The connectivity threshold of random geometric graphs with Cantor distributed vertices

Abstract

For the connectivity of random geometric graphs, where there is no density for the underlying distribution of the vertices, we consider n i.i.d. Cantor distributed points on [0, 1]. We show that for such a random geometric graph, the connectivity threshold, R n, converges almost surely to a constant 1-2φ where 0<φ<1/2, which for the standard Cantor distribution is 1/3. We also show that {norm of matrix}Rn-(1-2φ){norm of matrix}1~2C(φ)n-1/dφ where C(φ)>0 is a constant and d φ{colon equals}-log2/logφ is the Hausdorff dimension of the generalized Cantor set with parameter φ. © 2012 Elsevier B.V..