h-vector
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Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes Open
Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social a…
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Weighted growing simplicial complexes Open
Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial comp…
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Yoneda Lemma for Simplicial Spaces Open
We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure . In particular, we prove a recognition …
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Strong Collapse for Persistence Open
We introduce a fast and memory efficient approach 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 strong collapses, as introduced by …
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The amazing world of simplicial complexes Open
Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.
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Few-weight quaternary codes via simplicial complexes Open
In this paper, we construct quaternary linear codes via simplicial complexes and we also determine the weight distributions of these codes. Moreover, we present an infinite family of minimal quaternary linear codes, which also meet the Gri…
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Principled Simplicial Neural Networks for Trajectory Prediction Open
We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture:…
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Transcendental simplicial volumes Open
We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit family of (transcendental) real numbers that are not realised as the simplicial volume of a closed manifol…
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Locally standard torus actions and $h'$-numbers of simplicial posets Open
We consider the orbit type filtration on a manifold with a locally standard torus action and study the corresponding spectral sequence in homology. When all proper faces of the orbit space are acyclic and the free part of the action is tri…
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Higher-order connection Laplacians for directed simplicial complexes Open
Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of …
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Simplicial Complex Representation Learning Open
Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d s…
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Dist2Cycle: A Simplicial Neural Network for Homology Localization Open
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
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Learning Simplicial Complexes from Persistence Diagrams Open
Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that…
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Minimal volume entropy of simplicial complexes Open
This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present complemen…
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Embedding simply connected 2-complexes in 3-space -- IV. Dual matroids Open
We introduce dual matroids of 2-dimensional simplicial complexes. Under certain necessary conditions, duals matroids are used to characterise embeddability in 3-space in a way analogous to Whitney's planarity criterion. We further use dual…
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The uniform face ideals of a simplicial complex Open
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 r…
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Convolutional Learning on Simplicial Complexes Open
We propose a simplicial complex convolutional neural network (SCCNN) to learn data representations on simplicial complexes. It performs convolutions based on the multi-hop simplicial adjacencies via common faces and cofaces independently a…
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Extendability of simplicial maps is undecidable Open
We present a short proof of the Čadek-Krčál-Matoušek-Vokřínek-Wagner result from the title (in the following form due to Filakovský-Wagner-Zhechev). For any fixed even $l$ there is no algorithm recognizing the extendability of the identity…
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Strong Collapse for Persistence Open
We introduce a fast and memory efficient approach 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 strong collapses, as introduced by …
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ABHY Associahedra and Newton polytopes of F$F$‐polynomials for cluster algebras of simply laced finite type Open
A new construction of the associahedron was recently given by Arkani‐Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct general…
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Hodge Decompositions for Weighted Hypergraphs Open
Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the e…
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Spectral Sparsification of Simplicial Complexes for Clustering and Label Propagation Open
As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for…
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Isolated factorizations and their applications in simplicial affine semigroups Open
We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. …
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Integral foliated simplicial volume and $S^1$-actions Open
The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical mani…
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On the topology of a boolean representable simplicial complex Open
It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension >= 2. In the case …
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Computing Simplicial Depth by Using Importance Sampling Algorithm and Its Application Open
Simplicial depth (SD) plays an important role in discriminant analysis, hypothesis testing, machine learning, and engineering computations. However, the computation of simplicial depth is hugely challenging because the exact algorithm is a…
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On the graph products of simplicial groups and connected Hopf algebras Open
In this paper we study the classifying spaces of graph products of simplicial groups and connected Hopf algebras over a field, and show that they can be uniformly treated under the framework of polyhedral products. It turns out that these …
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Centrality measures in simplicial complexes: applications of TDA to Network Science Open
Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Using the recently introduced higher order notions of adjac…
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Regularity aspects for combinatorial simplicial surfaces Open
Combinatorial surfaces capture essential properties of continuous surfaces (like spheres and tori) in a discrete manner that lends itself more easily to a computational approach. They arise from triangulations of continuous surfaces and ar…
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Higher order degree in simplicial complexes, multi combinatorial Laplacian and applications of TDA to complex networks Open
Many real networks in social, biological or computer sciences have an inherent structure of a simplicial complex, which reflects the multi interactions among agents (and groups of agents) and constitutes the basics of Topological Data Anal…