Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

TL;DR

This paper explores the geometric and statistical limitations of learning dynamic latent structures in temporal networks modeled by Random Dot Product Graphs, proposing a framework to understand gauge ambiguities and recover underlying dynamics.

Contribution

It introduces a geometric framework using principal fiber bundles to formalize obstructions in learning RDPG dynamics and demonstrates how dynamics structure can resolve gauge ambiguities.

Findings

Identifies skew-symmetric generators as invisible dynamics.

Characterizes the realizable tangent space dimension as $nd - d(d-1)/2$.

Shows the spectral gap links curvature, injectivity, and Fisher information.

Abstract

Can we learn the differential equations governing the evolution of a temporal network? We investigate this within Random Dot Product Graphs (RDPGs), where each network snapshot is generated from latent positions evolving under unknown dynamics. We identify three fundamental obstructions: gauge freedom from rotational ambiguity in latent positions, realizability constraints from the manifold structure of the probability matrix, and trajectory recovery artifacts from spectral embedding. We develop a geometric framework based on principal fiber bundles that formalizes these obstructions. We characterize invisible dynamics as exactly the skew-symmetric generators, and show the realizable tangent space has dimension $nd - d(d-1)/2$. An holonomy dichotomy emerges: polynomial dynamics have commuting generators, stationary eigenvectors, and trivial holonomy, making gauge alignment purely statistical; Laplacian dynamics satisfy a non-commutativity criterion producing nontrivial holonomy, with curvature weighted by $1/(\lambda_\iota + \lambda_\gamma)$ linking gauge sensitivity to the spectral gap. In $d=2$ this yields full restricted holonomy $\mathrm{SO}(2)$; for $d \ge 3$ generic full $\mathrm{SO}(d)$ remains conjectural. Cram'er--Rao lower bounds reveal that the same spectral gap controlling curvature and injectivity simultaneously controls Fisher information, so geometric and statistical difficulty are inextricable. We prove an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve gauge ambiguity. We demonstrate this constructively with anchor-based alignment and a UDE pipeline recovering vector fields from noisy graph sequences. Yet finite-sample interactions between noise, gauge, and dynamics expressiveness remain beyond the asymptotic theory. We frame this gap as an open challenge.