Local Equivalence and Intrinsic Metrics between Reeb Graphs

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as {\em functional distortion} or {\em interleaving} that are provably more discriminative than the so-called {\em bottleneck distance}, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough {\em locally}, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the {\em intrinsic metrics} induced by these distances, which turn out to be all {\em globally} equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them

Local Equivalence and Intrinsic Metrics between Reeb Graphs

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them

Barcode Embeddings for Metric Graphs

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GH-dense subset. Moreover, we show that is globally injective on a full measure subset of metric graphs, in the appropriate sense.Comment: The newest draft clarifies the proofs in Sections 7 and 8, and provides improved figures therein. It also includes a results section in the introductio

Metric Graph Approximations of Geodesic Spaces

A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. In our work, given a compact geodesic metric space $X$, we define a sequence $(\delta^X_n)_{n \geq 0}$ of non-negative real numbers by $$\delta^X_n:=\inf \{d_{\mathrm{GH}}(X,G): G \text{ a finite metric graph, } \beta_1(G)\leq n \} .$$ By construction, and the above result, this is a non-increasing sequence with limit $0$. We study this sequence and its rates of decay with $n$. We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of $X$. Furthermore, we specifically analyze $\delta_0^X$ and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the function given by distance to a reference point

A Family of Metrics from the Truncated Smoothing of Reeb Graphs

In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter ?. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ? ? ? 2?, where ? is the smoothing parameter. Then, for the restriction of ? ? [0,?], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ? [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ? [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m\u27 ? [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs

Labeled Interleaving Distance for Reeb Graphs

Merge trees, contour trees, and Reeb graphs are graph-based topological descriptors that capture topological changes of (sub)level sets of scalar fields. Comparing scalar fields using their topological descriptors has many applications in topological data analysis and visualization of scientific data. Recently, Munch and Stefanou introduced a labeled interleaving distance for comparing two labeled merge trees, which enjoys a number of theoretical and algorithmic properties. In particular, the labeled interleaving distance between merge trees can be computed in polynomial time. In this work, we define the labeled interleaving distance for labeled Reeb graphs. We then prove that the (ordinary) interleaving distance between Reeb graphs equals the minimum of the labeled interleaving distance over all labelings. We also provide an efficient algorithm for computing the labeled interleaving distance between two labeled contour trees (which are special types of Reeb graphs that arise from simply-connected domains). In the case of merge trees, the notion of the labeled interleaving distance was used by Gasparovic et al. to prove that the (ordinary) interleaving distance on the set of (unlabeled) merge trees is intrinsic. As our final contribution, we present counterexamples showing that, on the contrary, the (ordinary) interleaving distance on (unlabeled) Reeb graphs (and contour trees) is not intrinsic. It turns out that, under mild conditions on the labelings, the labeled interleaving distance is a metric on isomorphism classes of Reeb graphs, analogous to the ordinary interleaving distance. This provides new metrics on large classes of Reeb graphs