Circular Max-Flow for Periodic Data via Reeb Graphs

We introduce a max-flow framework for data with periodic boundary conditions, motivated by the analysis of transport in atomistic materials. Starting from a space X equipped with a map into the circle encoding a chosen periodic direction, we use the associated Reeb graph to reduce the geometry of X to a directed one-dimensional tunnel network. We then augment this graph with capacity constraints derived from cross-sectional integrals with respect to Hausdorff measure, so that edge capacities represent bottlenecks in the corresponding level-set components. To obtain a scalar transport descriptor from this capacity-augmented directed Reeb graph, we define circular max-flow for directed graphs mapped to the circle. Unlike classical source-target max-flow, this formulation does not require choosing an inlet and an outlet, and is therefore intrinsic to the periodic setting. We show that circular max-flow can be computed through a linear optimization problem related to minimum-cost circulations, and we prove that its value agrees with the flow obtained on the periodically unrolled graph. We also prove the continuity results needed to justify the capacity construction and verify that the assumptions cover void spaces arising from finite thickened backbones in the torus. The appendix illustrates the framework on simulated periodic point clouds and reports results from a separate materials-science application to self-diffusion in glasses.