Structure of Growing Networks: Exact Solution of the Barabasi--Albert's Model

We generalize the Barab'{a}si–Albert’s model of growing networks accounting for initial properties of sites and find exactly the distribution of connectivities of the network $P(q)$ and the averaged connectivity $\bar{q}(s,t)$ of a site $s$ in the instant $t$ (one site is added per unit of time). At long times $P(q)\sim q^{-\gamma }$ at $q\to \mathrm{\infty }$ and $\bar{q}(s,t)\sim (s\mathrm{/}t{)}^{-\beta }$ at $s\mathrm{/}t\to 0$, where the exponent $\gamma$ varies from 2 to $\mathrm{\infty }$ depending on the initial attractiveness of sites. We show that the relation $\beta (\gamma -1)=1$ between the exponents is universal.