Related papers: Persistent Homology Guided Force-Directed Graph La…
Statisticians and quantitative neuroscientists have actively promoted the use of independence relationships for investigating brain networks, genomic networks, and other measurement technologies. Estimation of these graphs depends on two…
We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes: digital, position, and interaction, motivated by different capabilities of…
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…
Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set.…
Complex networks encountered in biology are often characterized by significant structural diversity. Whether it be differences in the three-dimensional structure of allosteric proteins, or the variation among the micro-scale structures of…
A large driver of the complexity of graph learning is the interplay between structure and features. When analyzing the expressivity of graph neural networks, however, existing approaches ignore features in favor of structure, making it…
Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…
Dimensionality effects pose major challenges in high-dimensional and non-Euclidean data analysis. Graph-based two-sample tests and change-point detection are particularly attractive in this context, as they make minimal distributional…
Three-dimensional dislocation networks control the mechanical properties such as strain hardening of crystals. Due to the complexity of dislocation networks and their temporal evolution, analysis tools are needed that fully resolve the…
We study the problem of learning representations with controllable connectivity properties. This is beneficial in situations when the imposed structure can be leveraged upstream. In particular, we control the connectivity of an…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
The (variational) graph auto-encoder and its variants have been popularly used for representation learning on graph-structured data. While the encoder is often a powerful graph convolutional network, the decoder reconstructs the graph…
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…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…
Recently, one has seen a surge of interest in developing such methods including ones for learning such representations for (undirected) graphs (while preserving important properties). However, most of the work to date on embedding graphs…
Persistent homology has been studied to better understand the structural properties and topology features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by non-pairwise interactions…
Embedding static graphs in low-dimensional vector spaces plays a key role in network analytics and inference, supporting applications like node classification, link prediction, and graph visualization. However, many real-world networks…
Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined…
Graph algorithms applied in many applications, including social networks, communication networks, VLSI design, graphics, and several others, require dynamic modifications -- addition and removal of vertices and/or edges -- in the graph.…