New study uses GNNs to predict group properties from Cayley graphs
131,406 Cayley graphs reveal how graph neural networks decode algebraic symmetries.
A new paper by Barket, Grimaldi, and colleagues explores how finite-group properties are encoded in the network geometry of Cayley graphs. They constructed a massive dataset of 131,406 Cayley graphs representing all groups of order at most 767 (excluding 512), enriching each graph with exact algebraic labels and a wide array of graph, cycle, distance, and spectral statistics. This census aims to provide benchmarks for studying the reflection of group-theoretic structure in graph observables.
Beyond enumeration—including new OEIS sequences for monolithic groups and groups generated by few elements—the authors conducted a rigorous comparison of classical models, MLPs, and graph neural networks (GCN, GIN) on the task of predicting algebraic properties directly from graph data. Results show that engineered graph statistics carry strong predictive signal, but GNNs—especially GIN and, in fixed-order settings, GCN—can recover structural signal from raw graph topology. The work identifies empirical regularities and phases of optimality for graph-aware architectures, opening the door to AI-assisted discovery in group theory.
- Dataset includes 131,406 Cayley graphs covering all groups up to order 767 (except 512) with full algebraic labels.
- Graph neural networks (GIN, GCN) outperform MLPs and classical models in predicting group properties from graph structure alone.
- New OEIS sequences contributed for monolithic groups and groups generated by ≤3,4,5 elements; empirical conjectures on diameter, clustering, and spectral gaps.
Why It Matters
Graph neural networks can now learn algebraic structure from group graphs, automating mathematical discovery and classification.