Mathematics – Probability
Scientific paper
2004-10-25
Mathematics
Probability
17 pages, no figures, submitted to Random Structures and Algorithms
Scientific paper
We extend the results of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and Mori. We consider a model of random tree growth, where at each time unit a new node is added and attached to an already existing node chosen at random. The probability with which a node with degree $k$ is chosen is proportional to $w(k)$, where $w$ is a fixed weight function. We prove that if $w$ fulfills some asymptotic requirements then the degree sequence converges in probability, we give the limit. In particular if $w$ is asymptotically linear then the degree sequence decays with power law. Our method of proof is analytic rather than combinatorial, having the advantage of robustness: only asymptotic properties of the weight function $w$ are used, while in the cited papers the explicit law $w(k)=ak+b$ is assumed.
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