Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

Learning dynamics from time-varying latent representations is a goal shared across many domains, from neural connectomics to social network analysis, yet it is fundamentally complicated by the non-identifiability inherent in latent variable models. When the likelihood is invariant under a symmetry group, some latent motions are invisible, and local estimates may not stitch into globally consistent trajectories. How should one formalize these obstructions, and what tools can quantify them? We address these questions concretely within the framework of Random Dot Product Graphs (RDPGs), where network snapshots are generated from latent positions evolving under unknown dynamics and the O(d) rotational symmetry is explicit. We identify three fundamental obstructions to recovering the governing differential equations: gauge freedom from rotational non-identifiability, realizability constraints from the manifold structure of the probability matrix, and trajectory recovery artifacts introduced by spectral embedding. We develop a geometric framework based on principal fiber bundles that formalizes these obstructions and reveals their interplay. Connections and curvature on the bundle quantify gauge velocity and explain when local alignment fails to globalize: polynomial dynamics have trivial holonomy, while Laplacian dynamics satisfy a proved non-commutativity criterion for nontrivial holonomy (full restricted SO(2) in dimension two; conditional in higher dimensions). Cramér-Rao bounds reveal that the spectral gap controlling curvature simultaneously controls Fisher information, so geometric and statistical difficulty are inextricable. An identifiability principle shows that symmetric dynamics cannot absorb skew-symmetric gauge contamination, yet significant practical obstacles remain in finite samples. We frame the gap between identifiability and constructive recovery as an open challenge.