相关论文: $\epsilon$-net Induced Lazy Witness Complexes on G…

Topological data analysis computes and analyses topological features of the point clouds by constructing and studying a simplicial representation of the underlying topological structure. The enthusiasm that followed the initial successes of…

Capitalizing on the intuitive premise that shape characteristics are more robust to perturbations, we bridge adversarial graph learning with the emerging tools from computational topology, namely, persistent homology representations of…

The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been…

Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from,…

Most real-world networks are embedded in latent geometries. If a node in a network is found in the vicinity of another node in the latent geometry, the two nodes have a disproportionately high probability of being connected by a link. The…

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…

Data samples collected for training machine learning models are typically assumed to be independent and identically distributed (iid). Recent research has demonstrated that this assumption can be problematic as it simplifies the manifold of…

Persistent homology is a cornerstone of topological data analysis, offering a multiscale summary of topology with robustness to nuisance transformations, such as rotations and small deformations. Persistent homology has seen broad use…

The effectiveness of Spatio-temporal Graph Neural Networks (STGNNs) in time-series applications is often limited by their dependence on fixed, hand-crafted input graph structures. Motivated by insights from the Topological Data Analysis…

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 propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…

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…

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…

In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…

Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…

Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a…

Representation learning on graphs is a fundamental problem that can be crucial in various tasks. Graph neural networks, the dominant approach for graph representation learning, are limited in their representation power. Therefore, it can be…

Topological features based on persistent homology capture high-order structural information so as to augment graph neural network methods. However, computing extended persistent homology summaries remains slow for large and dense graphs and…

This paper introduces a topological framework for interpreting the internal representations of Multilayer Perceptrons (MLPs). We construct a simplicial tower, a sequence of simplicial complexes connected by simplicial maps, that captures…

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the…