Polygon-based hierarchical planar networks based on generalized Apollonian construction

Experimentally observed complex networks are often scale-free, small-world and have unexpectedly large number of small cycles. Apollonian network is one notable example of a model network respecting simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. Here we present a similar construction based on consequential splitting of tetragons and other polygons with even number of edges. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs, in the limit of large number of splittings the degree distribution of the graph converges to a true power law with exponent, which is smaller than 3 in the case of tetragons, and larger than 3 for polygons with larger number of edges. We show that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.