scholarly journals Complexes of tournaments, directionality filtrations and persistent homology

2021 ◽  
Author(s):  
Dejan Govc ◽  
Ran Levi ◽  
Jason P. Smith
Keyword(s):  
Persistent Homology ◽  
Simplicial Complexes ◽  
Graph Data ◽  
Flag Complex ◽  
Digital Reconstruction ◽  
Graph Dynamics ◽  
Geometric Realisation

AbstractComplete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as “tournaplexes”, whose simplices are tournaments. In particular, given a digraph $${\mathcal {G}}$$ G , we associate with it a “flag tournaplex” which is a tournaplex containing the directed flag complex of $${\mathcal {G}}$$ G , but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project’s digital reconstruction of a rat’s neocortex.

scholarly journals Complexes of Tournaments, Directionality Filtrations and Persistent Homology

2020 ◽  
Author(s):  
Ran Levi ◽  
Dejan Govc ◽  
Jason Smith
Keyword(s):  
Persistent Homology ◽  
Simplicial Complexes ◽  
Graph Data ◽  
Flag Complex ◽  
Digital Reconstruction ◽  
Graph Dynamics ◽  
Geometric Realisation

Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as ``tournaplexes'', whose simplices are tournaments. In particular, given a digraph G, we associate with it a ``flag tournaplex'' which is a tournaplex containing the directed flag complex of G, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.

scholarly journals Simplicial Complexes are Game Complexes

10.37236/6958 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Sara Faridi ◽  
Svenja Huntemann ◽  
Richard J. Nowakowski
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Combinatorial Games ◽  
Natural Question ◽  
Flag Complex ◽  
Game Trees ◽  
Game Value ◽  
Flag Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

scholarly journals Shellability and the Strong gcd-Condition

10.37236/67 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Alexander Berglund
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Flag Complex ◽  
Alexander Dual ◽  
Golod Ring

Shellability is a well-known combinatorial criterion on a simplicial complex $\Delta$ for verifying that the associated Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, and he together with the author proved that it implies Golodness of the associated Stanley-Reisner ring. The two algebraic notions were earlier tied together by Herzog, Reiner and Welker, who showed that if $k[\Delta^\vee]$ is sequentially Cohen-Macaulay, where $\Delta^\vee$ is the Alexander dual of $\Delta$, then $k[\Delta]$ is Golod. In this paper, we present a combinatorial companion of this result, namely that if $\Delta^\vee$ is (non-pure) shellable then $\Delta$ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if $\Delta$ is a flag complex.

scholarly journals Determination of features based on the formation of persistent spectra of eigenvalues of Laplace matrices

2020 ◽  
pp. 49-64
Author(s):  
S. V. Lejhter ◽  
S. N. Chukanov
Keyword(s):  
Persistent Homology ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Wasserstein Distance ◽  
Computer Hardware ◽  
Euler Characteristics ◽  
Topological Features ◽  
Laplace Matrix ◽  
Definition Of

An algorithm for determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes has been developed in the paper. The spectrum of eigenvalues of the Laplace matrix is used as features in the data structure for image analysis. Similarly to the method of persistent homology, the filtering of embedded simplicial complexes is formed, approximating the image of the object, but the topological features at each stage of filtration is the spectrum of eigenvalues of the Laplace matrix of simplicial complexes. The spectrum of eigenvalues of the Laplace matrix allows to determine the Betti numbers and Euler characteristics of the image. Based on the method of finding the spectrum of eigenvalues of the Laplace matrix, an algorithm is formed that allows you to obtain topological features of images of objects and quantitative estimates of the results of image comparison. Software has been developed that implements this algorithm on computer hardware. The method of determining the spectrum of eigenvalues of the Laplace matrix has the following advantages: the method does not require a bijective correspondence between the elements of the structures of objects; the method is invariant with respect to the Euclidean transformations of the forms of objects. Determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes allows you to expand the number of features for machine learning, which allows you to increase the diversity of information obtained by the methods of computational topology, while maintaining topological invariants. When comparing the shapes of objects, a modified Wasserstein distance can be constructed based on the eigenvalues of the Laplace matrix of the compared shapes. Using the definition of the spectrum of eigenvalues of the Laplace matrix to compare the shapes of objects can improve the accuracy of determining the distance between images.

