Modelling Of Weighted Scale-Free Networks With The High Clustering

Complex networks describe a wide range of systems in nature and society. Forexample, World Wide Web, Internet, world wide airport networks, cellular networks,ecological networks, scientific collaboration networks, etc. Empirical research resultsshow that many of these real networks are scale-free and small-world: (1)the degreedistribution has a power-law tail, i.e., P(k)～Ak^-α for largeκ, where A,αareconstant,(2) the average path length of the network is short,(3) the network clus-tering coefficient is hight(especially social networks), the networks show a tendencyto cluster. Researchers have prgposed many evolving network models to investigatethe mechanisms responsible for the properties found in many natural networks.In this paper, we focus on weighted networks, propose three evolving networkmodels that can produce weighted scale-free networks with the high clustering coeffi-cient and analyze the networks by the combined numerical and analytical approach:(1)An evolving model of weighted network based on the random walks is pro-posed. At each time step, add a new node with some edges that link the new nodeto some exiting nodes preferentially, and a dynamical evolution also occurs amongexisting nodes: select some old nodes based on the weight-dependent walks, and linkthese nodes each other or strengthen their linking (i.e., increase the edge weight).The distributions of the strength, weight and degree are provided analytically andnumerically, results show that each distribution has a power-law tail, and the weight-dependent walk length will not influence the strength distribution. Particularly, theclustering coefficient is especially high when the weight-dependent walk length is 2,thus, this model can evolve into a scale-free network with high clustering.(2)A weighted network model based on the preferential selection of edges isproposed. In most previous network models, the preferential selection of the nodesare related directly to the quantities of the nodes, e.g., node's degree, strength,fitness, etc. Differentially, in this model, at each time step, add a new node withtwo edges that connect the new node to both ends of a preferentially selected link.Meanwhile, the weight of the link selected preferentially will be strengthened. Thepreferential probability that an edge will be selected is proportional to the edgeweight. Analytical results show that the model can produce a network with the power-law distributions of strength, weight and degree, and the clustering coefficientof the network shows a high value at the same time.(3)By the hierarchical networks, a weighted hierarchical network model is pro-posed. According to the algorithm, the topologies of the networks are analyzed.Results show that the network possesses power-law behavior of the strength, weightand degree distributions, meanwhile, the clustering coefficient of the network is high.