Journal of Applied Probability

The naming game in language dynamics revisited

Nicolas Lanchier

Abstract

In this article we study a biased version of the naming game in which players are located on a connected graph and interact through successive conversations in order to select a common name for a given object. Initially, all the players use the same word B except for one bilingual individual who also uses word A. Both words are attributed a fitness, which measures how often players speak depending on the words they use and how often each word is spoken by bilingual individuals. The limiting behavior depends on a single parameter, ϕ, denoting the ratio of the fitness of word A to the fitness of word B. The main objective is to determine whether word A can invade the system and become the new linguistic convention. From the point of view of the mean-field approximation, invasion of word A is successful if and only if ϕ > 3, a result that we also prove for the process on complete graphs relying on the optimal stopping theorem for supermartingales and random walk estimates. In contrast, for the process on the one-dimensional lattice, word A can invade the system whenever ϕ > 1.053, indicating that the probability of invasion and the critical value for ϕ strongly depend on the degree of the graph. The system on regular lattices in higher dimensions is also studied by comparing the process with percolation models.

Article information

Source
J. Appl. Probab., Volume 51A (2014), 139-158.

Dates
First available in Project Euclid: 2 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1417528472

Digital Object Identifier
doi:10.1239/jap/1417528472

Mathematical Reviews number (MathSciNet)
MR3317355

Zentralblatt MATH identifier
1322.60215

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 91D25: Spatial models [See also 91B72]

Keywords
Interacting particle system naming game language dynamics semiotic dynamics

Citation

Lanchier, Nicolas. The naming game in language dynamics revisited. J. Appl. Probab. 51A (2014), 139--158. doi:10.1239/jap/1417528472. https://projecteuclid.org/euclid.jap/1417528472


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References

  • Baronchelli, A., Dall'Asta, L., Barrat, A. and Loreto, V. (2006). Topology induced coarsening in language games. Phys. Rev. E 73, 015102.
  • Castellano, C., Fortunato, S. and Loreto, V. (2009). Statistical physics of social dynamics. Rev. Modern Phys. 81, 591–646.
  • Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.
  • Durrett, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Prob. 5, 127–152.
  • Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Saint-Flour, 1993; Lecture Notes Math. 1608), Springer, Berlin, pp. 97–201.
  • Harris, T. E. (1972). Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89.
  • Lanchier, N. (2013). Stochastic spatial model of producer-consumer systems on the lattice. Adv. Appl. Prob. 45, 1157–1181.
  • Steels, L. (1995). A self-organizing spatial vocabulary. Artif. Life J. 2, 319.