A central limit theorem for scaled eigenvectors of random dot product graphs
This provides theoretical guarantees for statistical inference in network analysis, but it is incremental as it builds on existing methodology for latent position estimation.
The paper proves a central limit theorem for the largest eigenvectors of adjacency matrices in random dot product graphs, showing that scaled differences between estimated and true latent positions converge to a mixture of Gaussian random variables, with a corollary applying this to Erdös-Renyi random graphs.
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 \citet{sussman2012universally} 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.