A Statistical Interpretation of Spectral Embedding: The Generalised Random Dot Product Graph

Abstract Spectral embedding is a procedure which can be used to obtain vector representations of the nodes graph. This paper proposes generalisation latent position network model known as random dot product graph, allow interpretation those estimates. The needed heterophilic connectivity (e.g. ‘opposites attract’) and cope with negative eigenvalues more generally. We show that, whether adjacency or normalised Laplacian matrix used, spectral produces uniformly consistent estimates asymptotically Gaussian error (up identifiability). standard mixed membership stochastic block models are special cases in positions take only K distinct values, representing communities, live (K ? 1)-simplex vertices respectively. Under model, our theory suggests clustering using mixture (rather than K-means) and, under membership, fitting minimum volume enclosing simplex, existing recommendations previously supported non-negative-definite assumptions. Empirical improvements link prediction (over graph), potential uncover richer structure (than posited models) demonstrated cyber-security example.