Strongly reinforced P\'olya urns with graph-based competition

TL;DR

This paper introduces a class of reinforced Polya urn models with graph-based competition, analyzing stability and phase transitions, relevant for neural connection formation.

Contribution

It develops a new reinforcement model with graph-based interactions and studies its stability and phase transitions, extending classical Polya urns.

Findings

Existence of phase transitions in the model

Stability analysis of equilibria

Application to neural connection modeling

Abstract

We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent 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 $\alpha>1$. 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.