Polygon-based hierarchical planar networks based on generalized Apollonian construction

TL;DR

This paper introduces a stochastic, polygon-based hierarchical network model generalizing Apollonian networks, producing planar scale-free graphs with tunable degree distribution exponents through splitting tetragons and other polygons.

Contribution

It extends Apollonian network construction to polygons with even edges, enabling tunable degree exponents and new planar scale-free network models.

Findings

Degree distribution follows a power law with exponent < 3 for tetragons.

Exponent > 3 for polygons with more than four edges.

Mixing tetragon and hexagon constructions tunes the degree exponent.

Abstract

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.