Apollonian Networks: Simultaneously Scale-Free, Small World, Euclidean, Space Filling, and with Matching Graphs

Abstract

We introduce a new family of networks, the Apollonian networks, that are simultaneously scale-free, small-world, Euclidean, space filling, and with matching graphs. These networks describe force chains in polydisperse granular packings and could also be applied to the geometry of fully fragmented porous media, hierarchical road systems, and area-covering electrical supply networks. Some of the properties of these networks, namely, the connectivity exponent, the clustering coefficient, and the shortest path are calculated and found to be particularly rich. The percolation, the electrical conduction, and the Ising models on such networks are also studied and found to be quite peculiar. Consequences for applications are also discussed.

On Certain Properties of Random Apollonian Networks

“…Specifically, the circle size distribution follows a power law with exponent around 1.3 [11]. Apollonian Networks (ANs) were introduced in [4] and independently in [20]. Zhou et al [35] introduced Random Apollonian Networks (RANs).…”

Ordinary percolation with discontinuous transitions

“…Indeed, there has been considerable interest in the properties of spatial networks, linking real-world geometry with small-world effects [3][4][5] . In particular, networks possessing hierarchical features 4,[6][7][8][9][10] relate to actual transport systems such as for air travel, routers and social interactions. Certain hierarchical networks, with a self-similar structure, have been shown to exhibit novel features 8,[11][12][13][14][15] .…”

Self-similar non-clustered planar graphs as models for complex networks

“…The first construction can be generalized to any d, see [10]. For d = 3, and related to the classical Apollonian packing of circles, Andrade et al and Doye and Massen introduced and studied the so called Apollonian networks [11,12]. These networks are also k-trees but new vertices are attached only to cliques which have never been selected in a former iteration.…”