In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements of SL(2, <C) as generators: R and T that are hyperbolic rotations of — and — (JV = 2,3,4,...), around two distinct points. The limit set of the discrete 3 N group generated by R and T provides, for N = 7,8,9,... a generalization of ...

The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. However, since it was introduced by Beardon in 1998, it has remained an open question what its isometries are. Beardon first raised this question and asked if the Apollonian isometries were just Möbius mappings. In this paper we show that this is the case in open sets with regular, f...

We propose a data replication scheme on a random apollonian P2P overlay that benefits from the small world and scale free properties. The proposed algorithm features a replica density estimation and a space filling mechanism designed to avoid redundant messages. Not only it provides uniform replication of the data stored into the network but it also improves on classical flooding approaches by ...

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to c log t, where t is the number of vertices, and c = c(k) is given as the solution to a t...