Maximal planar scale-free Sierpinski networks with small-world effect and power-law strength-degree correlation

Many real networks share three generic properties: they are scale-free, display a small-world effect, and show a power-law strength-degree correlation. In this paper, we propose a type of deterministically growing networks called Sierpinski networks, which are induced by the famous Sierpinski fractals and constructed in a simple iterative way. We derive analytical expressions for degree distribution, strength distribution, clustering coefficient, and strength-degree correlation, which agree well with the characterizations of various real-life networks. Moreover, we show that the introduced Sierpinski networks are maximal planar graphs. Introduction. – In the last few years, research of complex networks have become a focus of attention from the scientific community [1–5]. One of the main reasons behind the popularity of complex networks is their flexibility and generality for representing real systems in nature and society. Researchers have done a lot of empirical studies, uncovering that various real-life networks sharing some generic properties: power-law degree distribution [6], small-world effect including small average path length (APL) and high clustering coefficient [7]. Recently, many authors have described some real-world systems in terms of weighted networks, where an interesting empirical phenomenon has been observed that there exists a powerlaw scaling relation between the strength s and degree k of nodes, i.e. s ∼ k with β > 1 [8–11]. With the intention of studying the above properties of real-world systems, a wide variety of models have been proposed [1–4]. Watts and Strogatz, in their pioneering paper, introduced the famous small-world network model (WS model) [7], which exhibits small APL and high clustering coefficient. Another well-known model is Barabási and Albert’s scale-free network model (BA model) [6], which has a degree distribution of power-law form. How(a)[email protected] (Z.Z. Zhang) (b)[email protected] (S.G. Zhou) ever, in these two elegant models, scale-free feature and high clustering are exclusive. Driven by the two seminal papers [6, 7], a considerable number of other models have been developed that may represent processes more realistically taking place in real-world networks [12–18]. Very recently, Barrat, Barthélemy, and Vespignani have introduced a model (BBV) for the growth of weighted networks [19, 20], which is the first weighted network model that yields a scale-free behavior for strength and degree distributions. Enlightened by BBV’s remarkable work, various weighted network models have been proposed to explain the properties found in real systems [21–27]. These models may give some insight into the realities. Particulary, some of them presente all the above-mentioned three characteristics such as power-law degree distribution, small-world effect, and power-law strength-degree correlation [21–23]. Although great progresses have been made in the research of network topology, modeling complex networks with general structural properties is still of current interest. On the other hand, fractals are an important tool for the investigation of physical phenomena [28]. They were used to describe physical characteristics of things in nature and life systems such as clouds, trees, mountains, rivers, coastlines, waves on a lake, bronchi, and the human circulatory