Hierarchical coefficient of a multifractal based network

• There are many classes of networks in the market, each one with their peculiarities.
• We work with the Lucena network—the dual of a multifractal lattice.
• The multifractal is by construction a proper partition of a square according to vertical and horizontal sections.
• The Lucena network shows the hierarchical property, a power-law relation between clustering coefficient and connectivity.
• We work a mathematical demonstration connecting clustering coefficient and connectivity for any scale-free planar network.

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