Simplicial Complex Research Articles

Proper proximality in non-positive curvature

abstract: Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups $G$ acting properly nonelementarily by isometries on a proper ${\rm CAT}(0)$ space $X$. More precisely, proper proximality holds in the presence of rank one isometries or when $X$ is a locally thick affine building with a minimal $G$-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary ${\rm CAT}(0)$ cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean ${\rm CAT}(0)$ simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana, and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.

On Two- and Three-valued Semantics for Impure Simplicial Complexes

On Two- and Three-valued Semantics for Impure Simplicial Complexes

On the Lefschetz property for quotients by monomial ideals containing squares of variables

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A ( Δ ) = k [ x 1 , … , x n ] / 〈 x 1 2 , … , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ . In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A ( Δ ) in terms of the simplicial complex Δ . We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A ( Δ ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.

Vertex numbers of simplicial complexes with free abelian fundamental group

Vertex numbers of simplicial complexes with free abelian fundamental group

Tight Complexes Are Golod

Abstract Golodness of a simplicial complex is defined algebraically in terms of the Stanley–Reisner ring, and it has been a long-standing problem to find its combinatorial characterization. Tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into Euclidean space, and has been studied in connection to minimal manifold triangulations. In this paper, by using an idea from toric topology, we prove that tight complexes are Golod, and as a corollary, we obtain that a triangulation of a closed connected orientable manifold is Golod if and only if it is tight, which is a combinatorial characterization of Golodness for manifold triangulations.

PLPCA: Persistent Laplacian-Enhanced PCA for Microarray Data Analysis.

Over the years, Principal Component Analysis (PCA) has served as the baseline approach for dimensionality reduction in gene expression data analysis. Its primary objective is to identify a subset of disease-causing genes from a vast pool of thousands of genes. However, PCA possesses inherent limitations that hinder its interpretability, introduce class ambiguity, and fail to capture complex geometric structures in the data. Although these limitations have been partially addressed in the literature by incorporating various regularizers, such as graph Laplacian regularization, existing PCA based methods still face challenges related to multiscale analysis and capturing higher-order interactions in the data. To address these challenges, we propose a novel approach called Persistent Laplacian-enhanced Principal Component Analysis (PLPCA). PLPCA amalgamates the advantages of earlier regularized PCA methods with persistent spectral graph theory, specifically persistent Laplacians derived from algebraic topology. In contrast to graph Laplacians, persistent Laplacians enable multiscale analysis through filtration and can incorporate higher-order simplicial complexes to capture higher-order interactions in the data. We evaluate and validate the performance of PLPCA using ten benchmark microarray data sets that exhibit a wide range of dimensions and data imbalance ratios. Our extensive studies over these data sets demonstrate that PLPCA provides up to 12% improvement to the current state-of-the-art PCA models on five evaluation metrics for classification tasks after dimensionality reduction.

Large simplicial complexes: universality, randomness, and ampleness

The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality (Even-Zohar et al. in Eur J Math 8(1):1–32, 2022; Farber and Mead in Topol Appl 272(22):107065, 2020; Farber et al. in J Appl Comput Topol 5(2):339–356, 2021). In the first part of the paper we concentrate on r-ample simplicial complexes which are high dimensional analogues of the r-e.c. graphs introduced originally by Erdős and Rényi (Acta Math Acad Sci Hungar 14:295–315, 1963), see also Bonato (Contrib Discrete Math 4(1):40–53, 2009). The class of r-ample complexes is useful for applications since these complexes allow extensions of subcomplexes of certain type in all possible ways; besides, r-ample complexes exhibit remarkable robustness properties. We discuss results about the existence of r-ample complexes and describe their probabilistic and deterministic constructions. The properties of random simplicial complexes in medial regime (Farber and Mead 2020) are important for this discussion since these complexes are ample, in certain range. We prove that the topological complexity of a random simplicial complex in the medial regime satisfies TC(X)≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{TC}(X)\\le 4$$\\end{document}, with probability tending to 1 as n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. There exists a unique (up to isomorphism) ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\infty $$\\end{document}-ample complex on countable set of vertexes (the Rado complex), and the second part of the paper surveys the results about universality, homogeneity, indestructibility and other important properties of this complex. The Appendix written by J.A. Barmak discusses connectivity of conic and ample complexes.

