Eigenvectors of Random Graphs : Delocalization and Nodal Domains

We study properties of the eigenvectors of adjacency matrices of G(n, p) random graphs, for p = ω(logn)/n. This connects to similar investigations for other random matrix models studied in physics and mathematics. Motivated by the recent paper of Dekel, Lee and Linial we study delocalization properties of eigenvectors and their connection to nodal domains. We show the following for an eigenvector x (normalized s.t. ‖x‖2 = 1): 1. For any S ⊆ [n] and |S| = δn, we have ∑ i∈S xi ≥ δp log(1/pδ) w.h.p. A similar statement proved for δ > 1/2 by Dekel, Lee and Linial. 2. Let p > n−1/20. Then x has exactly two nodal domains whp. (i.e., the subgraph of vertices with xi ≥ 0 is connected and so is the subgraph of vertices with xi ≤ 0). Previously such a statement was not known even for p = 1/2, unless one is allowed to discard O(1/p) “exceptional” vertices of the graph. Our techniques involve using Wigner’s semicircle law on “short scales”, an idea previously used in mathematical physics by Erdős, Schlein and Yau.