High-dimensional random geometric graphs and their clique number

Abstract

We study the behavior of random geometric graphs in high dimensions. We show that as the dimension grows, the graph becomes similar to an Erdős-Rényi random graph. We pay particular attention to the clique number of such graphs and show that it is very close to that of the corresponding Erdős-Rényi graph when the dimension is larger than log3 n where n is the number of vertices. The problem is motivated by a statistical problem of testing dependencies. © 2011 Applied Probability Trust.