## Abstract

The voter model is a classical interacting particle system modelling how consensus is formed across a network. We analyze the time to consensus for the voter model when the underlying graph is a subcritical scale-free random graph. Moreover, we generalize the model to include a “temperature” parameter controlling how the graph influences the speed of opinion change. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. Finally, we also consider a discursive voter model, where voters discuss their opinions with their neighbors. Our proofs rely on the well-known duality to coalescing random walks and a detailed understanding of the structure of the random graphs.

Original language | English |
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Pages (from-to) | 376-429 |

Number of pages | 59 |

Journal | Random Structures and Algorithms |

Volume | 62 |

Issue number | 2 |

Early online date | 23 Jul 2022 |

DOIs | |

Publication status | Published - Mar 2023 |

### Bibliographical note

Funding Information:We would like to thank Peter Mörters and Alexandre Stauffer for many useful discussions. JF was supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1.

## Keywords

- inhomogeneous random graphs
- interacting particle systems
- random walks on random graphs
- scale-free networks
- voter model

## ASJC Scopus subject areas

- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics