Inverse mapping from persistence diagrams to graph elements in spatial graphs

Determine an inverse mapping from a persistence diagram of a spatial graph to specific graph elements (such as edges and nodes) so that each point in the diagram can be traced to the structural components generating that topological feature, thereby enabling identification of which edges should be collapsed to preserve specified topological features during topological spatial graph coarsening.

Background

The paper introduces a topological spatial graph coarsening method that collapses short edges while preserving key topological features, calibrated via persistence diagrams constructed from a triangle-aware graph filtration. The coarsening level is selected by minimizing a score that balances reduction in graph complexity with topological distortion measured by the bottleneck distance between persistence diagrams of the original and reduced graphs.

In the conclusion, the authors highlight a forward-looking goal: developing an inverse procedure to identify which edges to collapse in order to preserve certain topological features. Achieving this requires answering an explicit open question—how to invert a persistence diagram to map its points back to the graph elements responsible for those features—so that one can directly target structural modifications consistent with desired topological outcomes.