Statistical Properties Of Complex Networks Based On Quasiperiodic Ammann-beenker Tiling Model

Complex systems are ubiquitous in the human society and nature, and it is closely related with our life and material production. As an important tool to describe complex systems,complex network can be used to do extensive research from different perspectives, mainly including network model construction, characteristics research, empirical research of the real network and the specific application of network model. Among them, network modeling and characteristics are the most fundamental issues. In this thesis, we employ the Ammann-Beenker(AB) tiling, a two-dimensional quasicrystal model, as the basic structure to construct two kinds of evolutionary complex network models by adding some random links.Theoretical analysis and numerical calculations for the statistical properties and stability of various models are performed.Firstly, we introduce the theories of complex networks, including network graph theory,the commonly used characteristics parameters to describe network, such as degree and degree distribution, average path length, clustering coefficient and degree correlation. Then some other deterministic network models as well as the four classical network models are introduced in detail, including the regular, random, WS small world and BA scale free networks. Moreover, we also introduce three calculation methods of degree distribution based on BA network model and two calculation methods of the shortest path.Secondly, we specificly introduce three kinds of tiling models for two-dimensional quasicrystals, including the five-fold symmetric Penrose tiling, the eight-fold symmetric Ammann-Beenker(AB) tiling, and the twelve-fold symmetric Stampfli-Gahler tiling.Next, we focus on the characteristics of two-dimensional quasiperiodic AB tiling. The AB tiling is obtained from self-similar transformation and is used as the basic model of the deterministic complex network. Two different evolutionary models are generated by adding some random components. The statistical properties of each model are systematically studied,including the degree distribution, average path length, cluster coefficient and degree correlation. In the regular AB tiling network, there are a few fixed degree values only, so the degree distribution shows separete graphs. Because there is no triangle in this model, theclustering coefficient is zero. From the degree correlation, we can conclude that the network is globally uniform. In the evolutionary models, we mainly study the degree distribution. For the first evolutionary model, in which edge connection depends on the distance between two nodes, the degree distribution is approximately of Poisson with average degree ?? k. When the total number of nodes in the network is fixed and the probability of connection is changed,the range of the Poisson distribution figure is gradually widened, that is to say, the degree of the network is increased, and the average degree of the network is also increased. When the probability of connection is constant, the clustering coefficient decreases with the increase of the total number of nodes. When the total number of nodes is fixed, the clustering coefficient increases with the increase of the connection probability. According to the values of node degree classification, we construct another evolutionary model and find that the degree distribution is composed of seven approximately Poisson distribution sub-graphs. When studying the relationship between the degree and the clustering coefficient, it is found that there is a hierarchical structure in this model.Finally, we study the stability of the two evolutionary complex network models. In researching the stability of networks, the global efficiency, the relative size of maximum connected sub-graph and the connecting factor are used as evaluation indices. The numerical simulation figures show that evolutionary AB networks have good robustness to random failure and a certain vulnerability to the selective attack. This kind of phenomenon in the second evolutionary model appears more obviously.The combination of complex network and quasiperiodic structure has a certain reference value for the application of quasiperiodic structure in complex network research. The obtained results help us get a deeper understanding for the theoretical model of the quasicrystals, and also provide a new thought for the research of complex networks.