Planar unclustered graphs to model technological and biological networks

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases usually associated with topological restrictionstheir clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular technological and biological networks. PACS. 89.75.Hc Networks and genealogical trees – 05.45.Df Fractals – 89.75.Fb Structures and organization in complex systems