## Electronic Journal of Probability

### The naming game on the complete graph

Eric Foxall

#### 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
Accepted: 23 November 2018
First available in Project Euclid: 19 December 2018

https://projecteuclid.org/euclid.ejp/1545188693

Digital Object Identifier
doi:10.1214/18-EJP250

Mathematical Reviews number (MathSciNet)
MR3896863

Zentralblatt MATH identifier
07021682

#### 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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