Degree distributions in networks: beyond the power law

TL;DR

This paper introduces a new modeling framework combining generalized Pareto and Zipf-polylog distributions to better fit network degree data and address limitations of the traditional power law, including threshold selection and adequacy testing.

Contribution

It proposes a mixture distribution approach with Bayesian inference for more accurate modeling and hypothesis testing of network degree distributions beyond the power law.

Findings

Mixture models fit degree data well

Quantifies threshold uncertainty naturally

Provides a Bayesian model selection method

Abstract

The power law is useful in describing count phenomena such as network degrees and word frequencies. With a single parameter, it captures the main feature that the frequencies are linear on the log-log scale. Nevertheless, there have been criticisms of the power law, for example that a threshold needs to be pre-selected without its uncertainty quantified, that the power law is simply inadequate, and that subsequent hypothesis tests are required to determine whether the data could have come from the power law. We propose a modelling framework that combines two different generalisations of the power law, namely the generalised Pareto distribution and the Zipf-polylog distribution, to resolve these issues. The proposed mixture distributions are shown to fit the data well and quantify the threshold uncertainty in a natural way. A model selection step embedded in the Bayesian inference algorithm further answers the question whether the power law is adequate.