Computational Topology and its Applications in Geometric Design

2021 ◽  
Vol 15 ◽  
Author(s):  
Zhetong Dong ◽  
Hongwei Lin ◽  
Jinhao Chen
Keyword(s):  
Shape Optimization ◽  
Geometric Modeling ◽  
Persistent Homology ◽  
Original Data ◽  
Computational Topology ◽  
Geometric Design ◽  
Topological Invariants ◽  
Topological Design ◽  
Topological Features ◽  
Persistence Diagram

Background: In recent geometric design, many effective toolkits for geometric modeling and optimization have been proposed and applied in practical cases, while effective and efficient designing of shapes that have desirable topological properties remains to be a challenge. The development of computational topology, especially persistent homology, permits convenient usage of topological invariants in shape analysis, geometric modeling, and shape optimization. Persistence diagram, the useful topological summary of persistent homology, provides a stable representation of multiscale homology invariants in the presence of noise in original data. Recent works show the wide use of persistent homology tools in geometric design. Objective: In this paper, we review the geometric design based on computational topological tools in three aspects: the extraction of topological features and representations, topology-aware shape modeling, and topology-based shape optimization. Methods: By tracking the development of each aspect and comparing the methods using classical topological invariants, motivations, and key approaches of important related works based on persistent homology are clarified. Results : We review geometric design through topological extraction, topological design, and shape optimization based on topological preservation. Related works show the successful applications of computational topology tools of geometric design. Conclusion: Solutions for the proposed core problems will affect the geometric design and its applications. In the future, the development of computational topology may boost computer-aided topological design.

scholarly journals Persistence Diagrams to Visualise Damage to Biological Networks

2022 ◽  
Author(s):  
Matthew Bailey ◽  
Mark Wilson
Keyword(s):  
Temporal Resolution ◽  
Critical Points ◽  
Biological Networks ◽  
Persistent Homology ◽  
Sampling Frequency ◽  
Topological Features ◽  
Persistence Diagram ◽  
2D Network ◽  
Over Time ◽  
Persistence Diagrams

One of the critical tools of persistent homology is the persistence diagram. We demonstrate the applicability of a persistence diagram showing the existence of topological features (here rings in a 2D network) generated over time instead of space as a tool to analyse trajectories of biological networks. We show how the time persistence diagram is useful in order to identify critical phenomena such as rupturing and to visualise important features in 2D biological networks; they are particularly useful to highlight patterns of damage and to identify if particular patterns are significant or ephemeral. Persistence diagrams are also used to analyse repair phenomena, and we explore how the measured properties of a dynamical phenomenon change according to the sampling frequency. This shows that the persistence diagrams are robust and still provide useful information even for data of low temporal resolution. Finally, we combine persistence diagrams across many trajectories to show how the technique highlights the existence of sharp transitions at critical points in the rupturing process.

scholarly journals Using persistent homology and dynamical distances to analyze protein binding

2016 ◽  
Vol 15 (1) ◽  
Author(s):  
Violeta Kovacev-Nikolic ◽  
Peter Bubenik ◽  
Dragan Nikolić ◽  
Giseon Heo
Keyword(s):  
Conformational Changes ◽  
Permutation Test ◽  
Persistent Homology ◽  
Protein A ◽  
Support Vector ◽  
Active Site Residues ◽  
Topological Features ◽  
Rips Complex ◽  
Persistence Diagram ◽  
Anisotropic Network

AbstractPersistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic, the persistence landscape, was recently introduced by Bubenik. It is a functional summary, so it is easy to calculate sample means and variances, and it is straightforward to construct various test statistics. Implementing a permutation test we detect conformational changes between closed and open forms of the maltose-binding protein, a large biomolecule consisting of 370 amino acid residues. Furthermore, persistence landscapes can be applied to machine learning methods. A hyperplane from a support vector machine shows the clear separation between the closed and open proteins conformations. Moreover, because our approach captures dynamical properties of the protein our results may help in identifying residues susceptible to ligand binding; we show that the majority of active site residues and allosteric pathway residues are located in the vicinity of the most persistent loop in the corresponding filtered Vietoris-Rips complex. This finding was not observed in the classical anisotropic network model.

scholarly journals Effect of APOE ε4 on multimodal brain connectomic traits: a persistent homology study

2020 ◽  
Vol 21 (S21) ◽  
Author(s):  
Jin Li ◽  
◽  
Chenyuan Bian ◽  
Dandan Chen ◽  
Xianglian Meng ◽  
...  
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Genetic Risk ◽  
Brain Connectivity ◽  
Persistent Homology ◽  
Brain Network ◽  
Resonance Imaging ◽  
Modeling Framework ◽  
Apoe Ε4 ◽  
Topological Features

