Time-dependent balls and bins model with positive feedback

TL;DR

This paper studies a time-dependent balls and bins model with positive feedback, revealing phase transitions and conditions for monopoly or balanced distribution of balls among bins.

Contribution

It introduces a time-dependent version of the feedback balls and bins model, analyzing phase transitions and monopoly conditions based on the growth of balls added over time.

Findings

For >1, one bin dominates almost completely.

A phase transition exists depending on the growth rate of _n.

In the case =1, no dominance occurs, contrasting with models where all new balls go to one bin.

Abstract

Balls and bins models are classical probabilistic models where balls are added to bins at random according to a certain rule. The balls and bins model with feedback is a non-linear generalisation of the P\'olya urn, where the probability of a new ball choosing a bin with $m$ balls is proportional to $m^{\alpha}$, with $\alpha$ being the feedback parameter. It is known that if the feedback is positive (i.e. $\alpha>1$) then the model is monopolistic: there is a finite time after which one of the bins will receive all incoming balls. We consider a time-dependent version of this model, where $\sigma_n$ independent balls are added at time $n$ instead of just one. We show that if $\alpha>1$ then one of the bins gets all but a negligible number of balls, and identify a phase transition in the growth of $(\sigma_n)$ between the monopolistic and non-monopolistic behaviour. We also describe the critical regime, where the probability of monopoly is strictly between zero and one. Finally, we show that in the feedback-less case $\alpha=1$ no dominance occurs, that is, each bin gets a non-negligible proportion of balls eventually. This is in sharp contrast with a similar model where new balls added at time $n$ are all placed in the same bin rather than independently.