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

2013/05/31 by Athreya, Avanti, Lyzinski, Vince, Marchette, David J. +3
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1305.7388

Abstract

We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology outlined in \citetsussman2012universally to construct consistent estimates for the latent positions, and we show that the appropriately scaled differences between the estimated and true latent positions converge to a mixture of Gaussian random variables. As a corollary, we obtain a central limit theorem for the first eigenvector of the adjacency matrix of an Erdös-Renyi random graph.

Related

• Limit theorems for eigenvectors of the normalized Laplacian for random graphs
• Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes
• Two CLTs for Sparse Random Matrices
• Sparse random graphs: Eigenvalues and Eigenvectors
• Functional Central Limit Theorem for the principal eigenvalue of dynamic Erdős-Rényi random graphs