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

TL;DR

This paper investigates the invertibility of adjacency matrices of sparse random graphs, revealing a sharp transition based on edge probability and highlighting the role of zero rows or columns in non-invertibility.

Contribution

It establishes precise thresholds for invertibility of adjacency matrices in various sparse random graph models, linking invertibility to zero rows or columns and providing bounds on their condition numbers.

Findings

Invertibility sharply transitions at np ≈ log n

Zero rows or columns are primary cause of non-invertibility

Condition number is approximately n^{1+o(1)} when invertible

Abstract

We consider three different models of sparse random graphs:~undirected and directed Erd\H{o}s-R\'{e}nyi 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 right vertices in the latter case). If $np \le \log n -k(n)$ then these matrices are invertible with probability approaching zero, as $n \to \infty$. In the intermediate region, when $np=\log n + k(n)$, for a bounded sequence $k(n) \in \mathbb{R}$, the event $\Omega_0$ that the adjacency matrix has a zero row or a column and its complement both have non-vanishing probability. For such choices of $p$ our results show that conditioned on the event $\Omega_0^c$ the matrices are again invertible with probability tending to one. This shows that the primary reason for the non-invertibility of such matrices is the existence of a zero row or a column. The bounds on the probability of the invertibility of these matrices are a consequence of quantitative lower bounds on their smallest singular values. Combining this with an upper bound on the largest singular value of the centered version of these matrices we show that the (modified) condition number is $O(n^{1+o(1)})$ on the event that there is no zero row or column, with large probability. This matches with von Neumann's prediction about the condition number of random matrices up to a factor of $n^{o(1)}$, for the entire range of $p$.