Abstract
We study the following preferential attachment variant of the classical Erdős–Rényi 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.
Funding Statement
The first author (SJ) was supported by a grant from the Knut and Alice Wallenberg Foundation and a grant from the Simons foundation. The second author (LW) was supported by NSF Grant DMS-1703516, NSF CAREER grant DMS-1945481 and a Sloan Research Fellowship.
Acknowledgments
Part of this work was carried out during the authors’ visit to the Isaac Newton Institute for Mathematical Sciences during the programme Theoretical Foundations for Statistical Network Analysis (EPSCR Grant Number EP/K032208/1).
Citation
Svante Janson. Lutz Warnke. "Preferential attachment without vertex growth: Emergence of the giant component." Ann. Appl. Probab. 31 (4) 1523 - 1547, August 2021. https://doi.org/10.1214/20-AAP1610
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