A Geometric Preferential Attachment Model of Networks

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of Gn are n sequentially generated points x 1, x 2, . . . , x n chosen uniformly at random from the unit sphere in ℝ3. After generating xt , we randomly connect that point to m points from those points in x 1, x 2, . . . , xt -1 that are within distance r of xt . Neighbors are chosen with probability proportional to their current degree, and a parameter a biases the choice towards self loops. We show that if m is sufficiently large, if r ≥ ln n/n 1/2-β for some constant β, and if α > 2, then with high probabilty (whp) at time n the number of vertices of degree k follows a power law with exponent α + 1. Unlike the preferential attachment graph, this geometric preferential attachment graph has small separators, similar to experimental observations of [Blandford et al. 03]. We further show that if m ≥ K ln n, for K sufficiently large, then Gn is connected and has diameter O(ln n/r) whp.