Related papers: Simplicial Hausdorff Distance for Topological Data…
Quantum measurements often exhibit non-classical features, such as contextuality, which generalizes Bell's non-locality and serves as a resource in various quantum computation models. Existing frameworks have rigorously captured these…
Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a…
Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…
We study when co-evolving (or adaptive) higher-order networks defined on directed hypergraphs admit a simplicial description. Binary and triadic couplings are modelled by time-dependent weight tensors. Using representation theory of the…
Persistent homology has been studied to better understand the structural properties and topology features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by non-pairwise interactions…
Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be…
We introduce higher simplicial complexity of a simplicial complex $K$ and higher combinatorial complexity of a finite space $P$ (i.e. $P$ is a finite poset). We relate higher simplicial complexity with higher topological complexity of $|K|$…
In this paper, we introduce the higher analogues of contiguity distance and its relations with simplicial Lusternik-Schnirelmann category and discrete topological complexity. Also we study the effects of geometric realisation and…
This paper introduces a topological framework for interpreting the internal representations of Multilayer Perceptrons (MLPs). We construct a simplicial tower, a sequence of simplicial complexes connected by simplicial maps, that captures…
Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial…
Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description…
We construct new geometric realizations of simplicial and pre-simplicial sets where the standard $n$-simplex, viewed as the space of probability measures on $n+1$ elements, is replaced by the space of $(n+1)$-valued random variables, with…
Higher-order networks are able to capture the many-body interactions present in complex systems and to unveil new fundamental phenomena revealing the rich interplay between topology, geometry, and dynamics. Simplicial complexes are…
Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexity…
The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and…
The paper generalizes Thompson and Hilbert metric to the space of spectral densities. The resulting complete metric space has the differentiable structure of a Finsler manifold with explicit geodesics. The resulting distances are filtering…
The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying…
The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of…