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, P͑q͒, and the long-time limit of the average connectivity q͑s, t͒ of a site s at time t (one site is added per unit of time). At long times, P͑q͒ ϳ q 2g at q !à nd q͑s, t͒ ϳ ͑s͞t͒ 2b at s͞t ! 0, where the exponent g varies from 2 to`depending on the initial attractiveness of sites. We show that the relation b͑g 2 1͒ 1 between the exponents is universal. PACS numbers: 84.35.+i, 05.50.+q, 87.18.Sn It was observed recently that the distributions of several quantities in various growing networks have a power-law form. This scaling behavior was observed in the World Wide Web, in neural and social networks, in nets of citations of scientific papers, etc.; see [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] , and references therein. These observations challenge us to find the general reasons of such behavior. It is only recently that scientists became aware of the ever increasing impact of various evolving networks on everyone's life. Earlier studies [15] [16] [17] [18] [19] [20] concentrated on simple random networks, and it was recently discovered that many complex networks are hierarchically organized [5, 6, 21] . Mostly, the interest is concentrated on the distribution of shortest paths between the different sites of a network [1] and on the distribution of the number of connections with a site [2] [3] [4] [5] [6] . The second quantity is obviously simpler to obtain than the first one but even for it, in the case of the networks with scaling behavior, no exact results are known. The only known mechanism of self-organization of a growing network into a free-scale structure is preferential linking [7] [8] [9] , i.e., new links are preferentially attached to sites with high numbers of connections. A simple model of a growing network with preferential linking was proposed by Barabási and Albert [7] (BA model). At each time step a new site is added. It connects with old sites by a fixed number of links. The probability of an old site to get a new link is proportional to the total number of connections with this site. It was found in [7, 8] that the distribution of the number of links has a power-law form at long times. The value of the corresponding scaling exponent, g, obtained using a mean-field approach, equals 3. This value is close to that one observed in the network of citations [3] , but other examples of evolving networks show different values of g. Introduction of the aging of sites changes g [22] and may even break the scaling behavior [13, 22] . In the present Letter, we generalize the BA model and find the exact form of the distribution of incoming links of sites in the limit of large sizes of the growing network. We derive a scaling relation connecting the scaling exponent of the distribution of incoming links and the exponent of the temporal behavior of the average connectivity, and demonstrate that it applies for a large class of evolving networks. The model.-At each time step a new site appears (see Fig. 1 ). Simultaneously, m new directed links coming out from nonspecified sites are introduced. Let the connectivity q s be the number of incoming links to a site s, i.e., to a site added at time s. The new links are distributed between sites according to the following rule. The probability that a new link points to a given site s is proportional to the following characteristic of the site: thereafter called its attractiveness. All sites are born with some initial attractiveness A $ 0, but afterwards it increases because of the q s term. The introduced parameter A, the initial attractiveness, governs the probability for "young" sites to get new links. We emphasize that we do not specify sites from which the new links come out. They may come out from the new site, from old sites, or even from outside of the network. Our results do not depend on that. Therefore, the model describes also the particular case when every new site is the source of all the m new links like in the BA model. In this case, every site has m outgoing links and the total t t+1 0031-9007͞00͞85(21)͞4633(4)$15.00
In REF the authors overcome the restriction τ ≥ 3 in a different way, by choosing the endpoint of an edge proportional to the in-degree of a vertex plus some initial attraction A > 0.
Structure of growing networks with preferential linking.
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