Giulio Valentino Dalla Riva
5 min
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.
Alex: Horizontal strips out gauge for a clean path lift. Does it always work perfectly?
Sam: It works locally: from any starting point, there's a unique horizontal lift of an observed path, staying spin-free. But over loops or long paths, curvature twists things—like parallel train tracks curving and meeting rotated. This mismatch, called holonomy, builds up. The paper shows it blocks rule learning, especially when curvature spikes from close eigenvalues in sparse nets.
Alex: Curvature from eigenvalue closeness causes drift. Do different rule types trigger this differently?
Sam: Yes. Polynomial rules on link chances—like sums of powers—commute, keeping holonomy zero: lifts close perfectly. But Laplacian flows, mixing degrees and links, don't commute, building holonomy that scrambles recovery. In food webs, this makes evolving interactions hard to learn without gauge fixes.
Alex: Polynomials avoid the trap because they commute, but Laplacians tangle up. What about fixes like embedding all time points together to smooth gauge jumps?
Sam: Joint embedding stacks snapshots and finds patterns across time, assuming shared directions for hidden points. But in evolving RDPGs, directions rotate as positions change, so it distorts true motion into fake tweaks.
Alex: It warps paths. What about Bayesian smoothing for continuous trajectories?
Sam: Those favor smooth position changes, like steady velocities. It helps choppy estimates but misses dynamical consistency: real rules tie velocity strictly to current position. Smoothing creates fluid paths ignoring that—ones that wander without matching equations.
Alex: You get pretty paths not ruled by equations. Geometry matters because holonomy blocks clean recovery. Changes in link chances give partial clues without full gauges?
Sam: Yes. For linear rules, changes follow a structured equation from latent motion. Noise amplifies errors where spectral gaps are small, linking to curvature.
Alex: Small gaps fuzz estimation. Even with identifiability, practical fixes hit walls?
Sam: Yes. Aligning rotations while fitting rules blurs signal in noise. Holonomy adds unfixable global drift. One fix: anchor nodes that barely move—like stable species—align everything to their fixed spots.
Alex: Anchors pin the gauge. How do tests bear out the theory?
Sam: On simulated food web data, anchors keep alignment steady over long paths, while step-by-step matching drifts. For rule learning—like spirals around centers—anchors recover equations far better than alternatives.
Alex: The geometry spots hurdles like holonomy and guides fixes like anchors for evolving networks. Thanks for joining ResearchPod.