Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs

What the math gives to ML

The paper identifies an exact computational shortcut for Lin–Lu–Yau curvature: on weighted graphs, for every edge and every idleness parameter p greater than or equal to 1/2, Ollivier curvature is linear in 1-p, so the derivative-like LLY curvature is obtained exactly from a single Wasserstein computation at p=1/2. This is transferable to graph neural networks because curvature can become a precomputed, principled edge feature or message-passing bias without numerically taking a limit as p approaches 1. The most practical use is curvature-aware graph attention or edge gating, with curvature calculated once on the training graph and evaluated against degree-only and standard GNN baselines.