Power laws and power-of-two-choices

TL;DR

This paper explores a variation of the 'power of two choices' algorithm, focusing on selecting the largest among randomly chosen options, revealing a power-law distribution with predictable concentration properties.

Contribution

It introduces and analyzes the behavior of a novel selection process that results in a power-law distribution, extending the understanding of choice-based allocation algorithms.

Findings

The $i^{th}$-smallest value scales as $i^{d-1}$ with high probability.

The distribution is concentrated around its expected value.

Provides a formula for the expectation of the distribution.

Abstract

This paper analyzes a variation on the well-known "power of two choices" allocation algorithms. Classically, the smallest of $d$ randomly-chosen options is selected. We investigate what happens when the largest of $d$ randomly-chosen options is selected. This process generates a power-law-like distribution: the $i^{th}$-smallest value scales with $i^{d-1}$, where $d$ is the number of randomly-chosen options, with high probability. We give a formula for the expectation and show the distribution is concentrated around the expectation