High dimensional random Apollonian networks

We propose a simple algorithm which produces high dimensional Apollonian networks with both small-world and scale-free characteristics. We derive analytical expressions for the degree distribution, the clustering coefficient and the diameter of the networks, which are determined by their dimension.

We introduce a general deterministic model for Apollonian Networks in an iterative fashion. The networks have small-world effect and scale-free topology. We calculate the exact results for the degree exponent, the clustering coefficient and the diameter. The major points of our results indicate that (a) the degree exponent can be adjusted in a wide range, (b) the clustering coefficient of each individual vertex is inversely proportional to its degree and the average clusterin...

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 scale-free characteristics. To characterize further the structure of P-RAN, we introduce new parameters that we refer to as the parallel degree and the parallel coefficient, that determine locally and in average the number of vertices inside the (d+1)-c...

In this article, we investigate several properties of high-dimensional random Apollonian networks (HDRANs), including two types of degree profiles, the small-world effect (clustering property), sparsity, and three distance-based metrics. The characterizations of degree profiles are based on several rigorous mathematical and probabilistic methods, such as a two-dimensional mathematical induction, analytic combinatorics, and P\'{o}lya urns, etc. The small-world property is unco...

In this letter, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called {\bf Random Apollonian Networks}(RANs) as they can be considered as a variation of Apollonian networks. We obtain the analytic result of power-law exponent $\gamma =3$ and clustering coefficient $C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, which agree very well with the simulation results. We prove that the...

We propose two types of evolving networks: evolutionary Apollonian networks (EAN) and general deterministic Apollonian networks (GDAN), established by simple iteration algorithms. We investigate the two networks by both simulation and theoretical prediction. Analytical results show that both networks follow power-law degree distributions, with distribution exponents continuously tuned from 2 to 3. The accurate expression of clustering coefficient is also given for both networ...

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called {\bf Random Apollonian Networks}(RAN) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent $\gamma =3$ and clustering coefficient $C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, which agree very well with the simulation results. We prove that th...

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 expo...

In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-world, scale-free, Euclidean and space filling and highly clustered, the study of t...

In this paper, by both simulations and theoretical predictions we study two and three node (or degree) correlations in random Apollonian network (RAN), which have small-world and scale-free topologies. Using the rate equation approach under the assumption of continuous degree, we first give the analytical solution for two node correlations, expressed by average nearest-neighbor degree (ANND). Then, we revisit the degree distribution of RAN using rate equation method and get t...