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

Abstract

A generalisation of a latent position network model known as the random dot product graph model is considered. The resulting model may be of independent interest because it has the unique property of representing a mixture of connectivity behaviours as the corresponding convex combination in latent space. We show that, whether the normalised Laplacian or adjacency matrix is used, the vector representations of nodes obtained by spectral embedding provide strongly consistent latent position estimates with asymptotically Gaussian error. Direct methodological consequences follow from the observation that the well-known mixed membership and standard stochastic block models are special cases where the latent positions live respectively inside or on the vertices of a simplex. Estimation via spectral embedding can therefore be achieved by respectively estimating this simplicial support, or fitting a Gaussian mixture model. In the latter case, the use of K-means, as has been previously recommended, is suboptimal and for identifiability reasons unsound. Empirical improvements in link prediction, as well as the potential to uncover much richer latent structure (than available under the mixed membership or standard stochastic block models) are demonstrated in a cyber-security example.

Popularity Adjusted Block Models are Generalized Random Dot Product Graphs

“…the rows of the ASE of A also satisfy the subspace detection property. Theorem 4 builds upon existing work byRubin-Delanchy et al (2017) who describe the convergence behavior of the ASE of A to that of ΠXU , andWang and Xu (2016) who show the necessary conditions for the subspace detection property to hold in noisy cases where the points lie near subspaces. Finally we emphasize that while Noroozi, Rimal, and Pensky (2021+) also considered the use of SSC for community recovery in PABM, they instead applied SSC to the rows of A itself, foregoing the embedding step altogether.…”

“…The resulting procedure, named Orthogonal Spectral Clustering, is presented in Algorithm 1. The following result leverages existing theoretical properties of ASE for estimating of latent positions in a GRDPG(Rubin-Delanchy et al, 2017) to show that B converges almost surely to B; in particular Bij a.s.…”

“…Furthermore, by looking at the proof of Theorem 5 in (Rubin-Delanchy et al, 2017), we see that W * is also blocks diagonal with 2 blocks where the positive eigenvalues of M (k ) forming a block and the negative eigenvalues of M (k ) forming the remaining block. Because M (k ) has one positive eigenvalue and one negative eigenvalue, we see that W * is necessarily of the form W * = 1 0 0 −1 Using this form for W * , we obtain max{ λ(k…”

“…Remark 3. The latent vectors that produce XI p,q X = P are not unique (Rubin-Delanchy et al, 2017). More specifically, if P ij = x i I p,q x j , then we also have for any…”

“…Clearly, any Bernoulli Graph with positive semidefinite P is an RDPG. The positive definite Euclidean inner product in the RDPG model was replaced by an indefinite inner product in Rubin-Delanchy et al (2017), resulting in the Generalized RDPG (GRDPG).…”