Structure of Growing Networks with Preferential Linking

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, $P\left(q\right)$, and the long-time limit of the average connectivity $\overline{q}(s,t)$ of a site $s$ at time $t$ (one site is added per unit of time). At long times, $P\left(q\right)\sim q^{-\gamma }$ at $q\to \infty$ and $\overline{q}(s,t)\sim (s/t{)}^{-\beta }$ at $s/t\to 0$, where the exponent $\gamma$ varies from $2$ to $\infty$ depending on the initial attractiveness of sites. We show that the relation $\beta (\gamma -1\mathrm{})=1\mathrm{}$ between the exponents is universal.