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

Abstract

We generalize the Barabási--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 $P(q) \sim q^{-γ}$ at $q \to \infty$ and $\bar{q}(s,t) \sim (s/t)^{-β}$ at $s/t \to 0$, where the exponent $γ$ varies from 2 to $\infty$ depending on the initial attractiveness of sites. We show that the relation $β(γ-1)=1$ between the exponents is universal.