Edge Collapse and Persistence of Flag Complexes

In this article, we extend the notions of dominated vertex and strong collapse of a simplicial complex as introduced by J. Barmak and E. Miniam. We say that a simplex (of any dimension) is dominated if its link is a simplicial cone. Domination of edges appears to be a very powerful concept, especially when applied to flag complexes. We show that edge collapse (removal of dominated edges) in a flag complex can be performed using only the 1-skeleton of the complex. Furthermore, the residual complex is a flag complex as well. Next we show that, similar to the case of strong collapses, we can use edge collapses to reduce a flag filtration ? to a smaller flag filtration ?^c with the same persistence. Here again, we only use the 1-skeletons of the complexes. The resulting method to compute ?^c is simple and extremely efficient and, when used as a preprocessing for persistence computation, leads to gains of several orders of magnitude w.r.t the state-of-the-art methods (including our previous approach using strong collapse). The method is exact, irrespective of dimension, and improves performance of persistence computation even in low dimensions. This is demonstrated by numerous experiments on publicly available data

An Efficient Representation for Filtrations of Simplicial Complexes

A filtration over a simplicial complex $K$ is an ordering of the simplices of $K$ such that all prefixes in the ordering are subcomplexes of $K$. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [Algorithmica '17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes.Comment: A preliminary version appeared in SODA 201

Effondrements et homologie persistante

In this thesis, we introduce two new approaches to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using special types of collapses (strong and edge collapse) and to compute the PH of an induced sequence of smaller size that has the same PH as the initial one.Our first approach uses strong collapse which is introduced by J. Barmak and E. Miniam [DCG (2012)]. Strong collapse comprises of removal of special vertices called \textit{dominated} vertices from a simplicial complex.Our approach with strong collapse has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for othertypes of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot ofspace and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel.In the case of flag complexes strong collapse can be performed over the 1-skeleton of the complex and the resulting complex is also a flag complex. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the coreflag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremelyefficient. We extend the notions of dominated vertex to a simplex of any dimension. Domination of edges appear to be very powerful and we study it in the case of flag complexes in more detail. We show that edge collapse (removal of dominated edges) in a flag complex can be performed using only the 1-skeleton of the complex as well. Furthermore, the residual complex is a flag complex as well. Next we show that, similar to the case of strong collapses, we can use edge collapses to reduce a flag filtration F to a smaller flag filtration F^c with the same persistence. Here again, we only use the 1-skeletons of the complexes. As a result and as demonstrated by numerous experiments on publicly available data sets, our approaches are extremely fast and memory efficient in practice. In particular the method using edge collapse performs the best among all known methods including the strong collapse approach. Finally, we can compromizebetween precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.Dans cette thèse, nous introduisons deux nouvelles approches pour calculer l'homologie persistante(HP) d'une séquence de complexes simpliciaux. L'idée de base est de simplifier les complexes de la séquence d'entrée en utilisant des types spéciaux de collapses (effondrement), les collapses forts et les collapses d'arêtes, et de calculer l'HP d'une séquence réduite de plus petite taille qui a la même HP que la séquence initiale. Notre première approche utilise les collapses forts introduits par J. Barmak et E. Miniam [DCG (2012)]. Un collapse fort supprime les sommets dits dominés d'un complexe simplicial. Notre approche utilisant les collapses forts a plusieurs caractéristiques qui la distinguent des travaux antérieurs. La méthode n'est pas limitée aux filtrations (c'est-à-dire aux séquences de sous-complexes simpliciaux imbriqués) mais fonctionne pour d'autres types de séquences comme les tours et les zigzags. Par ailleurs, pour implémenter les collapses forts, il suffit de représenter les simplexes maximaux du complexe, et pas l'ensemble de tous ses simplexes, ce qui économise beaucoup d'espace et de temps. De plus, les complexes de la séquence peuvent être collapsés indépendamment et en parallèle.Dans le cas des complexes en drapeaux (flag complexes), les collapses forts peuvent être réalisés sur le 1-squelette du complexe et le complexe résultat est également un complexe en drapeau. Nous montrons que si l'on restreint la classe des complexes simpliciaux aux complexes en drapeaux, on peut améliorer la complexité en temps et en espace de facon décisive par rapport aux travaux antérieurs. Lorsque les collapses forts sont appliqués aux complexes d'une tour de complexes en drapeau, nous obtenons une séquence réduite qui est aussi une tour de complexes en drapeau que nous appelons le coeur de la tour. Nous convertissons ensuite le coeur de la tour en une filtration équivalente pour calculer son HP. Là encore, nous n'utilisons que les 1-squelettes des complexes. La méthode résultante est simple et extrêmement efficace.Nous étendons la notion de sommet dominé au cas de simplexes de dimension quelconque. Le concept d'arête dominée apparait très puissant et nous l'étudions dans le cas des complexes en drapeaux de faconplus détaillée. Nous montrons que les collapses d'arêtes (suppression des arêtes dominées) dans un complexe en drapeaux peut être effectué, comme précédemment, en utilisant uniquement le 1-squelette du complexe. En outre, le complexe résiduel est également un complexe de drapeaux. Ensuite, nous montrons que, comme dans le cas des collapses forts, on peut utiliser les collapses d'arêtes pour réduire une filtration de complexes en drapeaux en une filtration de complexes en drapeaux plus petite qui a la même HP. Là encore, nous utilisons uniquement le 1-squelettes des complexes.Comme l'ont démontré de nombreuses expériences sur des données publiques, les approches développées sont extrêmement rapides et efficaces en mémoire. En particulier, la méthode utilisant les collapses d'arêtes offre de meilleures performances que toutes les méthodes connues, y compris l'approche par collapses forts. Enfin, nous pouvons faire des compromis entre précision et temps de calcul en choisissant le nombre de complexes simpliciaux de la séquence à collapser

