Empirical Tests of Zipf's law Mechanism In Open Source Linux Distribution

TL;DR

This paper empirically tests the mechanisms behind Zipf's law in Linux distributions, confirming proportional growth, specific variance scaling, and tail behavior of package links over decades.

Contribution

It provides three empirical tests validating key assumptions of stochastic growth models explaining Zipf's law in open source Linux software.

Findings

Growth of package links follows Gibrat's law.

Variance of growth scales with the square root of time.

Tail of new package link distribution converges to Zipf's law exponent 1.

Abstract

The evolution of open source software projects in Linux distributions offers a remarkable example of a growing complex self-organizing adaptive system, exhibiting Zipf's law over four full decades. We present three tests of the usually assumed ingredients of stochastic growth models that have been previously conjectured to be at the origin of Zipf's law: (i) the growth observed between successive releases of the number of in-directed links of packages obeys Gibrat's law of proportional growth; (ii) the average growth increment of the number of in-directed links of packages over a time interval $\Delta t$ is proportional to $\Delta t$, while its standard deviation is proportional to $\sqrt{\Delta t}$; (iii) the distribution of the number of in-directed links of new packages appearing in evolving versions of Debian Linux distributions has a tail thinner than Zipf's law, with an exponent which converges to the Zipf's law value 1 as the time $\Delta t$ between releases increases.