Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework

Bridging geometry and topology, curvature is a powerful and expressiveinvariant. While the utility of curvature has been theoretically andempirically confirmed in the context of manifolds and graphs, itsgeneralization to the emerging domain of hypergraphs has remained largelyunexplored. On graphs, Ollivier-Ricci curvature measures differences betweenrandom walks via Wasserstein distances, thus grounding a geometric concept inideas from probability and optimal transport. We develop ORCHID, a flexibleframework generalizing Ollivier-Ricci curvature to hypergraphs, and prove thatthe resulting curvatures have favorable theoretical properties. Throughextensive experiments on synthetic and real-world hypergraphs from differentdomains, we demonstrate that ORCHID curvatures are both scalable and useful toperform a variety of hypergraph tasks in practice.<br

Differentially Describing Groups of Graphs

How does neural connectivity in autistic children differ from neuralconnectivity in healthy children or autistic youths? What patterns in globaltrade networks are shared across classes of goods, and how do these patternschange over time? Answering questions like these requires us to differentiallydescribe groups of graphs: Given a set of graphs and a partition of thesegraphs into groups, discover what graphs in one group have in common, how theysystematically differ from graphs in other groups, and how multiple groups ofgraphs are related. We refer to this task as graph group analysis, which seeksto describe similarities and differences between graph groups by means ofstatistically significant subgraphs. To perform graph group analysis, weintroduce Gragra, which uses maximum entropy modeling to identify anon-redundant set of subgraphs with statistically significant associations toone or more graph groups. Through an extensive set of experiments on a widerange of synthetic and real-world graph groups, we confirm that Gragra workswell in practice.<br