Entrywise Bounds for Eigenvectors of Random Graphs.

Let G be a graph randomly selected from G(n,p), the space of Erdos-Renyi Random graphs with parameters n and p, where p >= log(6)n/n. Also, let A be the adjacency matrix of G, and v(1) be the first eigenvector of A. We provide two short proofs of the following statement: For all i is an element of [n], for some constant c > 0 vertical bar v(1)(i) - 1/root n vertical bar <= c1/root nlogn/log(np)root logn/np with probability 1 - o(1). This gives nearly optimal bounds on the entrywise stability of the first eigenvector of (Erdos-Renyi) Random graphs. This question about entrywise bounds was motivated by a problem in unsupervised spectral clustering. We make some progress towards solving that problem.