### Abstract

We introduce a class of reinforcement models where, at each time step t, one first chooses a random subset At of colours (independently of the past) from n colours of balls, and then chooses a colour i from this subset with probability proportional to the number of balls of colour i in the urn raised to the power α>1. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph; a context which is a toy model for the formation and reinforcement of neural connections. We conjecture that for any graph G and all α sufficiently large, the set of stable equilibria is supported on so-called whisker-forests, which are forests whose components have diameter between 1 and 3.

Original language | English |
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Pages (from-to) | 2494-2539 |

Number of pages | 46 |

Journal | Annals of Applied Probability |

Volume | 26 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 |

### Keywords

- Reinforcement model
- Pólya urn
- stochastic approximation algorithm
- stable equilibria

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## Cite this

van der Hofstad, R., Holmes, M., Kuznetsov, A., & Ruszel, W. (2016). Strongly reinforced Pólya urns with graph-based competition.

*Annals of Applied Probability*,*26*(4), 2494-2539. https://doi.org/10.1214/16-AAP1153