Latent Distance Estimation for Random Geometric Graphs
This work addresses a fundamental challenge in network analysis for researchers and practitioners using latent space models, offering a novel spectral method that is efficient and does not require assumptions about the link function.
The paper tackles the problem of estimating pairwise distances between latent points in random geometric graphs without prior knowledge of the link function, achieving a convergence rate matching nonparametric function estimation up to a logarithmic factor and also enabling consistent estimation of the latent space dimension.
Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of $n$ points $X_1,X_2,\cdots,X_n$ on the Euclidean sphere~$\mathbb{S}^{d-1}$ which represents the latent positions of nodes of the network. The connection probabilities between the nodes are determined by an unknown function (referred to as the "link" function) evaluated at the distance between the latent points. We introduce a spectral estimator of the pairwise distance between latent points and we prove that its rate of convergence is the same as the nonparametric estimation of a function on $\mathbb{S}^{d-1}$, up to a logarithmic factor. In addition, we provide an efficient spectral algorithm to compute this estimator without any knowledge on the nonparametric link function. As a byproduct, our method can also consistently estimate the dimension $d$ of the latent space.