A Study of the Probability Distribution of the Balls in Bins Process with Power Law Feedback

Abstract

We analyse the balls in bins process with feedback with primary focus on the power law feedback function $f(\omega )=\eta {\omega }^{\gamma }$, $\eta >0$ $\gamma \ge 0$. Using the recursive solution to the master equation we find for power law feedback numerical evidence that the probability mass function for finite time scales asymptotically with ${\omega }^{-\gamma }$ for $\gamma >1$. We also provide simulations supporting a previous result by Oliveira (corollary to Theorem 4 in \cite{oliveira2009onset}) that the tail of the losers scale as ${\omega }^{-(\gamma -1)}$ but extending to $N\ge 2$ bins. We thus find evidence that the balls in bins process with power law feedback produces power law distributions as is common to many real world phenomena.