NONLINEAR STATISTICS OF HUMAN SPEECH DATA

2009 ◽  
Vol 19 (07) ◽  
pp. 2307-2319 ◽  
Author(s):  
KENNETH A. BROWN ◽  
KEVIN P. KNUDSON
Keyword(s):  
Time Delay ◽  
Betti Number ◽  
Persistent Homology ◽  
Point Clouds ◽  
Simplicial Complexes ◽  
Data Sets ◽  
Human Speech ◽  
Speech Data ◽  
Nonlinear Statistics

We study the structure of point clouds obtained as time delay embeddings of human speech signals by approximating the data sets with certain simplicial complexes and analyzing their persistent homology. Results for several different sounds are presented in embedding dimensions 3 and 4. The first Betti number allows a coarse classification of sounds into three groups: vowels, nasals and noise.

scholarly journals Approximate Triangulations of Grassmann Manifolds

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 172
Author(s):  
Kevin P. Knudson
Keyword(s):  
Euclidean Space ◽  
Persistent Homology ◽  
Simplicial Complexes ◽  
Grassmann Manifolds ◽  
The Family

We define the notion of an approximate triangulation for a manifold M embedded in Euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in M and use persistent homology to find a complex in the family whose homology agrees with that of M. Our key examples are various Grassmann manifolds G k ( R n ) .

scholarly journals Undesired Parking Spaces and Contractible Pieces of the Noncrossing Partition Link

10.37236/7212 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Michael Dougherty ◽  
Jon McCammond
Keyword(s):  
Simplicial Complexes ◽  
Order Complex ◽  
Parking Functions ◽  
Partition Lattice ◽  
Flag Complex ◽  
Noncrossing Partition ◽  
Noncrossing Partition Lattice

There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. The first author and his coauthors conjectured that various collections of simplices of the noncrossing partition link (determined by the undesired parking spaces in the corresponding parking functions) form contractible subcomplexes. In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.

scholarly journals Model comparison via simplicial complexes and persistent homology

2021 ◽  
Vol 8 (10) ◽  
Author(s):  
Sean T. Vittadello ◽  
Michael P. H. Stumpf
Keyword(s):  
Mathematical Models ◽  
Algebraic Topology ◽  
Model Comparison ◽  
Persistent Homology ◽  
A Priori ◽  
Positional Information ◽  
Simplicial Complexes ◽  
Statistical Model Selection ◽  
Model Equivalence ◽  
Compare And Contrast

In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori . Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

scholarly journals Persistent Homology of Geospatial Data: A Case Study with Voting

2019 ◽  
Author(s):  
Michelle Feng ◽  
Mason A. Porter
Keyword(s):  
Simplicial Complex ◽  
Persistent Homology ◽  
Point Clouds ◽  
Simplicial Complexes ◽  
Geospatial Data ◽  
Alternative Methods ◽  
Data Set ◽  
New Methods ◽  
Complex Construction

A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (in our case, a simplicial complex). Modern packages for persistent homology often construct Vietoris–Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute. We investigate alternative methods of constructing these complexes and the effects of making associated choices during simplicial-complex construction on the output of persistent-homology algorithms. We present two new methods for constructing simplicial complexes from two-dimensional geospatial data (such as maps). We apply these methods to a California precinct-level voting data set, demonstrating that our new constructions can capture geometric characteristics that are missed by distance-based constructions. Our new constructions can thus yield more interpretable persistence modules and barcodes for geospatial data. In particular, they are able to distinguish short-persistence features that occur only for a narrow range of distance scales (e.g., voting behaviors in densely populated cities) from short-persistence noise by incorporating information about other spatial relationships between precincts.

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