  We introduce a class of reinforcement models where, at each time step $t$,
one first chooses a random subset $A_t$ of colours (independent 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.
