Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

• URL: http://arxiv.org/abs/2603.05703v1
• Date: Thu, 05 Mar 2026 21:56:07 GMT
• Title: Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities
• Abstract summary: We investigate the differential equations governing the evolution of a temporal network within Random Dot Product Graphs.<n>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.<n>We prove an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve obstructions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $1/(λ_ι+ λ_γ)$ 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.

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