【作者】 黄亮；

【导师】 谭波；

【作者基本信息】 华中科技大学 ， 基础数学， 2022， 博士

【摘要】 复杂网络作为一门交叉学科,由于其对自然和人工系统可以灵活地描述,以及其通用性,在过去二十年里在自然科学和社会科学领域发挥了重要作用。而分形构造的复杂网络在其它学科中有非常多的实际应用,例如,基于分形构造的数据中心网络结构不仅易于扩展,而且十分容易研究其拓扑性质。本文主要研究了由分形构造的演化网络中的一些基本拓扑指标,如平均测地距离,累积度数分布,聚类系数以及分形标度等。全文共分为六章。在第一章中,我们首先给出本论文所研究问题的相关背景,并简单介绍了复杂网络发展的历史进程。然后我们介绍了一些复杂网络或图论中的基本概念以及现实生活中常见的网络模型。在第二章中,我们考虑了一种基于经典分形――Durer五边形建模的演化网络,其第t级网络的节点集是Durer五边形前t级构造中的所有实心正五边形构成的集合。在网络中,两个节点是邻居当且仅当它们对应的正五边形的交不是空集或单点集,而是一条直线段,并且我们用唯一的一条无向边连接它们。所有节点之间的边构成了网络的边集。我们利用网络的自相似性和基本更新定理,使用一种很巧妙的方法得到了该演化网络中平均测地距离的渐近公式。在第三章、第四章和第五章中,我们基于一种Sierpi′nski-like地毯来构建演化网络,其节点集与边集与之前由Durer五边形建模的演化网络类似。节点集是Sierpi′nski-like地毯的前t级构造中的所有实心正方形,而两个节点之间有唯一的一条无向边当且仅当它们对应的正方形的交是一条直线段。利用分形几何中常用的词编码方法以及网络的自相似性,我们证明了该演化网络具有无标度性质和小世界效应,但没有分形标度。我们在第三章给出了该网络的度数公式,并证明了网络的累积度数分布服从幂律分布,从而说明我们构造的网络是无标度网络。在第四章中,我们证明了网络的聚类系数大于某个正常数,这表明网络的聚类系数相对较高。并且还给出了平均路径长度的上界和下界,它们均正比于网络规模的对数。这表明网络的平均路径长度相对较小,从而我们构造的网络具有小世界效应。最后在第五章中,我们通过对网络进行盒覆盖,证明了我们构造的网络不具有分形标度。本论文的最后一章对本文的主要结果进行了总结,并提出了一些可进一步研究的问题。更多还原

【Abstract】 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.更多还原