Locality of random digraphs on expanders

Abstract

We study random digraphs on sequences of expanders with a bounded average degree which converge locally in probability. We prove that the relative size and the threshold for the existence of a giant strongly connected component as well as the asymptotic fraction of nodes with giant fan-in or nodes with giant fan-out are local, in the sense that they are the same for two sequences with the same 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 nontree-like graphs.

In the course of establishing these results, we generalize previous results on the locality of the size of the giant to expanders of bounded average degree with possibly nontree-like limits. We also show that, regardless of the local convergence of a sequence, the uniqueness of the giant and convergence of its relative size for unoriented percolation imply the bow-tie structure for directed percolation.

An application of our methods shows that the critical threshold for bond percolation and random digraphs on preferential attachment graphs is ${\mathit{p}}_{\mathit{c}}=0$ with an infinite order phase transition at ${\mathit{p}}_{\mathit{c}}$.