Abstract Background Although genetic risk factors and network-level neuroimaging abnormalities have shown effects on cognitive performance and brain atrophy in Alzheimer’s disease (AD), little is understood about how apolipoprotein E (APOE) ε4 allele, the best-known genetic risk for AD, affect brain connectivity before the onset of symptomatic AD. This study aims to investigate APOE ε4 effects on brain connectivity from the perspective of multimodal connectome. Results Here, we propose a novel multimodal brain network modeling framework and a network quantification method based on persistent homology for identifying APOE ε4-related network differences. Specifically, we employ sparse representation to integrate multimodal brain network information derived from both the resting state functional magnetic resonance imaging (rs-fMRI) data and the diffusion-weighted magnetic resonance imaging (dw-MRI) data. Moreover, persistent homology is proposed to avoid the ad hoc selection of a specific regularization parameter and to capture valuable brain connectivity patterns from the topological perspective. The experimental results demonstrate that our method outperforms the competing methods, and reasonably yields connectomic patterns specific to APOE ε4 carriers and non-carriers. Conclusions We have proposed a multimodal framework that integrates structural and functional connectivity information for constructing a fused brain network with greater discriminative power. Using persistent homology to extract topological features from the fused brain network, our method can effectively identify APOE ε4-related brain connectomic biomarkers.

scholarly journals A topological characterization of DNA sequences based on chaos geometry and persistent homology

2021 ◽  
Author(s):  
Dong Quan Ngoc Nguyen ◽  
Phuong Dong Tan Le ◽  
Lin Xing ◽  
Lizhen Lin
Keyword(s):  
Dna Sequences ◽  
Algebraic Topology ◽  
Influenza A ◽  
Graphical Representation ◽  
Dimensional Space ◽  
Persistent Homology ◽  
Point Clouds ◽  
Influenza A Viruses ◽  
Topological Characterization ◽  
Topological Features

AbstractMethods for analyzing similarities among DNA sequences play a fundamental role in computational biology, and have a variety of applications in public health, and in the field of genetics. In this paper, a novel geometric and topological method for analyzing similarities among DNA sequences is developed, based on persistent homology from algebraic topology, in combination with chaos geometry in 4-dimensional space as a graphical representation of DNA sequences. Our topological framework for DNA similarity analysis is general, alignment-free, and can deal with DNA sequences of various lengths, while proving first-of-the-kind visualization features for visual inspection of DNA sequences directly, based on topological features of point clouds that represent DNA sequences. As an application, we test our methods on three datasets including genome sequences of different types of Hantavirus, Influenza A viruses, and Human Papillomavirus.

Persistent Homology, Regimes and Climate Data

2021 ◽  
Author(s):  
Kristian Strommen ◽  
Nina Otter ◽  
Matthew Chantry ◽  
Joshua Dorrington
Keyword(s):  
Dynamical Systems ◽  
State Of The Art ◽  
Persistent Homology ◽  
Topological Invariants ◽  
Climate Data ◽  
Future Directions ◽  
Atlantic Sector ◽  
Linear Dynamical Systems ◽  
Markovian Dynamics ◽  
Shed Light

<p>The concept of weather or climate 'regimes' have been studied since the 70s, to a large extent because of the possibility they offer of truncating complicated dynamics to vastly simpler, Markovian, dynamics. Despite their attraction, detecting them in data is often problematic, and a unified definition remains nebulous. We argue that the crucial common feature across different dynamical systems with regimes is the non-trivial topology of the underlying phase space. Such non-trivial topology can be detected in a robust and explicit manner using persistent homology, a powerful new tool to compute topological invariants in arbitrary datasets. We show some state of the art examples of the application of persistent homology to various non-linear dynamical systems, including real-world climate data, and show how these techniques can shed light on questions such as how many regimes there really are in e.g. the Euro-Atlantic sector. Future directions are also discussed.</p>

scholarly journals Bot Detection on Social Networks Using Persistent Homology

2020 ◽  
Vol 25 (3) ◽  
pp. 58
Author(s):  
Minh Nguyen ◽  
Mehmet Aktas ◽  
Esra Akbas
Keyword(s):  
Machine Learning ◽  
Social Networks ◽  
Social Media ◽  
Persistent Homology ◽  
Structural Features ◽  
Higher Order ◽  
Ego Networks ◽  
Feature Extraction Method ◽  
Topological Features ◽  
Bot Detection

The growth of social media in recent years has contributed to an ever-increasing network of user data in every aspect of life. This volume of generated data is becoming a vital asset for the growth of companies and organizations as a powerful tool to gain insights and make crucial decisions. However, data is not always reliable, since primarily, it can be manipulated and disseminated from unreliable sources. In the field of social network analysis, this problem can be tackled by implementing machine learning models that can learn to classify between humans and bots, which are mostly harmful computer programs exploited to shape public opinions and circulate false information on social media. In this paper, we propose a novel topological feature extraction method for bot detection on social networks. We first create weighted ego networks of each user. We then encode the higher-order topological features of ego networks using persistent homology. Finally, we use these extracted features to train a machine learning model and use that model to classify users as bot vs. human. Our experimental results suggest that using the higher-order topological features coming from persistent homology is promising in bot detection and more effective than using classical graph-theoretic structural features.

scholarly journals Contractibility of a persistence map preimage

2020 ◽  
Vol 4 (4) ◽  
pp. 509-523
Author(s):  
Jacek Cyranka ◽  
Konstantin Mischaikow ◽  
Charles Weibel
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Time Series ◽  
Data Driven ◽  
Dimensional System ◽  
Multiple Time ◽  
Multiple Time Series ◽  
Topological Features ◽  
Persistence Diagram ◽  
Persistence Diagrams

Abstract This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.

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