Statistical Properties Of Complex Networks Based On Penrose Tiling

Starting from the quasiperiodic Penrose tiling, we construct complex network models and study their statistical properties in this thesis.First, we briefly introduce the basic theory and main methods in the study of complex networks, including the current status of the complex network, the knowledge of graph theory,the statistical description of network, the evolution models of network, and the main results in complex network. The statistical parameters include the degree of nodes, degree distribution and correlations, average distance,clustering coefficients,vulnerability, etc. The main complex network models include the regular network, the random network, the small-world network, and the Scale-free network.Second, we discuss the geometrical properties of Penrose tiling in more detail, including the construction methods, the types of vertices, the statistical distribution of various types. Using fat and thin rhombi as the basic structural unit, we generate finite sized Penrose tiling by self-similar transformation. The vertices of the Penrose tiling can be classified into eight types. In the infinite Penrose tiling, the proportion of each type of vertex can be obtained by analytical formula. There are a variety of types of neighboring configurations for each type of vertex and they are correlated with each other. Their geometric structure and topological properties are much different from those of a periodic lattice.Based on the quasiperiodic tiling, regular complex networks or partially random networks are constructed and their statistical properties are investigated analytically and numerically. The studied three models are(1) the original fat-thin rhombi Penrose tiling network,(2) the network evolved from the Penrose tiling by connecting certain vertices with the probability depending on the distance between vertices, and(3) the network evolved from the Penrose tiling by connecting certain vertices with the probability depending on the vertex types. We calculate the main statistical parameters of networks, such as the degree distribution, the degree correlation and the cluster coefficient. It is found that these networks have abundant properties.