STRONGLY REINFORCED POLYA URNS WITH GRAPH-BASED COMPETITION
Remco van der Hofstad, Mark Holmes, Alexey Kuznetsov, Wioletta Ruszel
ANNALS OF APPLIED PROBABILITY | INST MATHEMATICAL STATISTICS | Published : 2016
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 ..View full abstract
Awarded by Netherlands Organisation for Scientific Research (NWO) through VICI Grant
Awarded by Gravitation NETWORKS Grant
Supported in part by the Netherlands Organisation for Scientific Research (NWO) through VICI Grant 639.033.806 and the Gravitation NETWORKS Grant 024.002.003.Supported in part by the Marsden Fund, administered by RSNZ.Supported by the Natural Sciences and Engineering Research Council of Canada.