Formation of vocabularies in a decentralized graph-based approach to human language

Zipf's law establishes a scaling behavior for word frequencies in large text corpora. The appearance of Zipfian properties in vocabularies (viewed as an intermediate phase between referentially useless one-word systems and one-to-one word-meaning vocabularies) has been previously explained as an optimization problem for the interests of speakers and hearers. Remarkably, humanlike vocabularies can be viewed also as bipartite graphs. Thus, the aim here is double: within a bipartite-graph approach to human vocabularies, to propose a decentralized language game model for the formation of Zipfian properties. To do this, we define a language game in which a population of artificial agents is involved in idealized linguistic interactions. Numerical simulations show the appearance of a drastic transition from an initially disordered state towards three kinds of vocabularies. Our results open ways to study Zipfian properties in language, reconciling models seeing communication as a global minima of information entropic energies and models focused on self-organization.

Figure 1

Measures to describe the formation of vocabularies. Two quantities are shown against the ambiguity parameter $\wp$: average population clustering $cc\left(t_f\right)$ and effective vocabulary $V\left(t_f\right)$. Averages over 10 realizations. Vertical lines indicate standard deviation. As shown in the figure, three domains for the evolution of bipartite word-meaning mappings tend to appear: full, Zipfian-like, and single-word vocabularies.

Figure 2

Phase transition in bipartite graphs. Orange circles indicate meanings, whereas blue x's indicate words. From top to bottom, we choose three bipartite graphs corresponding, respectively, to $\wp$ in $\{0.1,0.52,1\}$. Node positions are based on a python [27] implementation of the Fruchterman-Reingold algorithm [29].

Figure 3

Average bipartite clustering vs time. For one player $k\in P$, the behavior of average clustering $cc\left(B^k\right)$ over time is described. Circles indicate measurements every ${10}^4$ speaker-hearer interactions (starting from $t=0$). For the initial condition, the player $k\in P$ is associated to a bipartite graph $B^k$, where $B_{ij}^k=1$ or $B_{ij}^k=0$ with probability 0.5. $\wp$ is varied from $\{0,0.4,0.5,0.6,1\}$.

Figure 4

Maximum degree over time. For one player $k\in P$, the behavior of the maximum degree for word nodes $d_W^{max}$ and the maximum degree for meaning nodes $d_M^{max}$ over time is described. Points indicate measurements every ${10}^4$ speaker-hearer interactions (starting from $t=0$). For the initial condition, the player $k\in P$ is associated to a bipartite graph $B^k$, where $B_{ij}^k=1$ or $B_{ij}^k=0$ with probability 0.5. $\wp$ is varied from $\{0.1,0.52,1\}$.