arXiv subjects

Shiquan Ren

Publications and source records attributed to Shiquan Ren.

At least 19 recordsLinked to original sources

arXiv2026

Hypergraphs on Riemannian manifolds and their double homology

In this paper, by constructing double homology for graded submanifolds of $Δ$-manifolds, we study the double homology of hyper(di)graphs on Riemannian manifolds. With the help of the canonical projection $π$ from ordered sequences to their underlying unordered sets, we prove a commutative diagram of the double homology of hyper(di)graphs on Riemannian manifolds and a commutative diagram of the spaces of differential forms of hyper(di)graphs on Riemannian manifolds. Besides, we give some relations between the fundamental goup(oid)s of hyper(di)graphs on surfaces and the (pure) braid groups on surfaces by certain homomorphisms induced from $π$. As applications, we study the homotopy types of filtrations for packings and coverings of balls on Riemannian manifolds with hypergraph constraints, and we characterize regular maps on Riemannian manifolds by geometric realizations of the independence complexes and hypergraphs on the manifolds.

math.AT↗

arXiv2026

Homological obstructions for regular embeddings of graphs

In [36, Section 8], the present author proposed the hypergraph obstruction for the existence of k-regular embeddings. In this paper, we develop the hypergraph obstruction concretely and give some homological obstructions for the k-regular embeddings of graphs by using the embedded homology of sub-hypergraphs of the (k-1)-skeleton of the independence complexes. Regular embeddings of graphs can be regarded equivalently as geometric realizations of the independence complexes and consequently be regarded equivalently as simplicial embeddings of the independence complexes into the vectorial matroids. We prove that if there exists a k-regular embedding of a graph, then there is an induced homomorphism from the embedded homology of the sub-hyper(di)graphs of the (k-1)-skeleton of the (directed) independence complexes to the homology of (directed) matroids. Moreover, if there exists certain triple of graphs where each graph has a k-regular embedding, then there are induced commutative diagrams of certain Mayer-Vietoris sequences of the embedded homology of hyper(di)graphs, the homology of (directed) independence complexes and the homology of matroids. Furthermore, if there exists certain couple of graphs where each graph has a k-regular embedding, then there are induced commutative diagrams of certain Kunneth type short exact sequences of the embedded homology of hyper(di)graphs, the homology of (directed) independence complexes and the homology of matroids.

math.AT↗

arXiv2025

Topological persistence of configuration spaces and independence complexes for digraphs

We study the topological persistence of the (path) configuration spaces and the (path) independence complexes for digraphs as well as their underlying graphs. We construct some canonical embeddings from the (path) independence complexes of the underlying graphs to the (path) independence complexes of the digraphs as well as some canonical embeddings between the (path) independence complexes induced by strong totally geodesic immersions and strong totally geodesic embeddings of (di)graphs. We apply the path homology to the path independence complexes of (di)graphs. As by-products, we derive some consequences about the Shannon capacities.

math.CO↗

arXiv2025

Persistent bundles over configuration spaces and obstructions for regular embeddings

We construct persistent bundles over configuration spaces of hard spheres and use the characteristic classes of these persistent bundles to give obstructions for embedding problems. The configuration spaces of $k$-hard spheres ${\rm Conf}_k(X,r)$, $r\geq 0$, give a $Σ_k$-equivariant filtration of the configuration space of $k$-points ${\rm Conf}_k(X)$. The filtered covering map from ${\rm Conf}_k(X,-)$ to ${\rm Conf}_k(X,-)/Σ_k$ gives a canonical persistent bundle $\boldsymbolξ(X,k,-)$. We use the Stiefel-Whitney class of $\boldsymbolξ(X,k,-)$, which is in the mod $2$ persistent cohomology ring of ${\rm Conf}_k(X,-)/Σ_k$, to give obstructions for $(k,r)$-regular embeddings and use the Chern class of $\boldsymbolξ(X,k,-)\otimes \mathbb{C}$, which is in the integral persistent cohomology ring of ${\rm Conf}_k(X,-)/Σ_k$, to give obstructions for complex $(k,r)$-regular embeddings. As applications, we discuss the geometric realizations of the independence complexes given by the regular embeddings. With the help of the persistent homology tools, the $k$-regular embedding problems of manifolds, the sphere-packing problems on manifolds, and the geometric realization problems of the independence complexes of graphs are prospectively to be computed approximately.

math.AT↗

arXiv2025

Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

Let $V$ be a finite set. Let $\mathcal{K}$ be a simplicial complex with its vertices in $V$. In this paper, we discuss some differential calculus on $V$. We construct some constrained homology groups of $\mathcal{K}$ by using the differential calculus on $V$. Moreover, we define an independent hypergraph to be the complement of a simplicial complex in the complete hypergraph on $V$. Let $\mathcal{L}$ be an independent hypergraph with its vertices in $V$. We construct some constrained cohomology groups of $\mathcal{L}$ by using the differential calculus on $V$.

math.AT↗

arXiv2025

Double complexes for configuration spaces and hypergraphs on manifolds

In this paper, we consider hypergraphs whose vertices are distinct points moving smoothly on a Riemannian manifold M. We take these hypergraphs as graded submanifolds of configuration spaces. We construct double complexes of differential forms on configuration spaces. Then we construct double complexes of differential forms on hypergraphs which are sub-double complexes of the double complex for the ambient configuration space. Among these double complexes for hypergraphs, the infimum double complex and the supremum double complex are quasi-isomorphic concerning the boundary maps induced from vertex deletion of the hyperedges. In particular, all the double complexes are identical if the hypergraph is a $Δ$-submanifold of the ambient configuration space.

