Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

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