Scaling properties of scale-free evolving networks: Continuous approach

The scaling behavior of scale-free evolving networks, arising in areas such as communications, scientific citations, collaborations, etc., is studied. We derive universal scaling relations describing properties of such networks, and indicate the limits of their validity. We show that the main properties of scale-free evolving networks may be described in the framework of a simple continuous approach. The simplest models of networks, growing according to a mechanism of preferential attachment of links to nodes, are used. We consider different forms of this preference, and demonstrate that the range of preferential attachments producing scale-free networks is wide. We also obtain scaling relations for networks with nonlinear, accelerating growth, and describe the temporal evolution of the arising distributions. Size effects-the cutoffs of these distributions-introduce restrictions for the observation of power-law dependences. Mainly we discuss the so-called degree distribution, i.e., the distribution of the number of connections of nodes. A scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied. We describe the effects of differences between nodes. The ``aging'' of nodes changes the exponents of the distributions. The appearance of a single node with high fitness changes the degree distribution of a network dramatically. If its fitness exceeds some threshold value, this node captures a finite part of all links of the network. We show that permanent random damage to a growing scale-free network-a permanent deletion of some links-radically changes the values of the scaling exponents. Results of other kinds of permanent damage are described.