Multifractal analysis of complex networks

TL;DR

This paper introduces a new multifractal analysis method for complex networks, revealing multifractality in certain network types and providing insights into their spatial heterogeneity.

Contribution

It presents a novel box covering algorithm for multifractal analysis and applies it to various theoretical and real-world networks.

Findings

Multifractality exists in scale-free and protein-protein interaction networks.

Multifractal behavior is less clear in small world and random networks.

Parameter changes in network models affect the generalized fractal dimensions.

Abstract

Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal dimensions. Multifractal analysis is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we introduce a new box covering algorithm for multifractal analysis of complex networks. This algorithm is used to calculate the generalized fractal dimensions $D_{q}$ of some theoretical networks, namely scale-free networks, small world networks and random networks, and one kind of real networks, namely protein-protein interaction networks of different species. Our numerical results indicate the existence of multifractality in scale-free networks and protein-protein interaction networks, while the multifractal behavior is not clear-cut for small world networks and random networks. The possible variation of $D_{q}$ due to changes in the parameters of the theoretical network models is also discussed.