SPD Ricci Flow: Graph Representation Learning through Metric Evolution

We introduce **SPD Ricci Flow**, a graph learning framework in which each node's symmetric positive definite (SPD) representation also serves as its local metric and evolves through Ricci-type dynamics. Unlike previous edge-based graph Ricci flows that evolve scalar edge lengths or weights, our framework directly evolves node-wise metric tensors using a matrix-valued Ricci-type curvature operator constructed from AIRM discrepancies between neighboring SPD metrics. This makes curvature-driven metric evolution itself the mechanism for updating node representations. The curvature also enables interpretable analysis of local geometric variation in node representations. We prove continuous-time energy dissipation and show that, under graph-Laplacian consistency, the linearized continuum limit matches the heat-type principal part of Ricci–DeTurck evolution. Experiments on node classification, molecular property prediction, and financial spillover prediction support metric evolution as a mechanism for graph representation learning.