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

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

2000-04-26

arxiv.org/abs/cond-mat/0004434v1

Phys. Rev. Lett. 85, 4633 (2000)

Physics

Condensed Matter

4 pages revtex (twocolumn, psfig), 1 figure

Scientific paper

10.1103/PhysRevLett.85.4633

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 $q \to \infty$ and $\bar{q}(s,t) \sim (s/t)^{-\beta}$ 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 $\beta(\gamma-1)=1$ between the exponents is universal.

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