Inverse-weight Ricci flow on graphs with cycles

Determine the behavior of the metric-dependent inverse-weight Lin–Lu–Yau Ricci flow on finite graphs containing cycles, extending the analysis beyond finite trees.

Background

The paper studies the continuous Lin–Lu–Yau Ricci flow in the inverse-weight case for finite trees, where the transition probabilities and path metric both depend on evolving edge lengths. It derives a curvature heat equation, proves convergence of individual edge curvatures, and establishes conditional convergence results for nondegenerate edge blocks.

The authors contrast this nonlinear tree-based flow with related flows on general weighted graphs and with models having fixed or uniform transition kernels. They explicitly identify the behavior on graphs with cycles as unresolved, indicating that the tree analysis does not yet extend to the cyclic setting.