The Truth about Power Laws: Theory and Reality

TL;DR

This paper challenges the universality of power laws in complex networks by proposing a theoretical framework, analyzing birth-and-death networks, and identifying factors affecting the observability of power law features.

Contribution

It introduces a generic theoretical framework for examining power laws, analyzes birth-and-death networks, and highlights the impact of network size and node disappearance on power law detection.

Findings

Birth-and-death networks exhibit power law tails in degree distributions.

Network size reduction and higher node disappearance probability hinder power law observation.

Observing the asymptotic behavior within effective intervals is an effective detection method.

Abstract

Consensus about the universality of the power law feature in complex networks is experiencing profound challenges. To shine fresh light on this controversy, we propose a generic theoretical framework in order to examine the power law property. First, we study a class of birth-and-death networks that is ubiquitous in the real world, and calculate its degree distributions. Our results show that the tails of its degree distributions exhibits a distinct power law feature, providing robust theoretical support for the ubiquity of the power law feature. Second, we suggest that in the real world two important factors, network size and node disappearance probability, point to the existence of the power law feature in the observed networks. As network size reduces, or as the probability of node disappearance increases, then the power law feature becomes increasingly difficult to observe. Finally, we suggest that an effective way of detecting the power law property is to observe the asymptotic (limiting) behaviour of the degree distribution within its effective intervals.