A Fractal Space-filling Complex Network

Soares, D. J. B.; Filho, J. Ribeiro; Moreira, A. A.; Moreira, D. A.; Corso, G.

A Fractal Space-filling Complex Network

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Description

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 $ρ$, in the limit of $ρ\to 1$ 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 $ρ\to 1$ we have $d_{f} \to 2$.
10 pages, 5 figures and 1 table