Apollonian networks
Hans J. Herrmann
ICA1, University of Stuttgart
Pfaffenwaldring 27, 70569 Stuttgart, Germany
We introduce a new family of networks, the Apollonian networks, that
are simultaneously scale-free, small world, Euclidean, space-filling and
matching graphs. These networks have a wide range of applications ranging
from the description of force chains in polydisperse granular packings and
geometry of fully fragmented porous media, to 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.
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