A central limit theorem for scaled eigenvectors of random dot product graphs

Spectral analysis of the adjacency and Laplacian matrices for graphs is of both theoretical (Chung, 1997) and practical (Luxburg, 2007) significance. For instance, the spectrum can be used to characterize the number of connected components in a graph and various properties of random walks on graphs, and the eigenvector corresponding to the second smallest eigenvalue of the Laplacian is used in the solution to a relaxed version of the min-cut problem (Fiedler, 1973). In our current work, we investigate the second-order properties of the eigenvectors corresponding to the largest eigenvalues of the adjacency matrix of a random graph. In particular, we show that under the random dot product graph model (Young and Scheinerman, 2007), the components of the eigenvectors are asymptotically normal and centered around the true latent positions (see Section 4).