A Fractal Space-filling Complex Network

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We study in this work the properties of the Qmf 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→ 1 the tiling degenerates into the square lattice that is associated with a regular network. The Qmf 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 Qmf network satisfy the relation N ∝ ℓᵈᶠ where ℓ is a typical length of the network (the average minimal distance) and N the network size. We call df the fractal dimension of the network. In tne limit case ρ→ 1 we have df → 2.