Greedy algorithms and Zipf laws

TL;DR

This paper analyzes a growth model based on greedy selection among entities, revealing conditions under which the size distribution follows Zipf's law, with analytical solutions for specific cases and insights into the dynamics of convergence.

Contribution

It provides analytical solutions for a multi-item growth model, showing how Zipf's law emerges under certain conditions and introducing a regularization method for the case of full item comparison.

Findings

Zipf's law arises when more than half of the items are considered in the selection.

No selection occurs and distributions are thin-tailed when half or fewer items are compared.

The model's convergence dynamics resemble an aging process described by Barrat & Me9zard.

Abstract

We consider a simple model of firm/city/etc. growth based on a multi-item criterion: whenever entity B fares better that entity A on a subset of $M$ items out of $K$, the agent originally in A moves to B. We solve the model analytically in the cases $K=1$ and $K \to \infty$. The resulting stationary distribution of sizes is generically a Zipf-law provided $M > K/2$. When $M \leq K/2$, no selection occurs and the size distribution remains thin-tailed. In the special case $M=K$, one needs to regularise the problem by introducing a small "default" probability $\phi$. We find that the stationary distribution has a power-law tail that becomes a Zipf-law when $\phi \to 0$. The approach to the stationary state can also been characterized, with strong similarities with a simple "aging" model considered by Barrat & M\'ezard.