Emergence of Zipf’s law in the evolution of communication

Zipf’s law seems to be ubiquitous in human languages and appears to be a universal property of complex communicating systems. Following the early proposal made by Zipf concerning the presence of a tension between the efforts of speaker and hearer in a communication system, we introduce evolution by means of a variational approach to the problem based on Kullback’s Minimum Discrimination of Information Principle. Therefore, using a formalism fully embedded in the framework of information theory, we demonstrate that Zipf’s law is the only expected outcome of an evolving communicative system under a rigorous definition of the communicative tension described by Zipf.

Figure 1

A growing communication system. In (a), possible meaning-signal associations made by the coder module $\mathbf{P}$, in which Eq. (2) holds, is depicted. In (b), we summarize the evolution rules of our communicative system. Suppose that symmetry between coder and decoder [i.e., Eq. (2)] holds for the step $n$ (above). At each step (below) a new element is added to the set $\Omega$ and Eq. (2) holds again for this new configuration. Furthermore, the new configuration is constrained by the MDIP, which introduces a path dependency in the evolutionary process.

Figure 2

Numerical simulation of the final distribution $q_n$ ($n={10}^4$) obtained by constraining the growth process with (i) the consequences of the symmetry of coding-decoding [see Eq. (2)] provided by Eq. (6) and (ii) the application of the MDIP at every step of the growth process. Different convergence values are studied: (a) $\nu =0.2$, (b) $\nu =0.3$, and (c) $\nu =0.5$. The dashed lines are the best-fit interpolations, which give estimated exponents $\gamma =1.06$, 1.04, and 1.01, respectively (all with correlation coefficients $r<-0.99$).