# Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

arXiv: https://arxiv.org/abs/2603.05703 · HTML: https://mlsanity.com/paper/2603.05703/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 27 references that are in the map.
- First posted on arXiv 5 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Spectral embedding for dynamic networks with stability guarantees](https://mlsanity.com/paper/2106.01282/) (2021)
- [Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral](https://mlsanity.com/paper/1710.10936/) (2017)
- [Efficient Estimation for Random Dot Product Graphs via a One-step Procedure](https://mlsanity.com/paper/1910.04333/) (2019)
- [The eigenvalues of stochastic blockmodel graphs](https://mlsanity.com/paper/1803.11551/) (2018)
- [Euclidean mirrors and dynamics in network time series](https://mlsanity.com/paper/2205.06877/) (2022)
- [Optimal Bayesian Estimation for Random Dot Product Graphs](https://mlsanity.com/paper/1904.12070/) (2019)

## Abstract (arXiv)

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_iota + λγ) linking gauge sensitivity to the spectral gap. In d=2 this yields full restricted holonomy SO(2); for d ≥ 3 generic full 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.