19 papers · ranked by Valyu relevance
Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod + 1 more
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input.…
Zhaoyang Wang, Xianghui Fu, Bo Deng, Yang Chen + 1 more
In algebraic topology, a k-dimensional simplex is defined as a convex polytope consisting of k + 1 vertices. If spatial dimensionality is not considered, it corresponds to the complete graph with k + 1 vertices in graph theory. The alternating sum of the number of simplices across dimensions yields a topological…
Nicholas W. Landry, Jean-Gabriel Young, Nicole Eikmeier
Higher-order networks are widely used to describe complex systems in which interactions can involve more than two entities at once. In this paper, we focus on inclusion within higher-order networks, referring to situations where specific entities participate in an interaction, and subsets of those entities also…
Ginestra Bianconi, Christoph Rahmede
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such…
Jean‐Daniel Boissonnat, C. S. Karthik, Sébastien Tavenas
The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its…
Oliver Knill
A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G0 , connecting two simplices if they intersect. We prove that the Poincar´e-Hopf value i ( x) = 1 − χ ( S ( x)), where χ ( S ( x)) is…
Colin Lynch, Kaitlin Baudier, Douglas Montgomery, Meghan Barrett
Animal nutritionists seek to understand how animals regulate the intake and balance of multiple nutrients, yet the design and analysis of such experiments are often limited by how nutrient spaces are represented. The geometric framework for nutrition (GFN) provides a powerful means to visualize nutrient interactions…
Denis Kleverov, Ekaterina Aladyeva, Alexey Serdyukov, Maxim N. Artyomov
Non-negative matrix factorization (NMF) is one of the most powerful linear algebra tools, which has found application in various areas of data analysis, including computational biology. Despite numerous optimization methods devised for NMF, our comprehension of the inherent topological structure within factorizable…
Udit Raj, Slobodan Maletić, Sudeepto Bhattacharya
Persistent homology has been studied to better understand the structural properties and topological features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by nonpairwise interactions in a network. Studying of higher-order interactions (HoIs) of a system…
Cesar A. Ipanaque Zapata, Ayşe Borat
We present the notion of facet-complexity, C(L;K), for two simplicial complexes L and K, along with basic results for this numerical invariant. This invariant C(L;K) quantifies the "complexity" of the following question: When does there exist a facet simplicial map L → K? A facet simplicial map is a simplicial map that…
Sean T. Vittadello, Michael P. H. Stumpf
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…
J. Kogan
For a given pair of numbers (d, k), we establish a lower bound on the number of vertices in pure d-dimensional simplicial complexes with non-trivial homology in dimension k, and prove that this bound is tight. Furthermore, we solve the problem under the additional constraint of strong connectivity with respect to any…
Daniel Hernández Serrano, Darío Sánchez Gómez
Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be…
Manuel Miranda, Gissell Estrada-Rodriguez, Ernesto Estrada, Ginestra Bianconi + 3 more
'Ginestra Bianconi' 'Rubén J. Sánchez-García' 'Anthony Baptista' 'Hanlin Sun'] Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of…
Mariana Gómez-Schiavon, Hana El-Samad
Mathematical models continue to be essential for deepening our understanding of biology. On one extreme, simple or small-scale models help delineate general biological principles. However, the parsimony of detail in these models as well as their assumption of modularity and insulation make them inaccurate for…
Adrian Krzyzanowski, Axel Pahl, Michael Grigalunas, Herbert Waldmann
The fraction of sp3 hybridised carbons (Fsp3) and the fraction of stereogenic carbons (FCstereo) are two widely employed scores of molecular complexity with a strong link to biologically relevant features such as frequency, potency and selectivity of protein binding. However, due to their simplistic nature, they do not…
Davide Bolognini, Paolo Sentinelli
We disprove a long-standing open conjecture due to Simon stating that all skeleta of simplices are extendably shellable. In particular, for every $d \geq 3$ we provide a pure $d$-dimensional shellable simplicial complex which is not shelling completable.
Jakub Nowosad, Tomasz F. Stepinski
Comparing a large number of landscapes calls for using the smallest possible set of landscape metrics. The overall complexity of land-scape pattern is the single most important metric, but the standard set of landscape metrics lacks the bona fide indicator of complexity. Demonstrate that information theory provides a…
Lionel Zoubritzky, François-Xavier Coudert
We present here an open-source Julia library for the topological identification of crystalline materials, with algorithmic and computational improvements over the previously available software in the field, resulting in a speed increase of one order of magnitude. This new algorithm and implementation can therefore be…