The study of constant width sets goes at least as far back as the time of Euler. The Apollonian metric, on the other hand, is a relatively new concept. It was introduced by Beardon in 1998 as a generalization of the hyperbolic metric of a ball to arbitrary domains [3]. Close connections between these concepts were established in [20] and [21]. In this paper, we study the Apollonian metric of do...

Let ̂ H = H ∪ {∞} denote the standard one-point completion of a real Hilbert space H. Given any non-trivial proper subset U ⊂ ̂ H one may define the so-called “Apollonian” metric dU on U . When U ⊂ V ⊂ ̂ H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let Δ = diamV (U) ∈ [0,+∞] be the diameter of the smaller subsets ...

We present and study in this paper a simple algorithm that produces so called growing Parallel Random Apollonian Networks (P-RAN) in any dimension d. Analytical derivations show that these networks still exhibit small-word and scalefree characteristics. To characterize further the structure of P-RAN, we introduce new parameters that we refer to as the parallel degree and the parallel coefficien...

Abstract –In our empirical works, we find that there exists the equivalency of the cumulative degree distribution and edge-cumulative distribution. Furthermore, we employ three network models of the recursive graphs, Sierpinksi networks and Apollonian networks to verify our conjecture: Both the cumulative degree distribution and the edge-cumulative distribution are equivalent to each other in d...