### Abstract

Similarly to the popular voter model, the Deffuant model describes opinion dynamics taking place in spatially structured environments represented by a connected graph. Pairs of adjacent vertices interact at a constant rate. If the opinion distance between the interacting vertices is larger than some confidence threshold ε > 0, then nothing happens, otherwise, the vertices' opinions get closer to each other. It has been conjectured based on numerical simulations that this process exhibits a phase transition at the critical value ε_{c} = 1/2. For confidence thresholds larger than one half, the process converges to a global consensus, whereas coexistence occurs for confidence thresholds smaller than one half. In this article, we develop new geometrical techniques to prove this conjecture.

Original language | English (US) |
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Pages (from-to) | 383-402 |

Number of pages | 20 |

Journal | Alea |

Volume | 9 |

Issue number | 2 |

State | Published - Dec 1 2012 |

### Keywords

- Interacting particle system
- Random walks
- Social dynamics.

### ASJC Scopus subject areas

- Statistics and Probability