Building complex networks with Platonic solids

We propose a unified model to build planar graphs with diverse topological characteristics which are of relevance in real applications. Here convex regular polyhedra (Platonic solids) are used as the building blocks for the construction of a variety of complex planar networks. These networks are obtained by merging polyhedra face by face on a tree-structure leading to planar graphs. We investigate two different constructions: (1) a fully deterministic construction where a self-similar fractal structure is built by using a single kind of polyhedron which is iteratively attached to every face and (2) a stochastic construction where at each step a polyhedron is attached to a randomly chosen face. These networks are scale-free, small-world, clustered, and sparse, sharing several characteristics of real-world complex networks. We derive analytical expressions for the degree distribution, the clustering coefficient, and the mean degree of nearest neighbors showing that these networks have power-law degree distributions with tunable exponents associated with the building polyhedron, and they possess a hierarchical organization that is determined by planarity.