Strongly reinforced Pólya urns with graph-based competition

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 $\alpha>1$. 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 $\alpha$ 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.