Electronic Journal of Probability

The naming game on the complete graph

Eric Foxall

Full-text: Open access

Abstract

We consider a model of language development, known as the naming game, in which agents invent, share and then select descriptive words for a single object, in such a way as to promote local consensus. When formulated on a finite and connected graph, a global consensus eventually emerges in which all agents use a common unique word. Previous numerical studies of the model on the complete graph with $n$ agents suggest that when no words initially exist, the time to consensus is of order $n^{1/2}$, assuming each agent speaks at a constant rate. We show rigorously that the time to consensus is at least $n^{1/2-o(1)}$, and that it is at most constant times $\log n$ when only two words remain. In order to do so we develop some useful estimates for semimartingales with bounded jumps.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 126, 42 pp.

Dates
Received: 1 March 2017
Accepted: 23 November 2018
First available in Project Euclid: 19 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1545188693

Digital Object Identifier
doi:10.1214/18-EJP250

Mathematical Reviews number (MathSciNet)
MR3896863

Zentralblatt MATH identifier
07021682

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

Keywords
interacting particle system naming game

Rights
Creative Commons Attribution 4.0 International License.

Citation

Foxall, Eric. The naming game on the complete graph. Electron. J. Probab. 23 (2018), paper no. 126, 42 pp. doi:10.1214/18-EJP250. https://projecteuclid.org/euclid.ejp/1545188693


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