Preferential attachment models were shown to be very effective in predicting such important properties of real-world networks as the power-law degree distribution, small diameter, etc. Many different models are based on the idea of preferential attachment: LCD, Buckley-Osthus, Holme-Kim, fitness, random Apollonian network, and many others. Although preferential attachment models reflect some im...

Thermodynamic and magnetic properties of Ising models defined on the triangular Apollonian network are investigated. This and other similar networks are inspired by the problem of covering a Euclidian domain with circles of maximal radii. Maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of tran...

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). T...

We propose a simple algorithm which produces a new category of networks, high dimensional random Apollonian networks, with small-world and scale-free characteristics. We derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network. The values obtained for these parameters are in good agreement with simulation resul...

The attractors of Iterated Function Systems in Euclidean space IFS fractals have been the subject of great interest for their ability to visually model a wide range of natural phenomena. Indeed computer-generated plants are often modeled using 3D IFS fractals, and thus their extent in virtual space is a fundamental question, whether for collision detection or ray tracing. A great variety of alg...