A review of topological descriptors for amorphous materials complementing graph neural networks

Amorphous systems lack long-range order, making their properties strongly dependent on short-range order (SRO) and medium-range order (MRO) structural motifs and their network connectivity. While graph neural networks (GNNs) excel for crystalline materials, their performance on amorphous systems is limited by graph construction ambiguities, message-passing bottlenecks, data scarcity, and transferability issues. This review surveys traditional descriptors and then focuses on topological approaches: complex network descriptors, persistent homology, rigidity theory (also known as topological constraint theory), and graph limit metrics. We summarize how rings, voids, and percolating backbones correlate with mechanics, transport, and thermodynamics across metallic glasses, oxide glasses, polymers, and organic semiconductors, among others, highlighting their potential to inform the exploration and design of amorphous energy materials. Finally, we provide a shortlist of topological variables with potential for property modeling and set out three directions in datasets, unified topology modeling and topology aware GNN architectures.