Topological Properties Of Several Scale-Free Small-World Networks Models

It is well known that the dynamical phenomenon of complex networks such as semantic and information flow are very difficult to predict from local information because of the abundant microstructure and relevant complex dynamics. It is difficult to calculate the random parameters of a large network withing thousands of nodes. Besides the topological structures of scale-free networks are not exactly figured up to now, so people can not detect specific paths of information dissemination in scale-free networks. However, it is required and useful that understand the topological structure of scale-free networks or small world networks for investigation of complex networks. Thus many scholars devote themselves to it. Some of them structure a series of growth networks with scale-free small-world property to approximate universality.It is an important way that build up deterministic network models having both scale-free and small-world property to picture some networks and to approximate real networks. It is obvious that network models with great characteristics may be useful for constructing real networks in the world. In this article,we first build up some kinds of growth networks and testify scale-free property about some of them, we give the specific growth mechanism and carry out the basic parameters, especially, our net-work model is becoming more and more random. Then we determine some properties of the models, such as small-world, power law distribution and clustering coefficient, and furthermore we propose some new methods such as the edge cumulative distri-bution for explore topological structures of the models. Besides we design several recursive algorithms in order to find spanning trees having maximal leaves for inves-tigating topological structures in the growing network models, and furthermore some important properties of ML-spanning trees have been determined. An application of ML-spanning trees is to determine balanced sets and dominating sets of the bound growing network models. Define and validate the spanning tree mentioned in paper is (??,??)-spanning tree. Finally we make a summary and look forward to the future work.