Related papers: Simplicial Complex Emergence on Directed Hypergrap…
Analyzing embedded simplicial complexes, such as triangular meshes and graphs, is an important problem in many fields. We propose a new approach for analyzing embedded simplicial complexes in a subdivision-invariant and isometry-invariant…
The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erd\H{o}s-R\'enyi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate…
The emergence of non-configurational symmetry is studied in a minimal example. The system under scrutiny consists of a dimeric hexagonal complex with configurational $C_3$ symmetry, formulated as a tight-binding model. An accidental…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can…
Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we…
Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations…
The paper is devoted to an approach to the bounded cohomology theory based on the theories of simplicial sets and Postnikov systems. In particular, the main results of the bounded cohomology theory of topological spaces are extended to…
Measurement data is often sampled irregularly, i.e., not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [1] and…
We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex $Y$ is then taken to be the minimal simplicial…
We construct a model structure on the category of ordered simplicial complexes, Quillen equivalent to the standard model structure on simplicial sets. This shows that simplicial complexes, which are fully combinatorial in nature, provide a…
We search for weakly interacting fixed points in extensions of the minimally supersymmetric standard model (MSSM). Necessary conditions lead to three distinct classes of anomaly-free extensions involving either new quark singlets, new quark…
We develop a framework for computing the homology of weighted simplicial complexes with coefficients in a discrete valuation ring. A weighted simplicial complex, $(X,v)$, introduced by Dawson [Cah. Topol. G\'{e}om. Diff\'{e}r. Cat\'{e}g. 31…
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…
For a fixed dimension $k\ge 1$, let us consider the randomly growing simplical complex on the vertex set $\{1,2,\dots,n\}$ defined as follows: We start with the empty complex, and for each $k+1$-element subset $\sigma$ of $\{1,2,\dots,n\}$,…
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group…
We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a ``small subsystem'' and an infinite dimensional one -- a ``reservoir''. The operator, which we call a…
The standard semantics of multi-agent epistemic logic S5 is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond…
Symmetries are an essential feature of complex networks as they regulate how the graph collective dynamics organizes into clustered states. We here show how to control network symmetries, and how to enforce patterned states of…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…