## Abstract

We introduce a class of reinforcement models where, at each time step t, one first chooses a random subset A_{t} 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 | The Annals of Applied Probability |

Volume | 26 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Aug 2016 |

## Keywords

- Pólya urn
- Reinforcement model
- Stable equilibria.
- Stochastic approximation algorithm