Viability of an elementary syntactic structure in a population playing naming games

We explore how the social dynamics of communication and learning can bring about the rise of a syntactic communication in a population of speakers. Our study is developed starting from a version of the Naming Game model in which an elementary syntactic structure is introduced. This analysis shows how the transition from nonsyntactic to syntactic communication is socially favored in communities which need to exchange a large number of concepts.

Figure 1

Example of the dynamics of the inventories for a one-concept model without syntax. On the left we present a successful game, on the right a failed one. We represent the 32 possible symbols with only five slots for the sake of clarity. When a symbol is not present in the inventory, the corresponding bit is set to 0; otherwise, when the symbol is present, the bit is set to 1. The shadowed elements are the ones transmitted by the speaker.

Figure 2

An example of the structure and evolution of the inventories for the model with syntax when four concepts ($S=2$) are exchanged. In this particular example, the number of inventories ($2S$) is equal to the number of concepts. We can think that the first two inventories represent nouns and the third and fourth stand for verbs. The starred inventories are the ones corresponding to the concept sorted out in a specific communication event. The shadowed elements are the ones transmitted by the speaker.

Figure 3

One-concept model. Top: model without syntax. We present the temporal evolution for the total number of words divided by the total population. The inset shows the success rate $R_S\left(t\right)$. Bottom: the same data for the syntactic model. All data are averaged over 1000 simulations.

Figure 4

We present the convergence time as a function of the population size for the one-concept model. Top: the model with no syntax is well fitted by the depicted relation $T_{ns}^1\approx -22.1P+7.6P\ln P$. The model with syntax is well fitted by the relation: $T_s^1\approx -213.2P+52.7P\ln P$. Bottom: rescaled convergence times for the model without syntax (on the left) and with syntax (on the right). The rescaled times are well fitted by the function $a_1+a_2\ln P$ (continuous lines). A power law fit of these data ($b_1{P}^{b_2}$), represented by the dashed lines, turns out to be moderately less accurate than the previous one, which can be derived from some theoretical considerations.

Figure 5

The convergence time as a function of $S$ for different populations. The linear regressions are very well approximated by the general expression $T_s^{\text{tot}}\approx 1.5S{T}_s^1\left(P\right)$ (the continuous lines). The inset shows the convergence time as a function of $P$ for different $S$ values. The convergence times for $S=4,9$ are rescaled by the factor $1.5S$. The continuous lines are the rescaled curves $c_{1}P+c_{2}P\ln P$ obtained from the regressions. The rescaling transformation produces a good collapse of data and curves. Data are averaged over up to 1000 simulations.

Figure 6

Temporal evolution of the success rate for a different number of exchanged concepts ($C$) for a model without syntax or with syntax. For $C\ge 81$, $T_s^{\text{tot}}<C{T}_{ns}^1$, which means a faster convergence for population using syntax. $P=25000$