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Transport-Free Lower Bounds for Ollivier--Ricci Curvature in Higher-Order Networks: Scope, Tightness, and Computational Applications
arXiv:2405.13302v2 Announce Type: replace-cross Abstract: We study how the Bauer--Jost--Liu curvature bound applies to hypergraph Ollivier--Ricci curvature (ORC) and how its accuracy depends on local mass and geometry. An explicit weighted-graph representation shows how the graph theorem applies. A transport construction transfers the bounds to monotone pairwise hyperedge aggregations. We analyze endpoint cancellation under laziness, quantify the exact improvement over the earlier mixture bound, and characterize the extremal geometry at fixed endpoint and overlap statistics. Further results quantify worst-case slack, stability, and exactness on isolated hyperedges. A capacity-constrained short-detour refinement uses additional geometric information and improves monotonically as the query budget increases. As a computational application, a threshold cascade preserves exact transport decisions. Numerical verification and public-hypergraph experiments examine tightness, walk dependence, and aggregation, including evaluations on 42,940 finite pairwise comparisons and 64,620 public-data instances. Together, the results specify the scope, equality mechanisms, and computational uses of BJL-type hypergraph bounds.
Abstract:We study how the Bauer--Jost--Liu curvature bound applies to hypergraph Ollivier--Ricci curvature (ORC) and how its accuracy depends on local mass and geometry. An explicit weighted-graph representation shows how the graph theorem applies. A transport construction transfers the bounds to monotone pairwise hyperedge aggregations. We analyze endpoint cancellation under laziness, quantify the exact improvement over the earlier mixture bound, and characterize the extremal geometry at fixed endpoint and overlap statistics. Further results quantify worst-case slack, stability, and exactness on isolated hyperedges. A capacity-constrained short-detour refinement uses additional geometric information and improves monotonically as the query budget increases. As a computational application, a threshold cascade preserves exact transport decisions. Numerical verification and public-hypergraph experiments examine tightness, walk dependence, and aggregation, including evaluations on 42,940 finite pairwise comparisons and 64,620 public-data instances. Together, the results specify the scope, equality mechanisms, and computational uses of BJL-type hypergraph bounds.
Machine Learning (stat.ML); Discrete Mathematics (cs.DM); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2405.13302 [stat.ML]
(or arXiv:2405.13302v2 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2405.13302 Focus to learn more arXiv-issued DOI via DataCite
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