P\'olya urns on hypergraphs

TL;DR

This paper investigates Pólya urn processes on hypergraphs, establishing almost sure convergence to a specific point under certain conditions related to the hypergraph's incidence matrix.

Contribution

It introduces new convergence results for Pólya urns on hypergraphs, especially highlighting the role of the incidence matrix's injectivity.

Findings

Convergence to a point v(H) when the incidence matrix is injective

Partial results for non-injective incidence matrices

Conditions for almost sure convergence of the process

Abstract

We study P\'olya urns on hypergraphs and prove that, when the incidence matrix of the hypergraph is injective, there exists a point $v=v(H)$ such that the random process converges to $v$ almost surely. We also provide a partial result when the incidence matrix is not injective.