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

A new arXiv preprint by Haruka Ezoe and Ryohei Hisano tackles a fundamental problem in dynamic network analysis: representing temporal network evolution in a geometrically meaningful way. Traditional spectral embedding methods suffer from ambiguity due to general linear transformations, which distort distances and invalidate temporal comparisons. The authors propose MENT (Multiscale Euclidean Network Trajectories), which uses second-moment geometry and isotropic normalization on anchor latent positions. This reduces the ambiguity to orthogonal transformations, preserving the core geometric structure. The framework defines a trace variation distance and mode-wise variation distances along orthogonal directions, then applies multidimensional scaling to produce low-dimensional trajectories at both global and mode-wise levels.

MENT supports multiple inference tasks: it enables mode-wise decomposition of temporal changes, attribution of those changes to specific nodes, and change point detection via 1D trajectories. The authors prove consistency of their unfolded spectral embedding and the induced trajectories. Experiments on two synthetic and two real dynamic networks demonstrate that MENT recovers temporal structure stably and interpretably, outperforming existing change point detection baselines. The work has implications for analyzing evolving social networks, biological interactions, and communication graphs where understanding when and how structural shifts occur is critical.