The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

Women and Stability: A Topological View of the Relationship between Women and Armed Conflict in West Africa

The relationship between women and stability, if any, is a topic of much debate and research. Several large and influential organizations have all researched women\u27s effect on stability. Furthermore, several of these world organizations, the United Nations, in particular, have declared gender equality to be a driving force in promoting stability and conflict prevention. Due to the United States active involvement in conflict prevention in such regions as West Africa, research concerning the relationship between women and stability is of particular interest to the United States Africa Command. As such, this research applied Topological Data Analysis, combined with other machine learning algorithms, to Demographic and Health Survey Program data combined with Armed Conflict Location and Event Data so as to observe the relationship between women\u27s status and armed conflicts in the West African region. While this team did not observe any direct correlation between women\u27s well-being and stability - defined as a lack of armed conflict events - the chosen methodologies and data usage have potential implications for future research concerning stability and conflict

Inference of Ancestral Recombination Graphs through Topological Data Analysis

The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relationships within and across species. Recombination, reassortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Ancestral recombination graphs represent potential histories that explicitly accommodate recombination and mutation events across orthologous genomes. However, they are computationally costly to reconstruct, usually being infeasible for more than few tens of genomes. Recently, Topological Data Analysis (TDA) methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. We build upon previous TDA developments for detecting and quantifying recombination, and present a novel framework that can be applied to hundreds of genomes and can be interpreted in terms of minimal histories of mutation and recombination events, quantifying the scales and identifying the genomic locations of recombinations. We implement this framework in a software package, called TARGet, and apply it to several examples, including small migration between different populations, human recombination, and horizontal evolution in finches inhabiting the Gal\'apagos Islands.Comment: 33 pages, 12 figures. The accompanying software, instructions and example files used in the manuscript can be obtained from https://github.com/RabadanLab/TARGe

Topological Parallax: A Geometric Specification for Deep Perception Models

For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deep-learning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset.Comment: 18 pages, 6 figures. Preprint submitted to NeurIPS 202

Experiment of Mapper Algorithm on High-Dimensional Data in Microseismic Monitoring

The objective of this research to utilize data driven methods to analyze microseismic monitoring, especially using Topological data analysis (TDA) with limited physically based approaches. Python Mapper (PM) is the tool of TDA for this study. Microseismic data has great characteristics of big data. Previous studies suggesting stage-by-stage microseismic analysis also avoid the limitation of current software, which can only process slightly over 10,000 data points. During this study, more TDA packages are constantly evolving to handle larger and more complex data such as Betti Mapper by Spark. PM is a tool by combining topology principles and machine learning methods into an integrated data analytic implementation. The high-dimensionality of microseismic data practically limits what classical statistical analyses can achieve. Machine learning techniques such as dimensionality reduction are required for such datasets. Where PM stands out is its ability to retain the raw feature of data set when machine-learning algorithm is applied. The first portion of the study is to observe the data point relation of microseismic data entirely and stage-by-stage. Dividing attributes into location and signal data reveals the relation within and between two different data types. The main discovery from location data of network is the high density areas are tend to be earlier events and could locate where high pressure start to build up, or the origins of the fracture networks. Origins that are far apart in the beginning grow into each other to result in one (most of the time) or more (rarely more than two) networks. The fracture growth with complex directions of extensions can be represented with a much simpler, single-directional network. Signal data reveals location-specific data quality trends. These trends are hardly visible if attributes are investigated in pairs but obvious when mapped altogether. Locational and geological characteristics may be an explanation, but this needs further information to prove the observations. In fracture growth softwares, these trends will allow researchers to ignore the location of the wellbore and focuses at the actual origins of the fracture network. An override including discontinuity of the network and confidence of stimulated reservoir volume could be manually added to improve the accuracy of the fracture simulation. A sensitivity analysis to PM parameters is carried out to test the robustness of the method and comparing raw data clustering method to prove the effectiveness and benefits of using TDA. TDA is a great method for data preprocesses, analyses, and has virtually infinite possibility, but should never be the end of a project. The results from PM could be used as input for many other studies