A geometric preferential attachment model of networks

Abstract

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 x1,x 2,... ,xn chosen uniformly at random from the unit sphere in R3. After generating xt, we randomly connect it to m points from those points in x1,x2,..., xt-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/n1/2-β for some constant β 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 Gn is connected and has diameter O(m/r) whp. © Springer-Verlag 2004.