The model of growing networks with the preferential attachment of new links is generalized to include initial attractiveness of sites. We find the exact form of the stationary distribution of the number of incoming links of sites in the limit of long times, P(q), and the long-time limit of the average connectivity q(s,t) of a site s at time t (one site is added per unit of time). At long times, P(q) approximately q(-gamma) at q-->infinity and q(s,t) approximately (s/t)(-beta) at s/t-->0, where the exponent gamma varies from 2 to infinity depending on the initial attractiveness of sites. We show that the relation beta(gamma-1) = 1 between the exponents is universal.

We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local treelike structure), exact equations are derived. These equations are generalized to the case of networks with correlations between neighboring vertices. The tail of the density of eigenvalues rho(lambda) at large /lambda/ is related to the behavior of the vertex degree distribution P(k) at large k. In particular, as P(k) approximately k(-gamma), rho(lambda) approximately /lambda/(1-2 gamma). We propose a simple approximation, which enables us to calculate spectra of various graphs analytically. We analyze spectra of various complex networks and discuss the role of vertices of low degree. We show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real-world networks, e.g., of the Internet.

We describe the anomalous phase transition of the emergence of the giant connected component in scale-free networks growing under mechanism of preferential linking. We obtain exact results for the size of the giant connected component and the distribution of vertices among connected components. We show that all the derivatives of the giant connected component size S over the rate b of the emergence of new edges are zero at the percolation threshold bc, and S ∝ exp{−d(γ)(b − bc) −1/2 }, where the coefficient d is a function of the degree distribution exponent γ. In the entire phase without the giant component, these networks are in a "critical state": the probability P(k) that a vertex belongs to a connected component of a size k is of a power-law form. At the phase transition point, P(k) ∼ 1/(k ln k)2 . In the phase with the giant component, P(k) has an exponential cutoff at kc ∝ 1/S. In the simplest particular case, we present exact results for growing exponential networks.

We show that the connectivity distributions P (k, t) of scale-free growing networks (t is the network size) have the generic scale -the cut-off at kcut ∼ t β . The scaling exponent β is related to the exponent γ of the connectivity distribution, β = 1/(γ − 1). We propose the simplest model of scale-free growing networks and obtain the exact form of its connectivity distribution for any size of the network. We demonstrate that the trace of the initial conditions -a hump at k h ∼ kcut ∼ t β -may be found for any network size. We also show that there exists a natural boundary for the observation of the scale-free networks and explain why so few scale-free networks are observed in Nature. 05.10.-a, 05-40.-a, 05-50.+q, 87.18.Sn A significant progress was made recently in the field of evolving networks [1][2][3][4][5][6][7]. It was observed that a number of growing networks in Nature (World-Wide Web, Internet, networks of scientific citations, collaboration nets, some networks in biology, etc.) are scale-free, i.e., their connectivity distribution is of a power-law form. Moreover, it was found that at least many of them have to be scale-free, otherwise growing networks are not resilient enough to random breakdowns [8,9]. The infinite scalefree network with the connectivity distribution exponent γ < 3 does not decay for any concentration (less than one) of randomly removed links [9].The proposed mechanism of self-organization of networks into scale-free structures, the preferential linking, is quite natural [10]. New links of the growing networks are preferentially attached to nodes which already have many connections (connectivity k). In fact, it is the realization of a general principle -popularity is attractive. Several types of preferential linking were proposed [10][11][12][13][14][15] which provide a variety of the γ exponent values between 2 and infinity.One should emphasize that only a few scale-free networks is known yet. The range of the values of the connectivity, in which the power-law behavior can be observed, is usually too narrow for a precise measurement of the exponent γ. It is unclear, why are so few scale-free networks observed? Why are the values of γ for all of them only between 2 and 3? (Note that not any network has to be resilient, e.g., neither nodes nor links of collaboration networks are removable by definition [17].) In the present Letter, we answer these questions.In previous papers, the connectivity distributions P (k, t) of scale-free networks were calculated in the "thermodynamic limit", i.e., in the limit of the large system size, t, which also plays the role of time, if one node is added at each increment of time. In this case, the distribution is stationary, and is of the form P (k) ∼ k −γ in all range of large enough k, k ≫ 1. Nevertheless, real networks are finite. The evolution of P (k, t) to the stationary distribution turns to be non trivial. We demonstrate below that, for finite networks, the power-law region of the connectivity distribution has the cut-off at k cut ∼ t β , wher...