## The Annals of Applied Probability

### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2494-2539.

Dates
Revised: July 2015
First available in Project Euclid: 1 September 2016

https://projecteuclid.org/euclid.aoap/1472745465

Digital Object Identifier
doi:10.1214/16-AAP1153

Mathematical Reviews number (MathSciNet)
MR3543903

Zentralblatt MATH identifier
1352.60132

#### Citation

van der Hofstad, Remco; Holmes, Mark; Kuznetsov, Alexey; Ruszel, Wioletta. Strongly reinforced Pólya urns with graph-based competition. Ann. Appl. Probab. 26 (2016), no. 4, 2494--2539. doi:10.1214/16-AAP1153. https://projecteuclid.org/euclid.aoap/1472745465

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