Multiscale Euclidean Network Trajectories: Second-Moment Geometry, Attribution, and Change Points

Understanding a dynamic network requires more than locating when it changes: we also want to know the magnitude and structural direction of change and which nodes contribute. Unfolded spectral embedding places network snapshots in a joint representation, but the underlying rectangular latent factorization is invariant under general linear transformations. Such transformations preserve edge probabilities while potentially distorting Euclidean second moments. We introduce Multiscale Euclidean Network Trajectories (MENT), whose central estimands are pairwise second-moment operators of the dynamic latent positions. Under an isotropic normalization of the shared anchor latent positions, these operators are identifiable up to simultaneous orthogonal conjugation. They yield a global trace variation distance and, through an aggregated operator, shared orthogonal directions for mode-wise distances. The squared global distance decomposes exactly across these modes. Classical multidimensional scaling turns the distances into global and mode-wise trajectories of time. A modified unfolded spectral estimator consistently recovers the operators and induced trajectories, and we establish bounds connecting node attributions to trajectory displacements and trajectory estimation to change point localization. Synthetic experiments validate geometric recovery and show improved detection of mode-specific changes against baselines. Analyses of two real networks illustrate interpretable mode- and node-level descriptions of temporal evolution.