525 research outputs found
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
Hochschild homology, and a persistent approach via connectivity digraphs
We introduce a persistent Hochschild homology framework for directed graphs.
Hochschild homology groups of (path algebras of) directed graphs vanish in
degree . To extend them to higher degrees, we introduce the notion of
connectivity digraphs and analyse two main examples; the first, arising from
Atkin's -connectivity, and the second, here called -path digraphs,
generalising the classical notion of line graphs. Based on a categorical
setting for persistent homology, we propose a stable pipeline for computing
persistent Hochschild homology groups. This pipeline is also amenable to other
homology theories; for this reason, we complement our work with a survey on
homology theories of digraphs.Comment: Comments are welcome
Forman's Ricci curvature - From networks to hypernetworks
Networks and their higher order generalizations, such as hypernetworks or
multiplex networks are ever more popular models in the applied sciences.
However, methods developed for the study of their structural properties go
little beyond the common name and the heavy reliance of combinatorial tools. We
show that, in fact, a geometric unifying approach is possible, by viewing them
as polyhedral complexes endowed with a simple, yet, the powerful notion of
curvature - the Forman Ricci curvature. We systematically explore some aspects
related to the modeling of weighted and directed hypernetworks and present
expressive and natural choices involved in their definitions. A benefit of this
approach is a simple method of structure-preserving embedding of hypernetworks
in Euclidean N-space. Furthermore, we introduce a simple and efficient manner
of computing the well established Ollivier-Ricci curvature of a hypernetwork.Comment: to appear: Complex Networks '18 (oral presentation
Simplicial Complexes From Graphs Toward Graph Persistence
Persistent homology is a branch of computational topology which uses geometry and topology for shape description and analysis. This dissertation is an introductory study to link persistent homology and graph theory, the connection being represented by various methods to build simplicial complexes from a graph. The methods we consider are the complex of cliques, of independent sets, of neighbours, of enclaveless sets and complexes from acyclic subgraphs, each revealing several properties of the underlying graph. Moreover, we apply the core ideas of persistence theory in the new context of graph theory, we define the persistent block number and the persistent edge-block number
Hardness of Approximation for Morse Matching
Discrete Morse theory has emerged as a powerful tool for a wide range of
problems, including the computation of (persistent) homology. In this context,
discrete Morse theory is used to reduce the problem of computing a topological
invariant of an input simplicial complex to computing the same topological
invariant of a (significantly smaller) collapsed cell or chain complex.
Consequently, devising methods for obtaining gradient vector fields on
complexes to reduce the size of the problem instance has become an emerging
theme over the last decade. While computing the optimal gradient vector field
on a simplicial complex is NP-hard, several heuristics have been observed to
compute near-optimal gradient vector fields on a wide variety of datasets.
Understanding the theoretical limits of these strategies is therefore a
fundamental problem in computational topology. In this paper, we consider the
approximability of maximization and minimization variants of the Morse matching
problem, posed as open problems by Joswig and Pfetsch. We establish hardness
results for Max-Morse matching and Min-Morse matching. In particular, we show
that, for a simplicial complex with n simplices and dimension , it is
NP-hard to approximate Min-Morse matching within a factor of
, for any . Moreover, using an L-reduction
from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is
both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial
complexes of dimension within certain explicit constant factors.Comment: 20 pages, 1 figur
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