6 M ar 2 00 8 epl draft Random Sierpinski network with scale - free small - world and mod - ular structure

Incompatibility graphs (networks) are abundant in the real world. In this paper, we define a stochastic Sierpinski gasket, on the basis of which we construct a random incompatibility network—random Sierpinski network (RSN). We investigate analytically or numerically the statistical characteristics of RSN. The obtained results reveal that the properties of RSN is particularly rich, it is simultaneously scale-free, small-world, uncorrelated, modular, and maximal planar. All obtained analytical predictions are successfully contrasted with extensive numerical simulations. Our network representation method could be applied to study the complexity of some real systems in biological and information fields. Introduction. – In the last few years, much attention has been paid to the study of complex networks as an interdisciplinary subject [1]. It is now established that network science is a powerful tool in the analysis of real-life complex systems by providing intuitive and useful representations for networked systems. Many real-world natural and man-made systems have been examined from the perspective of complex network theory. Commonly cited examples include the Internet [2], the World Wide Web [3], metabolic networks [4], protein networks in the cell [5], co-author networks [6], sexual networks [7], to name but a few. The empirical studies have uncovered the presence of several generic properties shared by a lot of real systems: power-law degree distribution [8], small-world effect including small average path length (APL) and high clustering coefficient [9], and community (modular) structure [10]. These new discoveries have inspired researchers to develop a variety of techniques and models in an effort to understand or predict the behavior of real systems [1]. It is still of current interest to reveal other different processes in real-life systems that may lead to above general characteristics. In the real world, there are a large variety of systems that can be described by a class of new complex networks, (a)[email protected] (b)[email protected] called incompatibility networks, since these networks are associated with contact relation. For instance, the navigational complexity of cities can be conveniently investigated from the viewpoint of incompatibility networks with roads mapped to nodes and intersections to edges between nodes [11]. Another example is RNA folding study, to which the incompatibility network representation is frequently applied [12, 13]. Moreover, previous connections relating incompatibility network to polymers have proven useful in the study of polymer physics [14, 15]. Although incompatibility networks are ubiquitous, relevant network models have been far less investigated. In our earlier paper, we have proposed a family of deterministic incompatibility networks based on the well-known Sierpinski fractals [16]. These networks posses good topological properties observed in some real systems. However, their deterministic construction are not in line with the randomness of many real-world systems. In this paper, we present a stochastic Sierpinski gasket, in relation to which a novel incompatibility network, named random Sierpinski network (RSN), is constructed. The obtained network is a maximal planar graph, it display the general topological features of real systems: heavy-tailed degree distribution, small-world effect, and modular structure. We also obtain the degree correlations of RSN. All theoretical predictions are successfully confirmed by numerical simulations.