Birth of a giant $(k_1,k_2)$-core in the random digraph

TL;DR

This paper determines the threshold edge-density for the emergence of a giant $(k_1,k_2)$-core in random directed graphs, generalizing core concepts and providing explicit formulas for the critical point.

Contribution

It establishes the existence and precise threshold for the giant $(k_1,k_2)$-core in random digraphs, extending core theory to directed graphs with explicit formulas.

Findings

Existence of a threshold $c^*$ for the $(k_1,k_2)$-core

Explicit formula for $c^*$ involving Poisson probabilities

Asymptotic almost sure presence or absence of the core

Abstract

The $(k_1,k_2)$-core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least $k_1$ and $k_2$ respectively. For $\max\{k_1, k_2\} \geq 2$, we establish existence of the threshold edge-density $c^*=c^*(k_1,k_2)$, such that the random digraph $D(n,m)$, on the vertex set $[n]$ with $m$ edges, asymptotically almost surely has a giant $(k_1,k_2)$-core if $m/n> c^*$, and has no $(k_1,k_2)$-core if $m/n<c^*$. Specifically, denoting $\text{P}(\text{Poisson}(z)\ge k)$ by $p_k(z)$, we prove that $c^*=\min\limits_{z_1,z_2}\max\left\{\tfrac{z_1}{p_{k_1}(z_1)p_{k_2-1}(z_2)}; \tfrac{z_2}{p_{k_1-1}(z_1)p_{k_2}(z_2)}\right\}$.