{ i-1,j}^{t-2} + {m}_{ i,i}^{t}{m}_{ i,i}^{t-1}{h}_{ i,j}^{t-2}& & \\ & &\end{array}$$

(14)

Fig. 7

A few steps showing the calculations of Eqs. 13 and 14

Since the first column of the matrix H _t encodes the degree distribution, it suffices to calculate the values of ${h}_{i,1}^{t}$ in order to estimate p _k, t . In fact, p _k, t (i.e., the ${(k + 1)}^{\mathrm{th}}$ entry of H _t ) is equal to ${h}_{k+1,1}^{t}$. In the following, we shall attempt to expand certain values of ${h}_{k+1,1}^{t}$ in order to detect the presence of a pattern (if any) in these values. In particular, let us investigate two cases of ${h}_{2,1}^{1}$ and ${h}_{2,1}^{2}$ from Fig. 7. We have

$${h}_{2,1}^{1} = {m}_{ 2,1}^{1}{h}_{ 1,1}^{0} + {m}_{ 2,2}^{1}{h}_{ 2,1}^{0} = \left (1 - \frac{1} {N}\right )\left ( \frac{1} {\gamma + N}\right ) + \left (\frac{N - 1} {\gamma + N}\right )\left ( \frac{1} {N}\right )$$

(15)

or,

$${h}_{2,1}^{1} = 2 \frac{(N - 1)} {(\gamma + N)N}$$

(16)

Similarly,

$${h}_{2,1}^{1} = {m}_{ 2,1}^{2}{m}_{ 1,1}^{1}{h}_{ 1,1}^{0} + {m}_{ 2,2}^{2}{m}_{ 2,1}^{1}{h}_{ 1,1}^{0} + {m}_{ 2,2}^{2}{m}_{ 2,2}^{1}{h}_{ 2,1}^{0}$$

(17)

or,

$${h}_{2,1}^{1} = 3\frac{(\gamma + N - 1)(N - 1)} {(2\gamma + N)(\gamma + N)N}$$

(18)

A closer inspection of Eqs. 16 and 18 reveals that the pattern of evolution of this row, in general, can be expressed as

$${ p}_{k,t} = \left (\begin{array}{c} t\\ k \end{array} \right )\frac{\prod \limits _{x=0}^{k-1}(\gamma x + 1)\prod \limits _{y=0}^{t-1-k}(N - 1 + \gamma y)} {\prod \limits _{w=0}^{t-1}(\gamma w + N)}$$

(19)

for 0 ≤ k ≤ t and p _k, t  = 0 otherwise. Further, we define the special case $\prod \limits _{z=0}^{-1}(\ldots \,) = 1$. Note that if we now put t = 2, k = 1 and t = 3, k = 1 in (19) we recover Eqs. 16 and 18 respectively.

Equation 19 is the exact solution of the Eq. 6 for the initial condition ${p}_{k,t=0} = {\delta }_{k,0}$. Therefore, this is the analytical expression for the degree distribution of the consonant nodes in PlaNet _theo for μ = 1.

In the limit $\gamma \rightarrow 0$ (i.e. when the attachments are completely random) Eq. 19 takes the form

$${ p}_{k,t} = \left (\begin{array}{c} t\\ k \end{array} \right ){\left ( \frac{1} {N}\right )}^{k}{\left (1 - \frac{1} {N}\right )}^{t-k}$$

(20)

for $0 \leq k \leq t$ and p _k, t  = 0 otherwise.

On the other hand, when $\gamma \rightarrow \infty$ (i.e., when the attachments are completely preferential) the degree distribution of the consonant nodes reduces to

$${p}_{k,t} = \left (1 - \frac{1} {N}\right ){\delta }_{k,0} + \frac{1} {N}{\delta }_{k,t}$$

(21)

1.2 Solution for μ > 1

In the previous section, we have derived an analytical solution for the degree distribution of the consonant nodes in PlaNet _theo specifically for μ = 1. However, note that the value of μ is greater than 1 (approximately 21) for the real network (i.e., PlaNet). Therefore, one needs to analytically solve for the degree distribution for values of μ greater than 1 in order to match the results with the empirical data. Here we attempt to generalize the derivations of the earlier section for μ > 1.

