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 x1, x2, . . . , 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. Neighbors are chosen with probability proportional to their current degree and a parameter α biasses the choice towards self loops. We show that if m is sufficiently large, if r ≥ ln n/n1/2−β for some constant β, and if α > 2, then whp at time n the number of vertices of degree

• Type: Conference
• Year: 2004
• Where: WAW
• Authors: Abraham Flaxman, Alan M. Frieze, Juan Vera
• Keywords: Algorithms, Geometric Random Graphs, Random Graph Gn, Random Graphs, WAW 2004