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Abstract Data consisting of a graph with a function mapping into$${\mathbb {R}}^d$$ $R^d$arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when$$d=1$$ $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts. 

Reeb graphs are widely used in a range of fields for the purposes of analyzing and comparing complex spaces via a simpler combinatorial object. Further, they are closely related to extended persistence diagrams, which largely but not completely encode the information of the Reeb graph. In this paper, we investigate the effect on the persistence diagram of a particular continuous operation on Reeb graphs; namely the (truncated) smoothing operation. This construction arises in the context of the Reeb graph interleaving distance, but separately from that viewpoint provides a simplification of the Reeb graph which continuously shrinks small loops. We then use this characterization to initiate the study of inverse problems for Reeb graphs using smoothing by showing which paths in persistence diagram space (commonly known as vineyards) can be realized by a path in the space of Reeb graphs via these simple operations. This allows us to solve the inverse problem on a certain family of piecewise linear vineyards when fixing an initial Reeb graph. While this particular application is limited in scope, it suggests future directions to more broadly study the inverse problem on Reeb graphs. 

This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks’ shapes that are missed by traditional connectivity and centrality graph statistics. 

Abstract PremiseThe selection ofArabidopsisas a model organism played a pivotal role in advancing genomic science. The competing frameworks to select an agricultural‐ or ecological‐based model species were rejected, in favor of building knowledge in a species that would facilitate genome‐enabled research. MethodsHere, we examine the ability of models based onArabidopsisgene expression data to predict tissue identity in other flowering plants. Comparing different machine learning algorithms, models trained and tested onArabidopsisdata achieved near perfect precision and recall values, whereas when tissue identity is predicted across the flowering plants using models trained onArabidopsisdata, precision values range from 0.69 to 0.74 and recall from 0.54 to 0.64. ResultsThe identity of belowground tissue can be predicted more accurately than other tissue types, and the ability to predict tissue identity is not correlated with phylogenetic distance fromArabidopsis.k‐nearest neighbors is the most successful algorithm, suggesting that gene expression signatures, rather than marker genes, are more valuable to create models for tissue and cell type prediction in plants. DiscussionOur data‐driven results highlight that the assertion that knowledge fromArabidopsisis translatable to other plants is not always true. Considering the current landscape of abundant sequencing data, we should reevaluate the scientific emphasis onArabidopsisand prioritize plant diversity. 

Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such graphs, can we rank how similar they are in such a way that we capture their geometric “shape” in the plane? We explore a method to compare two such embedded graphs, via a simplified combinatorial representation called a tail-less merge tree which encodes the structure based on a fixed direction. First, we examine the properties of a distance designed to compare merge trees called the branching distance, and show that the distance as defined in previous work fails to satisfy some of the requirements of a metric. We incorporate this into a new distance function called average branching distance to compare graphs by looking at the branching distance for merge trees defined over many directions. Despite the theoretical issues, we show that the definition is still quite useful in practice by using our open-source code to cluster data sets of embedded graphs. 

Abstract Graphs in metric spaces appear in a wide range of data sets, and there is a large body of work focused on comparing, matching, or analyzing collections of graphs in different ambient spaces. In this survey, we provide an overview of a diverse collection of distance measures that can be defined on the set of finite graphs immersed (and in some cases, embedded) in a metric space. For each of the distance measures, we recall their definitions and investigate which of the properties of a metric they satisfy. Furthermore we compare the distance measures based on these properties and discuss their computational complexity.