Singularity of the k-core of a random graph

TL;DR

This paper proves that the k-core of very sparse Erdős–Rényi random graphs is typically nonsingular, confirming a conjecture and revealing that low-degree dependencies are the main cause of singularity in such graphs.

Contribution

It establishes that for sparse Erdős–Rényi graphs with fixed k ≥ 3, the k-core is almost always nonsingular, resolving Vu's 2014 conjecture.

Findings

The k-core of sparse Erdős–Rényi graphs is typically nonsingular.

Low-degree dependencies are the primary cause of singularity in sparse random graphs.

A new technique uses high-degree vertices to improve rank estimates.

Abstract

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $k\ge 3$ and $\lambda > 0$, an Erd\H{o}s--R\'enyi random graph $G\sim\mathbb{G}(n,\lambda/n)$ with $n$ vertices and edge probability $\lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for "extremely sparse'' random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to "boost'' the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.