  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.

Statistical Mechanics