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 n p ≥ 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 n p ≤ log n - k ( n ) these matrices are invertible with probability approaching zero, as n → ∞ . In the intermediate region, when n p = 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 ) .