On the isolated vertices and connectivity in random intersection graphs (Q541188)

Summary: We study isolated vertices and connectivity in the random intersection graph \(G ( n,m,p)\) . A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When \(m = \lfloor n^a \rfloor \) and \(\alpha > 6\) , we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi random graphs.