Related papers: Persistent topological features of dynamical syste…
The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally…
We use the visibility algorithm to construct the time series networks obtained from the time series of different dynamical regimes of the logistic map. We define the simplicial characterisers of networks which can analyse the simplicial…
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal…
In this paper, we develop topological data analysis methods for classification tasks on univariate time series. As an application, we perform binary and ternary classification tasks on two public datasets that consist of physiological…
The analysis of nonlinear dynamics is an important issue in numerous fields of science. In this study, we propose a new method to analyze the time series data using persistent homology (PH). The key idea is the application of PH to the…
We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is…
Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphical representation of such information. A filtration might be determined by time in a set…
The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying…
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…
We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into…
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such…
Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…
To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
A topology preserving skeleton is a synthetic representation of an object that retains its topology and many of its significant morphological properties. The process of obtaining the skeleton, referred to as skeletonization or thinning, is…
We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families (visibility (natural and…
This work describes the way that topological mixing and chaos in continua, as induced by discrete dynamical systems, can or can't be understood through topological conjugacy with symbolic dynamical systems. For example, there is no symbolic…
Analyzing embedded simplicial complexes, such as triangular meshes and graphs, is an important problem in many fields. We propose a new approach for analyzing embedded simplicial complexes in a subdivision-invariant and isometry-invariant…