A Fractal Space-filling Complex Network

TL;DR

This paper introduces the $Q_{mf}$ network, a space-filling fractal network derived from a multifractal tiling, exhibiting power-law connectivity, high clustering, and fractal scaling properties.

Contribution

The study presents a novel space-filling fractal network model based on multifractal tiling, analyzing its structural properties and scaling behavior.

Findings

Network exhibits power-law degree distribution for k>7.

High clustering coefficient compared to random networks.

Network size scales with average distance as N ∝ ℓ^{d_f}.

Abstract

We study in this work the properties of the $Q_{mf}$ network which is constructed from an anisotropic partition of the square, the multifractal tiling. This tiling is build using a single parameter $\rho$, in the limit of $\rho \to 1$ the tiling degenerates into the square lattice that is associated with a regular network. The $Q_{mf}$ network is a space-filling network with the following characteristics: it shows a power-law distribution of connectivity for $k>7$ and it has an high clustering coefficient when compared with a random network associated. In addition the $Q_{mf}$ network satisfy the relation $N \propto \ell^{d_f}$ where $\ell$ is a typical length of the network (the average minimal distance) and $N$ the network size. We call $d_f$ the fractal dimension of the network. In tne limit case $\rho \to 1$ we have $d_{f} \to 2$.