一类确定性小世界网络模型及特性分析

A Kind of Deterministic Small-World Networks Model and Analysis of Their Characteristics

摘要: 确定性小世界网络是复杂网络中的一个重要的研究分支。2008年，章忠志等人(Eur.Phys.J.B 63)在复杂网络的视角下对确定性均匀递归树作了详尽地分析，得到了其拓扑属性。尽管确定性均匀递归树的平均路径长度表现出了网络大小的对数规模，但是它的聚类系数为零。2012年，陆哲明等人(Physica A 391)通过在确定性均匀递归树的基础上以一个简单的规则添加一些边得到一个确定性小世界网络模型。本文在新的视角下对文献Physica A 391中的模型的构造规律进行了探究，并自然地对其进行了推广。构造了一类确定性小世界网络模型，然后运用分析的方法给出了所构造网络模型的几个重要拓扑属性，包括呈幂律分布的累积度分布、高的聚类系数和小的直径。

Abstract: Deterministic small-world network is an important branch of study of complex networks. In 2008, Zhang et al. in Eur.Phys.J.B 63 have offered detailed topological characteristics of the deterministic uniform recursive tree from the viewpoint of complex network. They derived topological characteristics of the deterministic uniform recursive tree. It shows a logarithmic scaling with the size of the network; however its clustering coefficient is zero. In 2012, Lu et al. in Physica A 391, based on the deterministic uniform recursive tree and by a simple rule, added some edges and got a deterministic small-world network model. In this paper, we study the law of the structure on the basis of the model constructed in Physica A 391 from a new viewpoint, naturally, we generalize and change the rule to get an iterated class of small-world networks, and then give the analytic solutions to several important topological characteristics of these models proposed. The obtained vigorous results show that these networks have power-law cumulative degree distribution, high clustering coefficient and small diameter.