The naming game in language dynamics revisited

TL;DR

This paper analyzes a biased naming game on graphs, determining conditions under which a new word can invade and become the dominant linguistic convention, depending on the graph structure and fitness ratio.

Contribution

It provides a rigorous analysis of the biased naming game, establishing invasion thresholds on different graph structures, including complete graphs and lattices, using probabilistic methods.

Findings

Invasion threshold on complete graphs: phi > 3.

Invasion threshold on 1D lattice: phi > 1.053.

Graph structure significantly affects invasion probability.

Abstract

This article studies a biased version of the naming game in which players located on a connected graph interact through successive conversations to bootstrap 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 pronounced by bilingual individuals. The limiting behavior depends on a single parameter: phi = 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. In the mean-field approximation, invasion of word A is successful if and only if phi > 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 phi > 1.053 indicating that the probability of invasion and the critical value for phi 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.