High Degree Vertices and Eigenvalues in the Preferential Attachment Graph

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. For any constant $k$, let $\Delta_1 \geq \Delta_2 \geq \cdots \geq \Delta_k$ be the degrees of the $k$ highest degree vertices. We show that at time $t$, for any function $f$ with $f(t)\rightarrow\infty$ as $t\rightarrow\infty$, $\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq t^{1/2}f(t),$ and for $i = 2,\dots, k$, $\frac{t^{1/2}}{f(t)} \leq \Delta_i \leq \Delta_{i-1} - \frac{t^{1/2}}{f(t)},$ 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 $\lambda_k = (1\pm o(1))\Delta_k^{1/2}$ \whp.