We study random digraphs on sequences of expanders with bounded average
degree and weak local limit. The threshold for the existence of a giant
strongly connected component, as well as the asymptotic fraction of nodes with
giant fan-in or giant fan-out are local, in the sense that they are the same
for two sequences with the same weak local limit. The digraph has a bow-tie
structure, with all but a vanishing fraction of nodes lying either in the
unique strongly connected giant and its fan-in and fan-out, or in sets with
small fan-in and small fan-out. All local quantities are expressed in terms of
percolation on the limiting rooted graph, without any structural assumptions on
the limit, allowing, in particular, for non tree-like limits.
In the course of proving these results, we prove that for unoriented
percolation, there is a unique giant above criticality, whose size and critical
threshold are again local. An application of our methods shows that the
critical threshold for bond percolation and random digraphs on preferential
attachment graphs is pcβ=0, with an infinite order phase transition at pcβ.Comment: Added a proof on infinite order phase transition of PA graphs.
Revised introduction and moved the proof on applications to appendi