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1. Learning Simplicial Complexes from Persistence Diagrams

   Belton, Robin Lynne; Fasy, Brittany Terese; Mertz, Rostik; Micka, Samuel; Millman, David L.; Salinas, Daniel; Schenfisch, Anna; Schupbach, Jordan; Williams, Lucia (August 2018, Canadian Conference on Computational Geometry)

   Topological Data Analysis (TDA) studies the “shape” of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K = (V, E) in R^2, i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams. 

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2. Learning Simplicial Complexes from Persistence Diagrams

   Robin Lynne Belton, Brittany Terese (July 2018, CCCG '18: Thirtieth Canadian Conference on Computational Geometry)

   Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K=(V,E) in R^2 , i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams. 

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3. Temporal network analysis using zigzag persistence

   https://doi.org/10.1140/epjds/s13688-023-00379-5  

   Myers, Audun; Muñoz, David; Khasawneh, Firas A; Munch, Elizabeth (December 2023, EPJ Data Science)

   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. 

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4. Topological Data Analysis for Multivariate Time Series Data

   https://doi.org/10.3390/e25111509  

   El-Yaagoubi, Anass B.; Chung, Moo K.; Ombao, Hernando (November 2023, Entropy)

   Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach that can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent homology (PH), which can extract topological properties from data at various scales. The aim of this article is to introduce TDA concepts to a statistical audience and provide an approach to analyzing multivariate time series data. The application’s focus will be on multivariate brain signals and brain connectivity networks. Finally, this paper concludes with an overview of some open problems and potential application of TDA to modeling directionality in a brain network, as well as the casting of TDA in the context of mixed effect models to capture variations in the topological properties of data collected from multiple subjects. 

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5. Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences

   https://doi.org/10.4230/LIPIcs.ESA.2023.41  

   Ding, Ming; Zhang, Peng (August 2023, Leibniz International Proceedings in Informatics (LIPIcs):31st Annual European Symposium on Algorithms (ESA 2023))

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J.; Herman, Grzegorz (Ed.)

   We present efficient algorithms for solving systems of linear equations in 1-Laplacians of well-shaped simplicial complexes. 1-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in ℝ³ (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2014], Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]). Furthermore, Nested Dissection provides quadratic time solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in ℝ³. We generalize the specialized solvers for 1-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into "disjoint" and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC'2018]). 

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