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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 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 results and comparable to those coming from real networks. We estimate also analytically that the average path length of the networks increases at most logarithmically with the number of vertices. networks describe many systems in nature and society, such as Internet [6], World Wide Web [7], metabolic networks [8], protein networks in the cell [9], co-author networks [10] and sexual networks [11], most of which share three apparent features: power-law degree distribution, small average path length (APL) and high clustering coefficient. In recent years, many evolving models [3,4,5] have been proposed to describe real-life networks. The original BA model [2,12] captures the two main mechanisms responsible for the power-law degree distribution of degrees, namely growth and preferential attachment. Dorogovtsev, Mendes, and Samukhin [13] gave an exact solution for a class of growing network models thanks to the use of a "master-equation". Krapivsky, Redner, and Leyvraz [14] examined the effect of a nonlinear preferential attachment on network dynamics and topology. Amaral et al. [15] studied models that incorporate aging and cost and capacity constraints in order to explain the deviations from the power-law behavior in several reallife networks. Dorogovtsev and Mendes [16] also addressed the evolution of networks with aging of sites. Bianconi and Barabási [17] offered a model addressing the competitive aspect in many real networks such as World Wide Web. Additionally, in real systems a series of microscopic events shape the network evolution, including the addition or rewiring of new edges or the removal of vertices or edges. Albert and Barabási [18] discussed a model that incorporates new edges between existing vertices and the rewiring of edges. Dorogovtsev and Mendes [19] considered a class of undirected models in which new edges are added between old vertices and existing edges can be removed. Although it is now established that preferential attachment can explain the power-law characteristic of networks, there is a wide range of microscopic alternative mechanisms that could affect the evolution of growing networks and still lead to the observed scale-free topologies. Kleinberg et al. [20] and Kumar et al. [21,22] proposed copying mechanisms motivated by the desire to explain the power-law degree distribution of the World Wide Web. Chung et al. [23] introduced also a duplication model for biological networks. Krapivsky and Render [24] presented edge redirection mechanisms which are mathematically equivalent to the model of Kumar et al [21,22]. Inspired by citation networks Vázquez proposed in [25] the walking mechanism. Act...

Small-world networks are ubiquitous in real-life systems. Most previous models of small-world networks are stochastic. The randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections. Here we present a model that generates a small-world network in a simple deterministic way. Our model has a discrete exponential degree distribution. We solve the main characteristics of the model.

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