In age-biased attachment graphs, the descendant tree of a fixed root has a beta mixture limit for one attachment per vertex and reaches nearly the whole vertex set for multiple attachments, with matching and independent set greedy limits also derived.
Preferential attachment without vertex growth: emergence of the giant component
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abstract
We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the configuration model.
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2019 1verdicts
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Giant descendant trees, matchings and independent sets in the age-biased attachment graphs
In age-biased attachment graphs, the descendant tree of a fixed root has a beta mixture limit for one attachment per vertex and reaches nearly the whole vertex set for multiple attachments, with matching and independent set greedy limits also derived.