Hierarchical coefficient of a multifractal based network

The hierarchical property for a general class of networks stands for a power-law relation between clustering coefficient, CC and connectivity k: CC∝kβ. 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 CCimax of a vertex i in a planar graph. This condition expresses that the number of links among neighbour, N△, of a vertex i is equal to its connectivity ki, that means: N△=ki. The Lucena network fulfils the condition N△≃ki independent of ki and the anisotropy of ML. In addition, CCmax implies the threshold β=1 for the hierarchical property for any scale-free planar network.