Multifractality of complex networks is also due to geometry. The Geometric SandBox algorithm

TL;DR

This paper introduces a new multifractal analysis method for complex networks that incorporates node positions, revealing the impact of geometry on the network's fractal structure, unlike traditional methods.

Contribution

The authors propose a geometric multifractal analysis method that considers node coordinates, enhancing the understanding of network complexity beyond connection patterns.

Findings

Method is sensitive to changes in network geometry.

Reveals geometric influence on multifractal properties.

Effective on both synthetic and real-world networks.

Abstract

Over the past three decades, describing the reality surrounding us using the language of complex networks has become very useful and therefore popular. One of the most important features, especially of real networks, is their complexity, which often manifests itself in a fractal or even multifractal structure. As a generalisation of fractal analysis, multifractal analysis of complex networks is a useful tool for the identification and quantitative description of the spatial hierarchy of both theoretical and numerical fractal patterns. Nowadays, there are many methods of multifractal analysis. However, all these methods take into account only the fact of connection between nodes (and eventually the weight of edges) and do not take into account the real positions (coordinates) of nodes in space. However, intuition suggests that the geometry of network nodes' position should have a significant impact on its true fractal structure. Many networks identified in nature (e.g. air connection networks, energy networks, social networks, mountain ridge networks, networks of neurones in the brain, street networks) have their own often unique and characteristic geometry, which is not taken into account in the identification process of multifractality in commonly used methods. In this paper, we propose a multifractal network analysis method that takes into account both connections between nodes and the location coordinates of nodes (network geometry). We show the results for different geometrical variants of the same network and reveal that this method, contrary to the commonly used method, is sensitive to changes in network geometry. We carry out tests for synthetic as well as for real-world networks.