Some Questions In The Complex Networks Modeled On Fractals

Complex network,as an interdisciplinary,has played an important role in the fields of natural and social sciences in the past two decades because of their flexibility and generality for the description of natural and artificial systems.And the complex networks modeled by fractals have many practical applications in other disciplines,for example,the data center network structure based on fractal graphics is not only easy to expand,but also easy to study its topological characteristics.This dissertation mainly discusses some basic topological indices of the evolving complex networks modeled on fractals,such as the average geodesic distance,the cumulative degree distribution,the average clustering coefficient,the fractal scaling,and so on.The whole dissertation is divided into six chapters.The related research backgrounds and the historical process of the development of complex networks will be given in the first chapter.Then it also includes some basic concepts in complex networks or graph theory and some common network models in real life.In chapter 2,a kind of evolving networks modeled on the classical fractal–Durer Pentagon is introduced and we want to calculate the average geodesic distance of the network accurately.The node set of the network is the set consists of all solid regular pentagons in the construction of the Durer Pentagon up to stage t.In the t level network,two nodes are neighbors if and only if the intersection of their corresponding pentagons is neigher the empty set nor a singleton,but a line segment,and we connect them with a unique undirected edge.The edges between all nodes constitute the edge set of the network.Using the self similarity of the network and elementary renewal theorem,we obtain the asymptotic formula on the average geodesic distance of the evolving network by using a very ingenious method.In the following chapter 3,4 and 5,we use a kind of Sierpi ′nski-like carpet to construct evolving networks,and its node set and edge set are similar to the evolving network previously modeled by Durer Pentagon.The nodes of the network are all the solid squares in the construction of the Sierpi ′nski-like carpet up to stage t and there is a unique undirected edge between two nodes if and only if the intersection of their corresponding squares is a line segment.Using the word coding method commonly used in fractal geometry and the self similarity of the network,we prove that the evolving network we construct is scale-free and has small-world effect but has no fractal scale.In Chapter 3,we calculate the degree formula of the network,and show that the cumulative degree distribution of the network obeys the power-law distribution,which implies that the network we constructed has the scale-free property.In Chapter 4,we prove that the clustering coefficient of the network is greater than a positive constant,which indicates that the clustering coefficient of the network is relatively high.The upper and lower bounds of the average path length are also given,which are both proportional to the logarithm of the network scale,which shows that the average path length of the network is relatively small.Therefore the network we constructed has the small-world effect.Finally,in Chapter 5,we prove that the network we constructed does not have fractal scale by covering the network with diameter-based boxes.Finally,the last chapter is devoted to summarizing our main results in this dissertation and proposing some questions for further study.