High degree vertices and eigenvalues in the preferential attachment graph

TLDR: It is shown 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.

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→ ∞, \(\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq t^{1/2}f(t),\) and for i = 2,..., k, \(\frac{t^{1/2}}{f(t)} \leq \Delta_i \leq \Delta_{i-1} -- \frac{t^{1/2}}{f(t)},\) 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 λ k = (1± o(1))Δ k 1/2 whp.