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