On Topological Properties Of Several Complex Network Models:Small-world And Scale-free Feature,the Number Of Spanning Trees And Community Structure

As a new interdisciplinary research field,study of complex networks,both the-oretical modeling and computer simulation based on real data,have attracted more attention of a large number of researchers.Especially,the discovery of small-world property and scale-free feature makes research passion to a fresh peak,and even be-come more and more active.As time goes by,on the basis of the fact that the topological structure and intrin-sic characters of small-world scale-free networks are continuously changing,scholars,at home and abroad,have presented series of network models to describe and depict many real-life complex networks.Notice that,in Sec.2,we introduce a class of com-plex systems and discuss the relationship between this complex system and real-world phenomenon by establishing the relevant dynamic equation.Not only taking into account the evolution of networks being complicated and sophisticated,but also in order to uncover those mechanisms in the development process of networks,lots of theoretical network models are successively created and improved.Based on that,in Sec.3,we generate several kinds of network models,including stochastic and deterministic ones.Considering it not easy to capture the exact topological properties on stochastic models,we just prove them to be scale-free by virtue of vertex degree distribution.For deterministic ones,in Sec.4,we state that they have scale-free feature and even display small-world property by analytically computing the solutions both diameter and average path length.The total number of spanning trees,referred to as a structural invariant of graph,can be related to some important topological and structural characters,such as per-colation,epidemic spreading,synchronization and random walks and the like.We provide a new method for calculating the number of spanning trees on some models and apply our method to such existing network models in Sec.5.Compared with some current enumeration methods,our computation method is convenient and helpful to compute the close-form of the number of spanning trees.From another perspective,community structure,also called cluster(group),the topological property prevailing in real and virtual world,depicts the evolution mecha-nism of network internal structure.Compared with the problem of partitioning graph in classical graph theory,we debate the connection among those algorithms of detect-ing community structure and investigate the differences between them and the graph partition.At the end of this paper,we close our work by making a brief summary,focusing on both the next researching task and some interesting and unfound problems in the following future.