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 [Barabási and Albert 99]. For any constant k, let ∆1 ≥ ∆2 ≥ · · · ≥ ∆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 → ∞, t1/2 f(t) ≤ ∆1 ≤ tf(t), and for i = 2, . . . , k, t 1/2 f(t) ≤ ∆i ≤ ∆i−1 − t1/2 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 λk = (1± o(1))∆ k whp.