Strongly reinforced Pólya urns with graph-based competition

Remco van der Hofstad, Mark Holmes, Alexey Kuznetsov, Wioletta Ruszel

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)


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 languageEnglish
Pages (from-to)2494-2539
Number of pages46
JournalAnnals of Applied Probability
Issue number4
Publication statusPublished - 2016


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

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