Multiscale Euclidean Network Trajectories: Second-Moment Geometry, Attribution, and Change Points
Abstract
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.
Related
- Short-term solar eclipse predictions using Euclidean geometry and dynamical systems
- Rennala NSGD: Asynchronous Stochastic Optimization Beyond Euclidean Geometry
- A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media
- From Dataset Spectral Geometry to Network Weights: A Geometry-Aware Initialization for Sigmoidal MLPs in Image Classification
- Geometry and cooling effects on an induction self-heating insert for rapid mold temperature control: Experimental, Multiphysics, and artificial neural network-based analysis
BibTeX
@misc{ezoe2026multiscale,
title = {{Multiscale Euclidean Network Trajectories: Second-Moment Geometry, Attribution, and Change Points}},
author = {Haruka Ezoe and Ryohei Hisano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.04589},
eprint = {2605.04589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.04589},
}
RIS
TY - UNPB
TI - Multiscale Euclidean Network Trajectories: Second-Moment Geometry, Attribution, and Change Points
AU - Haruka Ezoe
AU - Ryohei Hisano
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2605.04589
UR - https://arxiv.org/abs/2605.04589
AB - 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.
ER -