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    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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    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.


    Item Type: Article
    School: School of Business, Economics & Informatics > Computer Science and Information Systems
    Depositing User: Administrator
    Date Deposited: 05 Jul 2016 09:41
    Last Modified: 05 Jul 2016 09:41


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