TITLE: Statistics of strongly connected components in random directed graphs
QUESTION [7 upvotes]: I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for.
By "random directed graph" I mean a digraph with $N$ nodes, where every possible edge has a uniform probability $p$ of being included in the graph, independently of the other edges. For undirected graphs this is sometimes known (possibly erroneously) as the Erdös-Rényi model, but for digraphs I don't know the correct term.
Equivalently, given a random (in general non-symmetric) matrix where every element has an independent probability of being non-zero, one can do a change of basis using a permutation matrix to put it in block upper triangular form; I'm interested in the statistics of the number and size of the resulting blocks on the diagonal.
I'm most interested in knowing the expected number of strongly connected components of size $>1$, for a given $N$ and $p$, but any results concerning the statistics of the strongly connected components (e.g. the size of the largest one) would be enormously helpful. I'm interested primarily in the limit of large $N$, for non-extreme values of $p$.

REPLY [2 votes]: First, this model is typically denoted $D(n,p)$. I'm not aware of a formal name, but personally I like to call it the Bernoulli random digraph. For $p=c/n$, note that the number of vertices with out-degree zero is going to be on the order of $n$. So in this range, the number of strong components will also be on the order of $n$. 
The phase transition for this model where a large strongly connected component emerges is due to Richard Karp and Tomasz Łuczak (Łuczak considers the analogous model $D(n,m)$ where $m$ directed edges are uniformly at random chosen to be present). 
Karp shows that for $p=c/n$ and $c<1$, and any $\omega \to \infty$ however slowly, w.h.p. each strong component of $D(n,p)$ has at most $\omega$ vertices. For $c>1$, there is a unique strong component with roughly $\Theta^2 n$ vertices where $\Theta$ is the unique positive root of $1-\Theta = e^{-c \Theta}$; in this case, w.h.p. all other strong components contain at most $\omega$ vertices. It is not a coincidence that the number of vertices in the giant component of $G(n,p=c/n)$ is roughly $\Theta n$ vertices. 
Boris Pittel and I recently proved that for $D(n, p=c/n), c>1$, the number of vertices in the largest strong component is asymptotically normal. In fact, we show that the pair of the numbers of vertices and arcs in the strong giant is jointly asymptotically 2-dim normal.