Scale-free Networks Well Done

TL;DR

This paper rigorously defines power-law distributions in networks, introduces statistically consistent estimators for their exponents, and demonstrates that real-world networks are not perfect power laws but exhibit deviations.

Contribution

It provides a rigorous definition of power-law distributions, introduces new consistent estimators based on extreme value theory, and applies them to real data to challenge previous assumptions.

Findings

Real-world networks deviate from pure power laws.

New estimators are statistically consistent.

Scale-free networks are more common than previously thought.

Abstract

We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distributions that are widely used in statistics and other fields. This definition allows the distribution to deviate from a pure power law arbitrarily but without affecting the power-law tail exponent. We then identify three estimators of these exponents that are proven to be statistically consistent -- that is, converging to the true value of the exponent for any regularly varying distribution -- and that satisfy some additional niceness requirements. In contrast to estimators that are currently popular in network science, the estimators considered here are based on fundamental results in extreme value theory, and so are the proofs of their consistency. Finally, we apply these estimators to a representative collection of synthetic and real-world data. According to their estimates, real-world scale-free networks are definitely not as rare as one would conclude based on the popular but unrealistic assumption that real-world data comes from power laws of pristine purity, void of noise and deviations.