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

The Degree-Rips Complexes of an Annulus with Outliers