  In this article, we consider the problem of testing whether two latent
position random graphs are correlated. We propose a test statistic based on the
kernel method and introduce the estimation procedure based on the spectral
decomposition of adjacency matrices. Even if no kernel function is specified,
the sample graph covariance based on our proposed estimation method will
converge to the population version. The asymptotic distribution of the sample
covariance can also be obtained. We design a procedure for testing independence
under permutation tests and demonstrate that our proposed test statistic is
consistent and valid. Our estimation method can be extended to the spectral
decomposition of normalized Laplacian matrices and inhomogeneous random graphs.
Our method achieves promising results on both simulated and real data.
