Reinforced Polya Urns

03/31/2016 - 16:00
03/31/2016 - 17:00
Alexey Kuznetsov, York University
McGill, Burnside Hall 1205

 Consider the following toy model for neural processing in the brain: a large number of neurons are interconnected by synapses, and the brain removes connections which are seldom or never used and reinforces those which are stimulated. We introduce a class of reinforced Polya urn models which aim to describe this dynamics. Our models work as follows: at each time step t, we first choose a random subset A_t of colours (independently of the past) from n colours of balls, and then choose a colour i from the subset A_t with probability proportional to the number of balls of colour i in the urn raised to the power alpha>1. We are mostly interested in stability of equilibria for such models studying phase transitions in a number of examples, including when the colours are the edges of a graph. 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.

This talk is based on joint work with Remco van der Hofstad, Mark Holmes and Wioletta Ruszel.

Last edited by on Thu, 03/24/2016 - 15:03