I am trying to find a rigorous statement, proof and reference, for a statement that looks as close as possible to the following (informal) one, if it is indeed correct, in which case I imagine it to be classical and well-known:
Start with a set of $N$ (large) nodes (=vertices). Create a directed graph on this set by adding edges from each given node pointing toward $n$ others uniformly chosen at random, where $n$ is itself random, following a distribution which is always the same and has expected value $\kappa$ (and perhaps finite variance?); all these random choices are independent. Then for most(?) nodes $x$ the number of nodes reachable from $x$ is approximately $\Gamma\cdot N$ when $N\to+\infty$, where $\Gamma = -W(-\kappa\,\exp(-\kappa))/\kappa$ is the unique solution strictly between $0$ and $1$ of $\Gamma = \exp(-\kappa(1-\Gamma))$.
The part which really interests me is that the number of nodes reachable from $x$ depends on the expected value $\kappa$ of $n$ and not the distribution. Note that it is crucial for this that we consider a directed graph (in the case of an undirected graph, the distribution of the degree of vertices will be important in computing the size of the giant connected component, because edges will tend to point more frequently toward highly-connected nodes, whereas the directed graph construction ensures that the outdegree is unrelated to the indegree).
I have an informal “proof” of the above informal statement, which I can explain, but I'm looking for a precise statement and proof, and more importantly, a reference where such questions are discussed.
Edit: I believe my question is strongly related to theorem 4 in Mathew D. Penrose's paper “The strong giant in a random digraph” (his $\mu_\infty$ is my $\kappa$, his $\sigma'(\mu_\infty)$ is my $\Gamma$, and his $\xi$ is the non-extinction event, i.e., the event that the chosen initial node $x$ reaches a giant number of nodes). I would still like confirmation that I'm reading this correctly, though, and that the theorem I just mentioned does indeed state what I described informally above.