Stochastic Analysis Seminar

Abstract:

We consider the random directed graph D(n, p) with vertex set {1, 2, . . . , n} in which each of the n(n − 1) possible directed edges is present independently with probability p. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at p = 1/n, with critical window p = 1/n + \lambda n^{-4/3} for \lambda \in \R. We show that, within this critical window, the strongly connected components of D(n, p), ranked in decreasing order of size and rescaled by n^{-1/3}, converge in distribution to a sequence of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops.  This is joint work with Robin Stephenson (Sheffield).