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We study the Projection of Lucena Network PLN : a planar network whose nodes coincide with the sites of the square lattice. The PLN has one internal parameter ρ , for ρ → 1 the network resembles a symmetric regular lattice with almost none highly connected node while ρ → 0 is more asymmetric and there are several highly connected nodes. We estimate P ( r ) , the probability of a node be connected to another node at distance r . The PLN follows the fractal scaling P ( r ) ∝ r − δ with 3 < δ < 4 according to ρ . For ρ → 1 we have δ close to 4 which is a signature of short range interactions, while ρ → 0 shows δ → 3 and has long range interactions. In addition, the fractal dimension, d f , of the PLN behaves as: for ρ → 1 , d f = 2 , the dimension of the embedding space while for ρ → 0 , d f increases and the PLN shares some similarities with a small world graph. [ABSTRACT FROM AUTHOR]

Abstract: The hierarchical property for a general class of networks stands for a power-law relation between clustering coefficient, and connectivity : . This relation is empirically verified in several biologic and social networks, as well as in random and deterministic network models, in special for hierarchical networks. In this work we show that the hierarchical property is also present in a Lucena network. To create a Lucena network we use the dual of a multifractal lattice ML, the vertices are the sites of the ML and links are established between neighbouring lattices, therefore this network is space filling and planar. Besides a Lucena network shows a scale-free distribution of connectivity. We deduce a relation for the maximal local clustering coefficient of a vertex in a planar graph. This condition expresses that the number of links among neighbour, , of a vertex is equal to its connectivity , that means: . The Lucena network fulfils the condition independent of and the anisotropy of ML. In addition, implies the threshold for the hierarchical property for any scale-free planar network. [Copyright &y& Elsevier]