High degree vertices and eigenvalues in the preferential attachment graph

Flaxman, A. and Frieze, A.M. and Fenner, Trevor (2003) High degree vertices and eigenvalues in the preferential attachment graph. In: Arora, S. and Jansen, K. and Rolim, J.D.P. and Sahai, A. (eds.) Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. Lecture Notes in Computer Science 2764. Springer, pp. 264-274. ISBN 9783540451983.

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Official URL: https://doi.org/10.1007/978-3-540-45198-3_23

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 [BA99]. For any constant k, let Δ1 ≥ Δ2 ≥ ⋯ ≥ Δ k be the degrees of the k highest degree vertices. We show that at time t, for any function f with f(t)→ ∞ as t→ ∞, t1/2f(t)≤Δ1≤t1/2f(t), and for i = 2,..., k, t1/2f(t)≤Δi≤Δi−1−−t1/2f(t), 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:	Book Section
School:	Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
Depositing User:	Sarah Hall
Date Deposited:	16 Mar 2021 20:13
Last Modified:	09 Aug 2023 12:50
URI:	https://eprints.bbk.ac.uk/id/eprint/43539

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