Scaling theory of fractal complex networks

TL;DR

This paper develops a comprehensive scaling theory for fractal complex networks, linking their local self-similarity and global scale-invariance through new exponents, verified across real-world and model networks.

Contribution

It introduces a unified framework connecting local and global properties of fractal networks using scale-invariant equations and new scaling exponents.

Findings

Fractality arises from geometric self-similarity in hierarchical structures.

New relationships between microscopic and macroscopic exponents are established.

The theory is validated on diverse real-world and model networks.

Abstract

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach - grounded in both scaling theory of phase transitions and renormalization group theory - leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information - the World Wide Web, biological - the human brain, and social - scientific collaboration networks) and in several fractal network models.