The Number of Sources and Isolated Vertices in Random Directed Acyclic Graphs

For a positive integer n and a real number p ∈ (0 , 1), a random directed acyclic digraph D ac ( n, p ) is obtained from the binomial random digraph model D ( n, p ) conditioned to be acyclic, i.e., directed cycles are forbidden. In the binomial random digraph model D ( n, p ), every possible directed edge (excluding loops) occurs independently with probability p . Sources and sinks are among the most natural characteristics of directed acyclic graphs. We investigate the distribution of the number of sources in D ac ( n, p ) when p is of the form λ/n , where λ is a fixed positive constant. Because of symmetry, the number of sinks will have the same distribution as the number of sources. Our main motivation is to understand how this distribution changes as we pass through the critical point p = 1 /n . Since we are in the sparse regime, it makes sense to include the number of isolated vertices as well. In a directed graph an isolated vertex can be regarded as a vertex that is both a source and a sink. We prove asymptotic normality for each of these parameters when p = λ/n . Our method is based on the analysis of a multivariate generating function from a work of Gessel.