Strongly reinforced Pólya urns with graph-based competition

Journal article (2016)

Authors

Remco van der Hofstad Eindhoven University of Technology

Mark Holmes The University of Auckland

Alexey Kuznetsov York University

W.M. Ruszel Applied Probability -

Research Group

Applied Probability () (TU Delft)

Reinforcement model Pólya urn Stochastic approximation algorithm Stable equilibria

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Published Date

2016

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

Abstract

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 so-called whisker-forests, which are forests whose components have diameter between 1 and 3.