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We pursue the problem of modelling and analysing latent space dynamics in collections of networks. Towards this end, we pose and study latent space generative models for signed networks that are amenable to inference via spectral methods. Permitting signs, rather than restricting to unsigned networks, enables richer latent space structure and permissible dynamic mechanisms that can be provably inferred via low rank truncations of observed adjacency matrices. Our treatment of and ability to recover latent space dynamics holds across different levels of granularity, namely, at the overall graph level, for communities of nodes, and even at the individual node level. We provide synthetic and real data examples to illustrate the effectiveness of methodologies and to corroborate accompanying theory. The contributions set forth in this paper complement an emerging statistical paradigm for random graph inference encompassing random dot product graphs and generalizations thereof.

 

This note revisits the classical orthogonal Procrustes problem and investigates the norm-dependent geometric behavior underlying Procrustes alignment for subspaces. It presents generic, deterministic bounds quantifying the performance of a specified Procrustes-based choice of subspace alignment. Numerical examples illustrate the theoretical observations and offer additional, empirical findings which are discussed in detail. This note complements recent advances in statistics involving Procrustean matrix perturbation decompositions and eigenvector estimation.