Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

Giulio Valentino Dalla Riva

stat.MEMar 5, 2026

Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

Authors: Giulio Valentino Dalla Riva

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 & tables

Result	Status	Scope
Theorem 1	Proved	Full-rank latent states
Proposition 9	Proved	Horizontal symmetric families
Proposition 11	Proved	Local nonzero commutator criterion
Corollary 3	Proved	Linear dynamics
Proposition 14	Proved	Polynomial dynamics + regularity assumptions
Theorem 4	Proved	Symmetric-generator identifiability

Table 1: Result status map: proved results vs conditional criteria vs conjecture.

Approach	Smoothness	Dynamical Consistency
Hierarchical GP prior	Yes ( $C^{k}$ )	No: interpolation only
SDE $dX=f(X)\,dt+\sigma\,dW$	No (Hölder $<1/2$ )	Yes, as $\sigma\to 0$
Neural ODE	Yes ( $C^{k}$ )	Yes (by construction)
GP-ODE (NPODE) ( Heinonen et al., 2018 )	Yes ( $C^{1}$ )	Yes: learns $f$ as GP
Structure-constrained (ours)	Yes	Yes: family $\mathcal{F}$ enforced

Table 2: Comparison of approaches: smoothness vs dynamical consistency.

Family	Form	Horizontal?	Parameters
Linear symmetric	$\dot{X}=NX$ , $N=N^{\top}$	Always	$n(n+1)/2$
Polynomial	$\dot{X}=(\sum_{k}\alpha_{k}P^{k})X$	Always	$K+1$
Laplacian	$\dot{X}=-LX$	Always	0 (fixed)
Message-passing	$\dot{x}_{i}=\sum_{j}P_{ij}g(x_{i},x_{j})$	If $g$ symmetric	$\|\theta\|$
Centroid circulation	$\dot{x}_{i}=(x_{i}-\bar{x})A$	No	$d(d-1)/2$
General linear	$\dot{X}=NX$	If $X^{\top}NX$ sym.	$n^{2}$

Table 3: Families of RDPG dynamics. The polynomial family is always horizontal, has few parameters (

K+1

independent of

n

), and is gauge-invariant (

N(P)

computable without alignment). Centroid circulation is observable but not horizontal.

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.

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