Modelling Of Stochastic Hierarchical Scale-Free Networks And Pseudofractal Scale-Free Networks

A good deal of systems in society and nature can be described by complex net-works, for example, World Wide Web, Internet,cellular networks, world wide airport networks, ecological networks, scientific collaboration networks, etc.where the nodes represent the elements of system and edges represent the interaction between them. One great important discovery of complex networks is the scale-free property of net-works. Empirical research results show that many real networks are scale-free:the degree distribution has a power-law tail, i.e., P(k)～k-α (for large k). In order to investigate the mechanisms responsible for this kind of network, many evolving network models have been proposed.In this paper, we focus on deterministic network with hierarchical structure and Pseudofractal Scale-free Web,propose four evolving network models.(1)We propose a model of hierarchy networks with all of offspring are equal.n=0:we start from a single node,which is the main root of the whole network.n＝1:μ new node is added with some edges that link the new node to the main root.n=2:μ2new node is added with some edges that every μ nodes connected to all of its root nodes.The whole process is repeated according to this law,until the desired size of the network is reached.And the analytical results show that the outdegree distribution has a power-law tail.(2)We propose a general model of hierarchy networks.It is the generalization of the first model.n=0:we still start from a single node,which is the main root of the whole network.n=1:μ1new node is added with some edges that link the new node to the main root.n=2:μ1μnew node is added with some edges that every μ2nodes connected to all of its root nodes. The whole process is repeated according to this law,until the desired size of the network is reached.Analytical results show that the model can produce a network with the power-law distributions of outdegree.(3)We propose a weighted model of hierarchy networks.The weight of a node link to the nodes t generations apart is assumed that al.Analytical results show that the model can produce a network with the power-law distributions of outdegree and outstrength.(4)We propose a random pseudofractal scale-free web. It start from a complete graph Kq+1(or q+1-clique)which has q+1nodes and q(q+1)/2edges, At t step, adding mt new nodes for each of Kq which is chosen from existing subgraphs isomorphic to a q-clique, and the new node is respectively connected to all the nodes of this subgraph. Make use of Master-equation approach,the distribution of the degree is provided and the analytical results show that the distribution has a power-law tail. And when q=2, q=3and q=4, the networks exhibit large clustering coefficient.