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

We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies np≥log n+k(n) with k(n)→∞ as n→∞ , then the adjacency matrix is invertible with probability approaching one ( n is the number of vertices in the two former cases and the same for each part in the latter case). For np≤log n-k(n) these matrices are invertible with probability approaching zero, as n→∞ . In the intermediate region, when np=log n+k(n) , for a bounded sequence k(n)∈ℝ , the event Ω _0 that the adjacency matrix has a zero row or a column and its complement both have a non-vanishing probability. For such choices of p our results show that conditioned on the event Ω _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. We further derive a bound on the (modified) condition number of these matrices on Ω _0^c , with a large probability, establishing von Neumann’s prediction about the condition number up to a factor of n^o(1) .