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 x(t), we randomly connect it to m points from those points in x(1), x(2),..., x(t-1) which are within distance r. Neighbours are chosen with probability proportional to their current degree. We show that if m is sufficiently large and if r greater than or equal to log n/n(1/2-beta) for some constant beta then whp at time n the number of vertices of degree k follows a power law with exponent 3. Unlike the preferential attachment graph, this geometric preferential attachment graph has small separators, similar to experimental observations of [7]. We further show that if m greater than or equal to K log n, K sufficiently large, then G(n) is connected and has diameter O(m/r) whp.