A Geometric Preferential Attachment Model of Networks

We study a random graph G(n) that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of G(n) are n sequentially generated points x(1), x(2),..., x(n) chosen uniformly at random from the unit sphere in R-3. After generating xt, we randomly connect that point to m points from those points in x(1), x(2),..., x(t-1) that are within distance r of x(t). Neighbors are chosen with probability proportional to their current degree, and a parameter alpha biases the choice towards self loops. We show that if m is sufficiently large, if r >= ln n/n(1/2-beta) for some constant beta, and if alpha > 2, then with high probabilty (whp) at time n the number of vertices of degree k follows a power law with exponent alpha + 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 G(n) is connected and has diameter O(ln n/r) whp.