Multiple colour interacting urns on complete graphs

TL;DR

This paper generalizes a graph interacting urn model to multiple colours, demonstrating that on complete graphs, the colour proportions in urns converge almost surely to fixed points of the reinforcement functions.

Contribution

It extends previous models to multiple colours and proves convergence of colour proportions on complete graphs for a broad class of reinforcement functions.

Findings

Colour proportions converge almost surely to fixed points

Results apply to a broad class of reinforcement functions

Convergence holds specifically on complete graphs

Abstract

We present a multiple colour generalisation of the model of graph interacting urns studied by Benaim et. al., Random Struct. Alg., 46: 614-634, 2015. We show that for complete graphs and for a broad class of reinforcement functions governing the addition of balls in the urns, the process of colour proportions at each urn converges almost surely to the fixed points of the reinforcement function.