Degree distributions in networks: Beyond the power law

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 preselected 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 modeling framework that combines two different generalizations of the power law, namely the generalized 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.