  The degree-Rips bifiltration is the most computable of the parameter-free,
density-sensitive bifiltrations in topological data analysis. It is known that
this construction is stable to small perturbations of the input data, but its
robustness to outliers is not well understood. In recent work, Blumberg-Lesnick
prove a result in this direction using the Prokhorov distance and homotopy
interleavings. Based on experimental evaluation, they argue that a more refined
approach is desirable, and suggest the framework of homology inference.
Motivated by these experiments, we consider a probability measure that is
uniform with high density on an annulus, and uniform with low density on the
disc inside the annulus. We compute the degree-Rips complexes of this
probability space up to homotopy type, using the Adamaszek-Adams computation of
the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the
limit objects for the Blumberg-Lesnick experiments. We argue that the homology
inference approach has strong explanatory power in this case, and suggest
studying the limit objects directly as a strategy for further work.
