Language structure in the $n$-object naming game

We examine a naming game with two agents trying to establish a common vocabulary for $n$ objects. Such efforts lead to the emergence of language that allows for an efficient communication and exhibits some degree of homonymy and synonymy. Although homonymy reduces the communication efficiency, it seems to be a dynamical trap that persists for a long, and perhaps indefinite, time. On the other hand, synonymy does not reduce the efficiency of communication but appears to be only a transient feature of the language. Thus, in our model the role of synonymy decreases and in the long-time limit it becomes negligible. A similar rareness of synonymy is observed in present natural languages. The role of noise, that distorts the communicated words, is also examined. Although, in general, the noise reduces the communication efficiency, it also regroups the words so that they are more evenly distributed within the available “verbal” space.

Figure 1

A communication attempt with $n=4$ objects. Speaker selected the second object and one of the words that are associated with this object. The selected word is communicated to the hearer that then decodes its meaning. Since the decoded object is the same as that chosen by the speaker, the above example is considered as a success. The selection of the communicated word and its decoding are stochastic in nature (see the main text) and controlled by weights $w$ associated with each word.

Figure 2

The time evolution of (a) the success rate; (b) the number of different largest-weight words; and (c) the fraction of second-largest-weight utterances. Calculations were made for $n=500$, $l=10$, and $r={10}^3$.

Figure 3

The distribution of largest-weight words and the largest-weight words after simulations of time $t={10}^3$. Calculations were made for $n=500$, $r={10}^3$, $ϵ={10}^{-5}$, and $l=10$. Different plotting symbols (circles; crosses) correspond to different agents. Quite often both agents have in some inventories (that usually corresponds to the same object) the same largest- and second-largest-weight words and in such a case the plotted symbols overlap.

Figure 4

Homonymy occurs when a word (1244) can be associated with more than one object. Synonymy occurs when an object is associated with more than one word.

Figure 5

The time evolution of the number of different largest-weight words for $n=500$, $l=10$, and $ϵ={10}^{-5}$. Small squares at the $t=0$ axis indicate the values for the randomly drawn words (see text). One can notice that during simulations a redistribution of largest-weight words takes place and that reduces the number of homonyms in the language. However, the number of largest-weight words saturates below $n$ and that shows that homonyms are a persistent feature of language. For large range $r$ the (almost) homonymy-free language is obtained.

Figure 6

The time evolution of the success rate of utterances with largest- and second-largest-weight words. Relatively large success rate of utterances with second-largest weight words indicates that more than one word can be associated with some objects; i.e., some words can be treated as synonyms. Increasing in time fluctuations of the second-largest weight data are due to poor statistics caused by the decreasing number of such utterances (i.e., synonymy decreases over time). Simulations were made for $n=500$, $r={10}^3$, $l=10$, and $ϵ={10}^{-5}$.

Figure 7

The average number of intervals $N\left(d\right)$ of a given distance $d$ between neighboring largest-weight words. Calculations were made for $n=500$, $r={10}^3$, $l=10$, and $ϵ={10}^{-5}$. These data show that noise greatly magnifies redistribution of words so that they are more evenly distributed within the available range (overlaps and large distances between words are much less likely, comparing to the distribution where they are selected independently).

Figure 8

The distribution of largest-weight words and the second-largest-weight words after $t={10}^3$ simulations. Calculations were made for $n=500$, $r={10}^3$, $l=10$, $ϵ={10}^{-5}$, and $\eta =0.05$. Different plotting symbols correspond to different agents.