math.AT↗

arXiv2024

Differential calculus for free algebra and constrained homology

In [Discrete differential calculus on simplicial complexes and constrained homology, Chin. Ann. Math. Ser. B 44(4), 615-640, 2023], the constrained (co)homology for simplicial complexes and independence hypergraphs is constructed via differential calculus on discrete sets. In this paper, we study the differential calculus for free algebra and subsequently study the homomorphisms of constrained (co)homology induced by inclusions of simplicial complexes and independence hypergraphs. We apply the differential calculus to hypergraphs. We realize simplicial complexes and independence hypergraphs as certain invariant traces of hypergraphs. As an application, we give the constrained persistent (co)homology for filtrations of simplicial complexes and filtrations of independence hypergraphs.

math.AT↗

arXiv2023

Differential Calculus on Hypergraphs and Mayer-Vietoris Sequences for the Constrained Persistent Homology

In this paper, we study the discrete differential calculus on hypergraphs by using the Kouzul complexes. We define the constrained (co)homology for hypergraphs and give the corresponding Mayer-Vietoris sequences. We prove the functoriality of the Mayer-Vietoris sequences for the constrained homology and the functoriality of the Mayer-Vietoris sequences for the constrained cohomology with respect to morphisms of hypergraphs induced by bijective maps between the vertices. Consequently, we obtain the Mayer-Vietoris sequences for the constrained persistent (co)homology for filtrations of hypergraphs. As applications, we propose the constrained persistent (co)homology as a tool for the computation of higher-dimensional persistent homology of large networks.

math.AT↗

arXiv2023

Weighted Analytic Torsion for Weighted Digraphs

In 2020, Alexander Grigor'yan, Yong Lin and Shing-Tung Yau [4] introduced the Reidemeister torsion and the analytic torsion for digraphs by means of the path complex and the path homology theory. Based on the analytic torsion for digraphs introduced in [4], we consider the notion of weighted analytic torsion for vertex-weighted digraphs. For any non-vanishing real functions $f$ and $g$ on the vertex set, we consider the vertex-weighted digraphs with the weights $(f,g)$. We calculate the $(f,g)$-weighted analytic torsion by examples and prove that the $(f,g)$-weighted analytic torsion only depend on the ratio $f/g$. In particular, if the weight is of the diagonal form $(f,f)$, then the weighted analytic torsion equals to the usual (un-weighted) torsion.

math.AT↗

arXiv2023

The Embedded Homology of Hypergraph Pairs

In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren, and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3)(2019) 479-500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer-Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs.

math.AT↗

arXiv2023

Weighted (Co)homology and Weighted Laplacian

In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $ϕ$-weighted coboundary operator induced by a weight function $ϕ$. Our weight function $ϕ$ is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.

math.AT↗

arXiv2023

Differential algebras on digraphs and parametrized homology

The theory of path homology for digraphs was developed by Alexander Grigor'yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. In this paper, we consider the differential algebras on digraphs and define the parametrized homology of digraphs as an analog of the path homology. We prove the functoriality of the parametrized homology of digraphs in Theorem 4.2. We also prove some Kunneth-type formulae for the parametrized homology of digraphs in Theorem 4.4.

math.CO↗

arXiv2023

The stability of persistent homology of hypergraphs

Hypergraph is the most general model for complex networks involving group interactions. Taking the ideas of path homology from Alexander Grigor'yan, Yong Lin, Yuri Muranov and Shing-Tung Yau [18-22], Stephane Bressan, Jingyan Li and the authors of this article introduced embedded homology of hypergraphs [6] in 2019, which has leaded to successful applications in protein-ligand binding network [24, 25] in 2021. A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persistent embedded homology as well as the persistent homology of the associated simplicial complex with respect to perturbations of the filtration on a hypergraph. We apply the persistent homology methods to morphisms of hypergraphs and prove the stability with respect to perturbations of the filtrations. We prove the constancy of the persistent Betti numbers under some conditions on the simple-homotopy types of hypergraphs.

math.AT↗

arXiv2022

Maps on random hypergraphs and random simplicial complexes

Let $L$ be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of $L$. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some maps on the space of probability functions on sub-hypergraphs of $L$. We study the compositions of these maps as well as their actions on the space of probability functions.

math.AT↗

arXiv2021

On The Discrete Morse Functions for Hypergraphs

A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We generalize the discrete Morse functions on simplicial complexes. We study the discrete Morse functions on hypergraphs as well as the discrete Morse functions on the (lower-)associated simplicial complexes of the hypergraphs.

math.CO↗

arXiv2021

Simplicial-like Identities for The Paths and The Regular Paths on Discrete Sets

Simplicial identities play an important and fundamental role in simplicial homotopy theory. On the other hand, the study of the paths and the regular paths on discrete sets is the foundation for the path-homology theory of digraphs. In this paper, by investigating some weighted face maps and weighted co-face maps on the space of the paths as well as the space of the regular paths, we prove some simplicial-like identities for the paths and the regular paths on discrete sets.

math.AT↗

arXiv2021

Weighted Simplicial Complexes and Weighted Analytic Torsions

A weighted simplicial complex is a simplicial complex with values (called weights) on the vertices. In this paper, we consider weighted simplicial complexes with $\mathbb{R}^2$-valued weights. We study the weighted homology and the weighted analytic torsion for such weighted simplicial complexes.

math.CO↗

arXiv2021

A Künneth Formula of Hypergraphs

In this paper, based on the embedded homology groups of hypergraphs defined in \cite{h1}, we define the product of hypergraphs and prove the corresponding Künneth formula of hypergraphs which can be generalized to the Künneth formula for the embedded homology of graded subsets of chain complexes with coefficients in a principal ideal domain.

math.AT↗