Counting strongly connected $(k_1,k_2)$-directed cores

TL;DR

This paper analyzes the probability that large, dense directed graphs with minimum in-degree and out-degree constraints are strongly connected, identifying thresholds for the emergence of giant strongly connected cores.

Contribution

It establishes a sharp threshold for the emergence of a giant strongly connected $(k_1,k_2)$-core in random digraphs and shows that such cores are highly likely to be strongly connected.

Findings

Fraction of $k$-strongly connected digraphs approaches 1 for large graphs.

Identifies a sharp edge-density threshold for the birth of a giant $(k_1,k_2)$-core.

Giant $(k_1,k_2)$-core exists and is $k$-strongly connected above the threshold.

Abstract

Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\min\{k_1,k_2\}\ge 2$ and $M/N>\max\{k_1,k_2\}$, $M=\Theta(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core exists and is $k$-strongly connected.