Computing Persistent Homology of Flag Complexes via Strong Collapses

In this article, we focus on the problem of computing Persistent Homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. We show that strong collapses of flag complexes can be computed in time O(k^2v^2) where v is the number of vertices of the complex and k is the maximal degree of its graph. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient

The uniform face ideals of a simplicial complex

We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear resolution, as do all of its powers, if and only if the colouring satisfies a certain nesting property. In the case when the colouring is nested, we give a minimal cellular resolution supported on a cubical complex. From this, we give the graded Betti numbers in terms of the face-vector of the underlying simplicial complex. Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both the ideal and its quotient. We also give explicit formul\ae\ for the codimension, Krull dimension, multiplicity, projective dimension, depth, and regularity. Further still, we describe the associated primes, and we show that they are persistent.Comment: 34 pages, 8 figure

Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT

Topological data analysis (TDA) delivers invaluable and complementary information on the intrinsic properties of data inaccessible to conventional methods. However, high computational costs remain the primary roadblock hindering the successful application of TDA in real-world studies, particularly with machine learning on large complex networks. Indeed, most modern networks such as citation, blockchain, and online social networks often have hundreds of thousands of vertices, making the application of existing TDA methods infeasible. We develop two new, remarkably simple but effective algorithms to compute the exact persistence diagrams of large graphs to address this major TDA limitation. First, we prove that $(k+1)$-core of a graph $\mathcal{G}$ suffices to compute its $k^{th}$ persistence diagram, $PD_k(\mathcal{G})$. Second, we introduce a pruning algorithm for graphs to compute their persistence diagrams by removing the dominated vertices. Our experiments on large networks show that our novel approach can achieve computational gains up to 95%. The developed framework provides the first bridge between the graph theory and TDA, with applications in machine learning of large complex networks. Our implementation is available at https://github.com/cakcora/PersistentHomologyWithCoralPrunitComment: Spotlight paper at NeurIPS 202