Corpus ID: 44395802

Computational Topology in

  title={Computational Topology in},
  author={Bernadette J. Stolz},
Computational topology is a set of algorithmic methods developed to understand topological invariants such as loops and holes in high-dimensional data sets. In particular, a method know as persistent homology has been used to understand such shapes and their persistence in point clouds and networks. It has only been applied to neuronal networks in recent years. While most tools from network science focus solely on local properties based on pairwise connections, the topological tools reveal more… Expand
The importance of the whole: Topological data analysis for the network neuroscientist
An introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as the authors move through a combinatorial object such as a weighted network. Expand
Two’s company, three (or more) is a simplex
It is proposed that the use of simplicial complexes, a structure developed in the field of mathematics known as algebraic topology, of increasing applicability to real data due to a rapidly growing computational toolset, has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition. Expand
Topological Data Analysis for Systems of Coupled Oscillators
Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto,Expand


jHoles: A Tool for Understanding Biological Complex Networks via Clique Weight Rank Persistent Homology
Complex networks equipped with topological data analysis are one of the promising tools in the study of biological systems (e.g. evolution dynamics, brain correlation, breast cancer diagnosis,Expand
Persistent Homology — a Survey
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematicalExpand
Discriminative persistent homology of brain networks
This paper applies the Rips filtration to construct the FDG-PET based functional brain networks out of 24 attention deficit hyperactivity disorder (ADHD) children, 26 autism spectrum disorder (ASD) children and 11 pediatric control subjects and visually shows the topological evolution of the brain networks using the barcode and performs statistical inference on the group differences. Expand
Barcodes: The persistent topology of data
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. TheExpand
javaPlex: A Research Software Package for Persistent (Co)Homology
A new software package for topological computation, replacing previous jPlex and Plex, enables researchers to access state of the art algorithms for persistent homology, cohomology, hom complexes, filtered simplicial complex, filtered cell complexes, witness complex constructions, and many more essential components of computational topology. Expand
Networks: An Introduction
This book brings together for the first time the most important breakthroughs in each of these fields and presents them in a coherent fashion, highlighting the strong interconnections between work in different areas. Expand
Topology and data
This paper will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data, particularly high throughput data from microarray or other sources. Expand
Weighted Functional Brain Network Modeling via Network Filtration
The proposed network filtration framework can discriminate the local and global differences of the brain networks of 24 attention deficit hyperactivity disorder (ADHD), 26 autism spectrum disorder (ASD) and 11 pediatric control (PedCon) children obtained through the FDG-PET data. Expand
A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology
Using a computational algorithm based on recently developed tools from Persistent Homology theory in the field of algebraic topology, it is found that the patterns of neuronal co-firing can, in fact, convey topological information about the environment in a biologically realistic length of time. Expand
Topological persistence and simplification
A notion of topological simplification is formalized within the framework of a filtration, which is the history of a growing complex, and a topological change that happens during growth is classified as either a feature or noise, depending on its life-time or persistence within the filTration. Expand