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Abstract
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 of a model network 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. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.
Highlights
It is often convenient to present big volumes of data as a graph, i.e., as a set of objects and binary relations between them
The behavior of the distribution for the small and large k is controlled by the behavior of the generating function in the vicinity of λ = 0 and λ = 1, respectively
This paper presents one possible class of planar random graphs constructed from polygons with an even number of edges using a procedure similar to the construction of Apollonian graphs [18]
Summary
It is often convenient to present big volumes of data as a graph, i.e., as a set of objects and binary relations (bonds) between them. Despite being such a beautiful and well-studied object, the Apollonian network has certain drawbacks as a model of real networks Most importantly, it is a single deterministic object with certain fixed properties, e.g., a fixed degree distribution with a fixed power law exponent γ = ln 3/ ln 2. To the best of our knowledge, random Apollonian graphs remain the only scale-free planar graph model with a continuously growing size for which the exact degree distribution exponent is known. Another way of generalizing the network is to consider the k-simpliceswith k > 3 as building blocks of the network construction procedure. Summarizes the results of the paper and discusses further open questions and possible generalizations
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