Locality of Random Digraphs on Expanders

TL;DR

This paper investigates the local properties of random directed graphs on expander graphs, establishing the conditions for the emergence of a giant strongly connected component and describing its structure in terms of local limits and percolation theory.

Contribution

It generalizes the concept of locality of the giant component to expanders with non-tree-like limits and connects percolation thresholds to the structure of directed graphs.

Findings

The threshold for a giant strongly connected component is local and depends on the limit graph.

The digraph exhibits a bow-tie structure with most nodes in the giant component or small fan-in/out sets.

Percolation on the limit graph determines local quantities and the structure of the directed graph.

Abstract

We study random digraphs on sequences of expanders with bounded average degree {which converge locally in probability}. We prove that 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 non tree-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 non-tree 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 $p_c=0$, with an infinite order phase transition at $p_c$.