Related papers: Exploring the Topology of Dynamical Reconstruction…
A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicial complex preserves all pertinent topological…
Reconstructing the equation of motion and thus the network topology of a system from time series is a very important problem. Although many powerful methods have been developed, it remains a great challenge to deal with systems in high…
Inferring network topology from dynamical observations is a fundamental problem pervading research on complex systems. Here, we present a simple, direct method to infer the structural connection topology of a network, given an observation…
Methods for reconstructing the topology of complex networks from time-resolved observations of node dynamics are gaining relevance across scientific disciplines. Of biggest practical interest are methods that make no assumptions about…
Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…
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…
Many dynamical processes of complex systems can be understood as the dynamics of a group of nodes interacting on a given network structure. However, finding such interaction structure and node dynamics from time series of node behaviours is…
We propose a topological framework for the detection of Hopf bifurcations directly from time series, based on persistent homology applied to phase space reconstructions via Takens embedding within the framework of Topological Data Analysis.…
In this paper we develop a novel Topological Data Analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a…
Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…
Computation of persistent homology of simplicial representations such as the Rips and the C\v{e}ch complexes do not efficiently scale to large point clouds. It is, therefore, meaningful to devise approximate representations and evaluate the…
Prediction models that capture and use the structure of state-space dynamics can be very effective. In practice, however, one rarely has access to full information about that structure, and accurate reconstruction of the dynamics from…
Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena…
Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and…
In this paper, we propose a novel framework for dynamical analysis of human actions from 3D motion capture data using topological data analysis. We model human actions using the topological features of the attractor of the dynamical system.…
The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary two-dimensional faceted surfaces $z = h(x,y)$. Our method uses a three-component facet/edge/junction storage model,…
In this work, we explore the state-space formulation of a network process to recover, from partial observations, the underlying network topology that drives its dynamics. To do so, we employ subspace techniques borrowed from system…
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (in our case, a simplicial complex). Modern packages for persistent homology often construct Vietoris--Rips or other…