High Degree Vertices and Eigenvalues in the Preferential Attachment Graph

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 Δ 1 ≥ Δ 2 ≥ ... ≥ 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 → ∞, t /f(t) ≤ Δ 1 ≤ t 1/2 f(t), and for i = 2,..., k, t /f(t) ≤ Δ i ≤ Δ i-1 - t /f(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 1/2 k whp.