Know What You Don't Know: Cohomological Obstruction Theory For Equivariant Graph Neural Networks

Bisiriyu Maqsudhat Kofoworola

Keywords: graph neural networks, equivariance, group cohomology, obstruction theory, algebraic topology, geometric deep learning, expressivity, message passing, symmetry, sheaf theory, spectral graph theory, molecular property prediction

TL;DR: We prove that the existence of globally equivariant GNNs is controlled by a class in $H^2(G, \mathcal{F})$, derive exact approximation lower bounds when this class is non-trivial, and validate the theory on synthetic and molecular benchmarks.

Abstract:

We develop a cohomological obstruction theory for equivariant graph neural
networks (GNNs), establishing rigorous conditions under which globally
G-equivariant architectures cannot be assembled from locally equivariant
message-passing operations. Framing GNN layers as sections of associated
vector bundles over a graph, we show that the obstruction to lifting local
equivariance to a globally consistent structure is measured by a class in
the second group cohomology H^2(G, F), where G is the symmetry group and F
is the G-module of feature representations. We prove three main theorems:
(i) a Vanishing Theorem showing that a globally G-equivariant GNN exists if
and only if the obstruction class o(G, F, Gamma) vanishes in H^2(G, F);
(ii) an Expressivity Obstruction Theorem proving that when the obstruction
is non-trivial, any message-passing neural network suffers a quantifiable
approximation gap against the class of truly equivariant functions; and
(iii) a Spectral Realization Theorem decomposing the obstruction class in
terms of the graph Laplacian eigenbasis, showing that expander graphs yield
smaller obstructions and hence smaller expressivity gaps. As corollaries, we
recover the Weisfeiler-Leman expressivity barrier, explain chirality-blindness
in DimeNet via a Z-valued obstruction, and show that E(n)-equivariant GNNs
avoid obstruction by acting freely on generic point configurations.
Experiments on synthetic graphs with prescribed obstruction rank and on the
QM9 molecular benchmark validate our predictions: MPNN error scales linearly
with obstruction rank, and chiral molecules exhibit approximately 1.9x larger
error, matching our theoretical prediction of 2.0x.

Submission Number: 126