We assume that $\mu \ll N$ (which is true for PlaNet) and expect Eq. 6 to be a good approximation for the case of μ > 1 after replacing $\widetilde{P}(k,t)$ by $\widehat{P}(k,t)$ where $\widehat{P}(k,t)$ is defined as

$$\widehat{P}(k,t) = \left \{\begin{array}{ccc} \frac{(\gamma k+1)\mu } {\mu \gamma t+N} & \mbox{ for} &0 \leq k \leq \mu t \\ 0 &\mbox{ otherwise}& \\ \end{array} \right.$$

(22)

The term μ appears in the denominator of the Eq. 22 for $0 \leq k \leq \mu t$ because, in this case the total degree of the consonant nodes in PlaNet _theo at any point in time is μt rather than t as in Eq. 7. The numerator contains a μ since at each time step there are μ edges that are being incorporated into the network rather than a single edge.

The solution of Eq. 6 with the attachment kernel defined in Eq. 22 can be expressed as

$${ p}_{k,t} = \left (\begin{array}{c} t\\ k \end{array} \right )\frac{\prod \limits _{x=0}^{k-1}(\gamma x + 1)\prod \limits _{y=0}^{t-1-k}(\frac{N} {\mu } - 1 + \gamma y)} {\prod \limits _{w=0}^{t-1}(\gamma w + \frac{N} {\mu } )}$$

(23)

for $0 \leq k \leq \mu t$ and p _k, t  = 0 otherwise.

Given that $\mu \ll N$ we can neglect the term containing μ ∕ N in the Eq. 23 and express the rest using factorials as

$${p}_{k,t} = \frac{t!\eta !(t - k + \eta - {\gamma }^{-1})!(k - 1 + {\gamma }^{-1})!{\gamma }^{-1}} {(t - k)!k!(t + \eta )!(\eta - {\gamma }^{-1})!({\gamma }^{-1})!}$$

(24)

where $\eta = N/\mu \gamma$. Approximating the factorials using Stirling’s formula (see [1] for a reference), we get

$${p}_{k,t} =\widetilde{ A}(t,\gamma ,\eta )\frac{{(k - 1 + {\gamma }^{-1})}^{k-1+{\gamma }^{-1}+0.5 }{(t - k + \eta - {\gamma }^{-1})}^{t-k+\eta -{\gamma }^{-1}+0.5 }} {{k}^{k+0.5}{(t - k)}^{t-k+0.5}}$$

(25)

where

$$\widetilde{A}(t,\gamma ,\eta ) = \frac{{t}^{t+0.5}{\eta }^{\eta +0.5}{\gamma }^{{\gamma }^{-1} -0.5}e} {\sqrt{2\pi }{(t + \eta )}^{t+\eta +0.5}{(\eta - {\gamma }^{-1})}^{\eta -{\gamma }^{-1}+0.5}}$$

(26)

is a term independent of k.

Since we are interested in the asymptotic behavior of the network such that t is very large, we may assume that $t \gg k \gg \eta > {\gamma }^{-1}$. Under this assumption, we can re-write the Eq. 25 in terms of the fraction k ∕ t and this immediately reveals that the expression is approximately a β-distribution in k ∕ t. More specifically, we have

$${p}_{k,t} \approx \widehat{ A}(t,\eta ,\gamma )\mathrm{B}(k/t;{\gamma }^{-1},\eta - {\gamma }^{-1}) =\widehat{ A}(t,\eta ,\gamma ){(k/t)}^{{\gamma }^{-1}-1 }{(1 - k/t)}^{\eta -{\gamma }^{-1}-1 }$$

(27)

where B(z; α, β) refers to a β-distribution over variable z. We can generate different distributions by varying the value of γ in Eq. 27. We can further compute P _k, t (i.e. the cumulative degree distribution) using Eqs. 1 and 27 together.