The Transitive Closure of a Random Digraph

In a random n-vertex digraph, each arc is present with probability p, independently of the presence or absence of other arcs. We investigate the structure of the strong components of a random digraph and present an algorithm for the construction of the transitive closure of a random digraph. We show that, when n is large and np is equal to a constant c greater than 1, it is very likely that all but one of the strong components are very small, and that the unique large strong component contains about Θ2n vertices, where Θ is the unique root in [0,1] of the equation 1 − x − e −cx = 0. Nearly all the vertices outside the large strong component lie in strong components of size 1. Provided that the expected degree of a vertex is bounded away from 1, our transitive closure algorithm runs in expected time O (n). For all choices of n and p, the expected execution time of the algorithm is O (w (n) (n log n)4/3), where w (n) is an arbitrary nondecreasing unbounded function. To circumvent the fact that the size of the transitive closure may be Ω(n 2) the algorithm presents the transitive closure in the compact form (A × B) ∪ C, where A and B are sets of vertices, and C is a set of arcs.