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原文摘要

Balanced Forman (BF) curvature is used in graph learning as a cheap stand-in for Ollivier-Ricci (OR) curvature, yet it does not lower-bound OR curvature. We show that the BF value of an edge certifies a two-sided interval for its lazy OR curvature without solving any optimal transport problem, that BF coincides with a quadrangle-augmented Jost-Liu (JL) bound with its positive parts removed, and that when the 2-hop neighborhood is available, the interval can be sharpened at a cost of $\mathcal O(d_{\max}^{2.5})$ per edge. On 23 benchmark graphs, our quadrangle-augmented bound is tighter than the bounds of JL and Tian-Lubberts-Weber (TLW) on 13.8-100% of the edges and reduces their mean gap to the exact curvature by a median of 31.2%; intersected with the bounds of TLW, the intervals settle the curvature sign of more than half of the edges on 21 of the 23 graphs. On 15 real networks with up to 1,049,866 edges, a certify-then-solve procedure answers sign queries exactly while solving a median of 5.9 times fewer transport problems, for median speedups of 2$\times$ over exact evaluation, measured against a min-cut solver that is itself about two orders of magnitude faster than the widely used $\mathtt{GraphRicciCurvature}$. The main results are machine-checked in Lean 4.