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 . 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 ≥ log n / n 1/2– β for some constant 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 ≥ K log n, K sufficiently large, then G n is connected and has diameter O ( m/ r) whp .