Title: Spatial Graph Neural Networks: Architectures, Invariance, and Computational Trade-offs

Presented by Derek Bagagem, Computing PhD student, Data Science emphasis

Abstract

Graph representation learning has become a central tool for reasoning over complex relational systems. In many domains of practical interest, however, graphs are not purely combinatorial objects: nodes are embedded in a metric space and the geometry of that embedding critically shapes the underlying phenomena. This paper examines how modern spatial graph neural networks attempt to encode this geometric structure while respecting basic symmetry requirements and remaining computationally tractable.

We analyze four representative architectures along a single line of development: the Gated Graph Sequence Neural Network (GGS-NN), which operates purely on topology; the Spatial Graph Convolutional Network (SGCN), which introduces coordinate-dependent filters but relies on data augmentation for rotational robustness; SchNet, which enforces Euclidean invariance through continuous-filter convolutions on interatomic distances; and Spatial Graph Message Passing (SGMP), which attains SE(3)-invariant and information-complete local representations via higher-order geometric invariants.

Across these models we compare how geometry is encoded, which invariances are guaranteed by construction, and how expressive power trades off against computational cost. This analysis exposes three persistent gaps: weak generalization to higher-dimensional spaces, a tension between geometric completeness and scalability, and limited theoretical guarantees beyond specific constructions. Motivated by these shortcomings, we outline a research program aimed at E($n$)-invariant, provably injective, and linearly scalable spatial GNNs. The goal is a framework in which geometry and graph structure are treated in a unified, symmetry-aware manner for graphs embedded in arbitrary dimension.

Advisor: Dr. Edoardo Serra

Committee Members: Dr. Francesco Gullo, Dr. Oliviero Andreussi

External Examiner: Dr. Matt Williamson