  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.
