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 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.

A persistent homological analysis of network data flow malfunctions

2017 ◽  
Vol 5 (6) ◽  
pp. 884-892 ◽  
Author(s):  
Nicholas A Scoville ◽  
Karthik Yegnesh
Keyword(s):  
Data Flow ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Network Data ◽  
Packet Delivery ◽  
Topological Features ◽  
Algorithmic Construction ◽  
The Stability ◽  
Novel Applications ◽  
Persistence Diagrams

Abstract Persistent homology has recently emerged as a powerful technique in topological data analysis for analysing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. In this article, we develop an application of ideas from the theory of persistent homology and persistence diagrams to the study of data flow malfunctions in networks with a certain hierarchical structure. In particular, we formulate an algorithmic construction of persistence diagrams that parameterize network data flow errors, thus enabling novel applications of statistical methods that are traditionally used to assess the stability of persistence diagrams corresponding to homological data to the study of data flow malfunctions. We conclude with an application to network packet delivery systems.

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 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 Determining clinically relevant features in cytometry data using persistent homology

2021 ◽  
Author(s):  
Soham Mukherjee ◽  
Darren Wethington ◽  
Tamal K. Dey ◽  
Jayajit Das
Keyword(s):  
T Cells ◽  
Data Analysis ◽  
Cd8 T Cells ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Healthy Controls ◽  
Topological Features ◽  
Healthy Donors ◽  
Structural Differences ◽  
Persistence Diagrams

AbstractCytometry experiments yield high-dimensional point cloud data that is difficult to interpret manually. Boolean gating techniques coupled with comparisons of relative abundances of cellular subsets is the current standard for cytometry data analysis. However, this approach is unable to capture more subtle topological features hidden in data, especially if those features are further masked by data transforms or significant batch effects or donor-to-donor variations in clinical data. We present that persistent homology, a mathematical structure that summarizes the topological features, can distinguish different sources of data, such as from groups of healthy donors or patients, effectively. Analysis of publicly available cytometry data describing non-naïve CD8+ T cells in COVID-19 patients and healthy controls shows that systematic structural differences exist between single cell protein expressions in COVID-19 patients and healthy controls.Our method identifies proteins of interest by a decision-tree based classifier and passes them to a kernel-density estimator (KDE) for sampling points from the density distribution. We then compute persistence diagrams from these sampled points. The resulting persistence diagrams identify regions in cytometry datasets of varying density and identify protruded structures such as ‘elbows’. We compute Wasserstein distances between these persistence diagrams for random pairs of healthy controls and COVID-19 patients and find that systematic structural differences exist between COVID-19 patients and healthy controls in the expression data for T-bet, Eomes, and Ki-67. Further analysis shows that expression of T-bet and Eomes are significantly downregulated in COVID-19 patient non-naïve CD8+ T cells compared to healthy controls. This counter-intuitive finding may indicate that canonical effector CD8+ T cells are less prevalent in COVID-19 patients than healthy controls. This method is applicable to any cytometry dataset for discovering novel insights through topological data analysis which may be difficult to ascertain otherwise with a standard gating strategy or in the presence of large batch effects.Author summaryIdentifying differences between cytometry data seen as a point cloud can be complicated by random variations in data collection and data sources. We apply persistent homology used in topological data analysis to describe the shape and structure of the data representing immune cells in healthy donors and COVID-19 patients. By looking at how the shape and structure differ between healthy donors and COVID-19 patients, we are able to definitively conclude how these groups differ despite random variations in the data. Furthermore, these results are novel in their ability to capture shape and structure of cytometry data, something not described by other analyses.

scholarly journals C3VFC: A Method for Tracing and Quantification of Microglia in 3D Temporal Images

2021 ◽  
Vol 11 (13) ◽  
pp. 6078
Author(s):  
Tiffany T. Ly ◽  
Jie Wang ◽  
Kanchan Bisht ◽  
Ukpong Eyo ◽  
Scott T. Acton
Keyword(s):  
Vector Field ◽  
Critical Points ◽  
State Of The Art ◽  
Temporal Data ◽  
Entire Structure ◽  
Tracing Algorithm ◽  
Vector Field Convolution ◽  
Multiple Cells ◽  
Tracing Method ◽  
Over Time

Automatic glia reconstruction is essential for the dynamic analysis of microglia motility and morphology, notably so in research on neurodegenerative diseases. In this paper, we propose an automatic 3D tracing algorithm called C3VFC that uses vector field convolution to find the critical points along the centerline of an object and trace paths that traverse back to the soma of every cell in an image. The solution provides detection and labeling of multiple cells in an image over time, leading to multi-object reconstruction. The reconstruction results can be used to extract bioinformatics from temporal data in different settings. The C3VFC reconstruction results found up to a 53% improvement on the next best performing state-of-the-art tracing method. C3VFC achieved the highest accuracy scores, in relation to the baseline results, in four of the five different measures: Entire structure average, the average bi-directional entire structure average, the different structure average, and the percentage of different structures.

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 On attracting sets in artificial networks: cross activation

2018 ◽  
Vol 16 ◽  
pp. 01005
Author(s):  
Felix Sadyrbaev
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Critical Points ◽  
Regulatory Networks ◽  
Main Diagonal ◽  
Sigmoidal Function ◽  
Genomic Regulatory Networks ◽  
Over Time

Mathematical models of artificial networks can be formulated in terms of dynamical systems describing the behaviour of a network over time. The interrelation between nodes (elements) of a network is encoded in the regulatory matrix. We consider a system of ordinary differential equations that describes in particular also genomic regulatory networks (GRN) and contains a sigmoidal function. The results are presented on attractors of such systems for a particular case of cross activation. The regulatory matrix is then of particular form consisting of unit entries everywhere except the main diagonal. We show that such a system can have not more than three critical points. At least n–1 eigenvalues corresponding to any of the critical points are negative. An example for a particular choice of sigmoidal function is considered.

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.

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.

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