Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

@article{Basak2018SharpTO,
  title={Sharp transition of the invertibility of the adjacency matrices of sparse random graphs},
  author={Anirban Basak and Mark Rudelson},
  journal={arXiv: Probability},
  year={2018}
}
We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we show that if the edge connectivity probability $p \in (0,1)$ satisfies $n p \ge \log n + k(n)$ with $k(n) \to \infty$ as $n \to \infty$, then the adjacency matrix is invertible with probability approaching one (here $n$ is the number of vertices in the two former cases and the number of left and… 
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References

SHOWING 1-10 OF 79 REFERENCES
On the singularity of adjacency matrices for random regular digraphs
We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming $$\min (d,n-d)\ge C\log
Circular law for sparse random regular digraphs
Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices.
Adjacency matrices of random digraphs: singularity and anti-concentration
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We
Invertibility of adjacency matrices for random d-regular graphs
Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*}
The smallest singular value of a shifted d-regular random square matrix
We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2
Invertibility of adjacency matrices for random d-regular directed graphs
Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random
Cokernels of adjacency matrices of random $r$-regular graphs
We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of M\'esz\'aros. We
The rank of random regular digraphs of constant degree
TLDR
It is shown that A_n has rank at least at least $n-1$ with probability going to one as $n$ goes to infinity.
The Circular Law for random regular digraphs
  • Nicholas A. Cook
  • Mathematics
    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
  • 2019
Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends
Singularity of sparse Bernoulli matrices
Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$,
...
1
2
3
4
5
...