RKHS-Based Latent Position Random Graph Correlation

We study the problem of evaluating the correlation between two latent position random graphs, which is a challenging issue because the latent positions are generally unobservable in random graphs. We introduce a correlation coefficient that is defined in the reproducing kernel Hilbert space and merely uses spectral decomposition of adjacency matrices. We show that, remarkably, the sample graph covariance converges in probability to its population version even when no kernel function is specified. We further design a permutation procedure to test the independence between two random graphs. We assess the performance of our proposal via simulations and real data analysis, and extend our proposal to spectral decomposition of normalized Laplacian matrices and inhomogeneous random graphs.