A statistical interpretation of spectral embedding: the generalised random dot product graph

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