LNCS 2764 High Degree Vertices and Eigenvalues in the Preferential Attachment Graph 1st Edition by Abraham Flaxman, Alan Frieze, Trevor Fenner ISBN 3540380450 9783540380450

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Product details:

ISBN 10: 3540380450

ISBN 13: 9783540380450

Author:  Abraham Flaxman, Alan Frieze, Trevor Fenner 

LNCS 2764 – High Degree Vertices and Eigenvalues in the Preferential Attachment Graph 1st Edition: 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→ ∞, <span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="t1/2f(t)≤Δ1≤t1/2f(t),”>t1/2f(t)≤Δ1≤t1/2f(t), and for i = 2,…, k, <span id="MathJax-Element-2-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; font-size-adjust: none; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="t1/2f(t)≤Δi≤Δi−1−−t1/2f(t),”>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.

 

LNCS 2764 – High Degree Vertices and Eigenvalues in the Preferential Attachment Graph 1st Edition Table of contents:

1 Introduction

2 Proof of Theorems
2.1 Proof of Theorem 1

2.2 Proof of Theorem 2

 

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