Related papers: Towards Homological Methods in Graphic Statics
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
Graphs are used to represent and analyze data in domains as diverse as physics, biology, chemistry, planetary science, and the social sciences. Across domains, random graph models relate generative processes to expected graph properties,…
Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d=2), and strictly contains the class of…
Topological signals, i.e., dynamical variables defined on nodes, links, triangles, etc. of higher-order networks, are attracting increasing attention. However the investigation of their collective phenomena is only at its infancy. Here we…
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally,…
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…
We develop a theory of graph algebras over general fields. This is modeled after the theory developed by Freedman, Lov\'asz and Schrijver in [22] for connection matrices, in the study of graph homomorphism functions over real edge weight…
Context: Edge graphs are graphs whose edges are labelled with identifiers, and nodes can have multiple edges between them. They are used to model a wide range of systems, including networks with distances or degrees of connection and…
A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of…
Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides…
Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings…
We establish a new theory which gives a unified topological approach to data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and…
The idea of a graph theoretical approach to modeling the emergence of a quantized geometry and consequently spacetime, has been proposed previously, but not well studied. In most approaches the focus has been upon how to generate a…
The theory uses methods and language of linear algebra to study nonlinear spaces. These techniques can be used particularly to describe analytic geometry of non-linear elliptic, hyperbolic, De Sitter and Anti de Sitter spaces. The main…
An effort has been made to show mathematicians some new ideas applied to image analysis. Gray images are presented as tilings. Based on topological properties of the tiling, a number of gray convex hulls: maximal, minimal, and oriented ones…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
Rigid body interactions are fundamental to numerous scientific disciplines, but remain challenging to simulate due to their abrupt nonlinear nature and sensitivity to complex, often unknown environmental factors. These challenges call for…
Graph pattern matching is often defined in terms of subgraph isomorphism, an NP-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be…
This work examines the problem of topology inference over discrete-time nonlinear stochastic networked dynamical systems. The goal is to recover the underlying digraph linking the network agents, from observations of their state-evolution.…
The aim of this article is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and…