High Degree Vertices and Eigenvalues in the Preferential Attachment Graph

Flaxman, A. and Frieze, A. and Fenner, Trevor (2005) High Degree Vertices and Eigenvalues in the Preferential Attachment Graph. Internet Mathematics 2 (1), pp. 1-19. ISSN 1542-7951.

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Official URL: http://doi.org/10.1080/15427951.2005.10129097

Abstract

The preferential attachment graph is a random graph formed by adding a new vertex at each time-step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every m steps the most recently added m vertices are contracted into a single vertex, so at time t there are roughly t/m vertices and exactly t edges. This process yields a graph which has been proposed as a simple model of the World Wide Web [Barabási and Albert 99]. For any constant k, let Δ1 ≥ Δ2 ≥ … ≥ Δk be the degrees of the k highest degree vertices. We show that at time t, for any function ƒ with ƒ(t)→ ∞ as , and for , with high probability (whp). We use this to show that at time t the largest k eigenvalues of the adjacency matrix of this graph have λ k = (1 ± o(1))Δ k 1/2 whp.

Metadata

Item Type:	Article
School:	Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
Depositing User:	Administrator
Date Deposited:	05 Jul 2016 09:41
Last Modified:	09 Aug 2023 12:38
URI:	https://eprints.bbk.ac.uk/id/eprint/15678

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