Connectivity of inhomogeneous random graphs
  We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when p = c log n/n. We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i,j) is added with probability min(1, \K(X_i,X_j) log n/n), where \K \ge 0 is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. 
Luc Devroye and Nicolas Fraiman
math.PR math.CO