Relationship between the topological diffuseness index and persistence diagrams

Characterize the relationship between the topological diffuseness index of a diffuse plate-boundary cover and the persistence diagram of that cover, thereby determining how the fixed-threshold component-counted Euler characteristic relates to the multi-scale information captured by persistent homology.

Background

The paper defines the topological diffuseness index as the absolute deviation of a component-counted weighted Euler characteristic from the Euler characteristic of the sphere. This index is evaluated at a fixed threshold and is designed as a computationally simple measure of topological ambiguity in diffuse plate boundaries.

Persistent homology instead tracks the birth and death of homological features across thresholds. The authors explicitly identify the relationship between these two descriptions—the fixed-threshold TDI and the full persistence diagram of the plate-boundary cover—as unresolved.