Steinerberger Curvature On Digraphs -- Discrete Bonnet-Myers and Lichnerowicz Theorems

What the math gives to ML

The paper turns a graph's global directed distance geometry into node-wise signed curvature by solving a distance-matrix linear system, rather than using only local degree or adjacency statistics. This provides a principled positional signal for graph neural networks and directed transformers: nodes can be represented according to how they participate in the entire asymmetric reachability metric, with separate incoming and outgoing roles. The most direct transfer is to compute in/out curvature once per graph, normalize it robustly, and inject it into node embeddings or attention logits. Its value should be tested against degree, PageRank, Laplacian, and shortest-path positional encodings, with preprocessing cost measured explicitly.