How rare are power-law networks really?

I Artico,E C Wit,V Vinciotti,I Smolyarenko

doi:10.1098/rspa.2019.0742

Abstract

The putative scale-free nature of real-world networks has generated a lot of interest in the past 20 years: if networks from many different fields share a common structure, then perhaps this suggests some underlying ‘network law’. Testing the degree distribution of networks for power-law tails has been a topic of considerable discussion. Ad hoc statistical methodology has been used both to discredit power-laws as well as to support them. This paper proposes a statistical testing procedure that considers the complex issues in testing degree distributions in networks that result from observing a finite network, having dependent degree sequences and suffering from insufficient power. We focus on testing whether the tail of the empirical degrees behaves like the tail of a de Solla Price model, a two-parameter power-law distribution. We modify the well-known Kolmogorov–Smirnov test to achieve even sensitivity along the tail, considering the dependence between the empirical degrees under the null distribution, while guaranteeing sufficient power of the test. We apply the method to many empirical degree distributions. Our results show that power-law network degree distributions are not rare, classifying almost 65% of the tested networks as having a power-law tail with at least 80% power.

Highlights

Networks play an important role in many fields, from epidemiology and ecology to engineering and sociology
(i) we will introduce the exact asymptotic definition of a power-law degree distribution and relate this to the problem of observing only finite networks; (ii) we explain how the dependency of a single empirical degree sample affects the distribution of a KS test statistic and (iii) we show how asymptotic tests must balance the delicate equilibrium between power of the test and the asymptotic power-law property
We have developed a tail testing procedure, taking into account a host of issues related to testing degree distributions of a single empirical network

Summary

Introduction

Networks play an important role in many fields, from epidemiology and ecology to engineering and sociology. Various forms of preferential attachment rules have been shown to result in network structures whereby the degree sequences are generally described by ratios of gamma functions [9], i.e. power-laws This putative universality of the power-law degree distributions sets it up as a natural paradigm for falsification [10], i.e. as a natural null hypothesis. Even though a number of studies have considered testing for power-law degree distributions in empirical networks, the final verdict is still open This current paper takes a complementary view to Voitalov et al [8]: we make stronger parametric assumptions about the asymptotic form of the tail of the degree distribution, avoiding the impossibility arguments [8, Section V], in order to get a lower-bound on the fraction of empirical networks that exhibit power-law behaviour.

Issues in testing empirical degree distributions

Testing framework

Testing 4482 network for power-law degree distributions

Findings

Conclusion