Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

TL;DR

It is proved an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve gauge ambiguity, and finite-sample interactions between noise, gauge, and dynamics expressiveness remain beyond the asymptotic theory.

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.

Figures (3)

• Figure 1: Anchor-based alignment experiment ($n = 200$, $d = 2$, polynomial dynamics $\dot{X} = (\alpha_0 I + \alpha_1 P) X$ with $m = 3$ Bernoulli samples per time step). (a) Alignment error vs. number of anchor nodes: below $n_a = d = 2$ (dashed vertical line) Procrustes is underdetermined and alignment fails; with sufficient anchors, error stabilizes at the ASE noise floor. (b) Error accumulation over trajectory length: anchor-based alignment (blue) remains bounded while sequential Procrustes (coral, dashed) grows with $T$, consistent with $O(\sqrt{T})$ drift accumulation. (c) Effect of anchor drift rate $\epsilon$: with larger $\epsilon$, systematic bias from drifting anchors becomes visible. (d) Per-timestep alignment error for the $T = 200$ trajectory: the anchor-based error (blue) remains approximately flat, while sequential Procrustes (coral) accumulates drift, with the gap widening toward later time steps. Shaded bands show $\pm 1$ standard deviation across 20 Monte Carlo repetitions.
• Figure 2: (a) Alignment error as a function of the initial embedding norm scale (proxy for signal strength): smaller norms yield weaker signals and less informative Bernoulli observations, increasing ASE noise and degrading alignment for both methods. Note that uniform scaling preserves the condition number $\sigma_1 / \sigma_2$ (constant at $\approx 2.2$); this experiment varies signal magnitude rather than the spectral gap ratio. (b) Phase portrait with $n_a = 15$ anchor nodes (red diamonds, stationary) and non-anchor nodes (gray trajectories evolving under polynomial dynamics) in $B_+^2$.
• Figure 3: UDE pipeline experiment ($n = 200$, $d = 3$, damped spiral dynamics with rotation around $(1,1,1)/\sqrt{3}$, $m = 10$ Bernoulli samples per frame). (a) True trajectories (gray) and anchor-aligned ASE (blue) in $B_+^3$; anchor nodes shown as red diamonds. (b) Learned NN residual $f_\theta(\delta)$ vs. true residual $f_u(\delta)$: anchor-aligned data (left) produces tight agreement along the diagonal; sequential Procrustes (center) and unaligned data (right) degrade progressively. (c) Symbolic regression Pareto fronts (loss vs. expression complexity): anchor-aligned data achieves $1$--$2$ orders of magnitude lower loss at each complexity level. (d) Total dynamics MSE (log scale) by alignment condition: anchor alignment achieves MSE $\approx 6 \times 10^{-4}$, sequential Procrustes $\approx 8 \times 10^{-3}$ ($13\times$ worse), and unaligned $\approx 0.44$ ($\approx 700\times$ worse). Error bars show $\pm 1$ standard deviation across 5 repetitions.

Theorems & Definitions (61)

• Remark
• Definition 1: Observable and Invisible Dynamics
• Theorem 1: Characterization of Invisible Dynamics
• proof
• Proposition 1: Tangent Space Constraint
• proof
• Corollary 1: Dimension Count
• Remark
• Definition 2: Valid Latent Positions
• Definition 3: RDPG Principal Bundle