Persistent Homology of Morse Decompositions in Combinatorial Dynamics
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Abstract:
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to vector cloud data. We present experimental results on two numerical examples.Keywords:
Persistent Homology
Topological data analysis
Discrete Morse theory
Persistence (discontinuity)
Dynamics
Homology
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This thesis proposes a combinatorial generalization of a nilpotent operator on a vector space. The resulting object is highly natural, with basic connections to a variety of fields in pure mathematics, engineering, and the sciences. For the purpose of exposition we focus the discussion of applications on homological algebra and computation, with additional remarks in lattice theory, linear algebra, and abelian categories. For motivation, we recall that the methods of algebraic topology have driven remarkable progress in the qualitative study of large, noisy bodies of data over the past 15 years. A primary tool in Topological Data Analysis [TDA] is the homological persistence module, which leverages categorical structure to compare algebraic shape descriptors across multiple scales of measurement. Our principle application to computation is a novel algorithm to calculate persistent homology which, in certain cases, improves the state of the art by several orders of magnitude. Included are novel results in discrete, spectral, and algebraic Morse theory, and on the strong maps of matroid theory. The defining theme throughout is interplay between the combinatorial theory matroids and the algebraic theory of categories. The nature of these interactions is remarkably simple, but their consequences in homological algebra, quiver theory, and combinatorial optimization represent new and widely open fields for interaction between the disciplines.
Homological algebra
Algebraic Theory
Algebraic structure
Category theory
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Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley--Morse graphs. A Conley--Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.
Discrete Morse theory
Persistent Homology
Circle-valued Morse theory
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We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of fast/slow variable decompositions or as abstractions in the computer science sense of the word.
Surjective function
Topological conjugacy
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We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems.Topological data analysis provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points.The methods, based on notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the data.We present strengths and weaknesses of current methods, as well as a range of empirical studies relevant to the field of complex systems, before identifying future methodological challenges to help understand the emergence of collective phenomena.
Relevance
Topological data analysis
Complex system
Strengths and weaknesses
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These notes follow my articles [1, 6], and give some new important details. We propose here a new combinatorial method of encoding of measure spaces with measure preserving transformations, (or groups of transformations) in order to give new, mostly locally finite geometrical models for investigation of dynamical properties of these objects.
Dynamics
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Modern applications of algebraic topology to point cloud data analysis have motivated active investigation of combinatorial clique complexes -- high-dimensional extensions of combinatorial graphs. We show that meaningful invariants of such spaces are reflected in the combinatorial properties of an associated family of linear matroids, and discuss matroid-theoretic approaches to several problems in computational topology. Our results allow us to derive estimates of the summary statistics of related constructs for random point cloud data, which we discuss for several sampling distributions in $\mathbb{R}^2$ and $\mathbb{R}^3$.
Clique
Algebraic topology
Topological data analysis
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Using elementary concepts from algebraic topology, a program is outlined for the application of topological notions to the problem of characterizing the connective structure of large-scale systems. The basic approach is to generate appropriate mappings which associate a given system with a simplicial complex. Topological tools are then used in a non-standard manner to investigate the connectivity pattern of the system. Auxiliary notions such as eccentricity of a simplex, patterns on a complex, and the homological structure of a complex are also shown to have system- theoretic relevance.
The general ideas of the paper are illustrated on a number of simple examples, including the standard linear system set-up. The paper concludes with a discussion of several research topics to be carried out in an attempt to connect the algebraic-topological ideas with other branches of mathematical system theory.
Algebraic topology
Simplex
Algebraic structure
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In geometric modeling, numerous combinatorial structures have been proposed for modeling the topology of subdivisions. Often, these structures are deduced from analyses of the topology of cellular subdivisions of E3 when main combinatorial structures defined in combinatorial topology are simplicial. Here, we present a general framework for the definition of combinatorial structures, for the representation of the topology of triangulable cellular objects. Starting from semi-simplicial sets, which are well known structures in combinatorial topology, we define a simple mechanism, inspired by barycentric triangulation, leading to the definition of a wide class of combinatorial cellular complexes. Other combinatorial structures are deduced for particular subclasses of cellular complexes, using other mechanisms. Our approach is a purely combinatorial one. Advantages and drawbacks are discussed. For instance, relations between these combinatorial structures and geometric objects are deduced from that with semi-simplicial sets. Many important topological properties can be computed on these structures. For particular subclasses of cellular complexes, we can deduce equivalent structures, where less information is explicitly represented, decreasing thus the data storage complexity, and, sometimes, the time complexity of algorithms. One of the main drawbacks is the fact that cells of so-defined cellular complexes are always contractible into a vertex, due to the mechanism inspired from barycentric triangulation. Relations with other works are also discussed.
Computational topology
Contractible space
Discrete Geometry
Abstract simplicial complex
Barycentric coordinate system
Simplicial homology
Discrete Morse theory
Simplicial approximation theorem
Representation
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We create a framework for studying symmetric chain decompositions of families of finite posets based on the geometry of polytopes. Our framework unifies almost all known results regarding symmetric chain decompositions of the Young posets $L(m,n)$ --- arising as cells in the Bruhat decomposition of quotients of $SL_{m+n+1}$ --- and yields unexpected new results. The methods we provide are geometric in nature, systematic, and totally amenable to human analysis. This allows us to discover new phenomena which are impenetrable to casework and brute force computer search. In particular, our method yields perfect and near perfect decompositions of various families of posets, which are intractable by known methods. A fundamental tool we use is geometrical projection, which in our framework cleanly unifies many different types of induction; as we move a point from which we project between faces of our polytope, we alter the type of induction. Moreover, projection allows us to decrease dimension and therefore obtain a clear geometric intuition. We also provide additional tools for producing decompositions, and discuss how the various decompositions behave under products.
Intuition
Chain (unit)
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