The total Betti number of the independence complex of ternary graphs

Given a graph G , the independence complex I(G) is the simplicial complex whose faces are the independent sets of V(G) . Let \tilde{b}_i denote the i -th reduced Betti number of I(G) , and let b(G) denote the sum of the \tilde{b}_i(G) ’s. A graph is ternary if it does not contain induced cycles with length divisible by 3. Kalai and Meshulam conjectured that b(G)\le 1 whenever G is ternary. We prove this conjecture. This extends a recent result proved by Chudnovsky, Scott, Seymour and Spirkl that for any ternary graph G , the number of independent sets with even cardinality and the number of independent sets with odd cardinality differ by at most 1.

Online-Offline Higher-Order Rumor Propagation Model Based on Quantum Cellular Automata Considering Social Adaptation

With the development of the Internet, the influence of rumors is more widespread. Studying the propagation law of rumors is an effective way to reduce the damage of rumors. We therefore propose a new two-layer rumor propagation model, which simultaneously introduces simplicial complexes into the online and offline network layers and considers the social adaptation of individuals. We derive the basic regeneration number, the rumor-free equilibrium, and the rumor-prevalence equilibrium using mean-field theory, and verify their accuracy through extensive computer simulations. The results show that the inclusion of higher-order interactions in both layers leads to richer dynamical phenomena. Specifically, higher-order interactions lead to bistable states, discontinuous transitions, and the coexistence of multiple rumor-prevalence equilibria, and increasing the infection parameter within a certain range reduces the outbreak size of rumors in another layer. In addition, a higher speed and degree of social adaptation, a stronger strength of coupled propagation enhancement effect, and a weaker strength of coupled recovery enhancement effect significantly increase the spread speed and size of rumors. Finally, we present and analyze two intervention strategies: improving public media literacy and building a truth space. We find that both intervention strategies can reduce the spread size and speed of rumors, but the synergy of two intervention strategies is more effective.

Some generalized centralities in higher-order networks represented by simplicial complexes

Abstract Higher-order interactions, that is, interactions among the units of group size greater than two, are a fundamental structural feature of a variety of complex systems across the scale. Simplicial complexes are combinatorial objects that can capture and model the higher-order interactions present in a given complex system and thus represent the complex system as a higher-order network comprising simplices. In this work, a given simplicial complex is viewed as a finite union of d-exclusive simplicial complexes. Thus, to represent a complex system as a higher-order network given by a simplicial complex that captures all orders of interactions present in the system, a family of symmetric adjacency tensors A(d) of dimension d + 1 and appropriate order has been used. Each adjacency tensor A(d) represents a d-exclusive simplicial complex and for d≥2 it represents exclusively higher-order interactions of the system. For characterizing the structure of d-exclusive simplicial complexes, the notion of generalized structural centrality indices namely, generalized betweenness centrality and generalized closeness centrality has been established by developing the concepts of generalized walk and generalized distance in the simplicial complex. Generalized centrality indices quantify the contribution of δ-simplices in any d-exclusive simplicial complex Δ, where δ<d and if d≥2, it describes the contribution of δ-faces to the higher-order interactions of Δ. These generalized centrality indices provide local structural descriptions, which lead to mesoscale insights into the simplicial complex that comprises the higher-order network. An important theorem providing a general technique for the characterization of connectedness in d-exclusive simplicial complexes in terms of irreducibility of its adjacency tensor has been established. The concepts developed in this work together with concepts of generalized simplex deletion in d-exclusive simplicial complexes have been illustrated using examples. The effect of deletions on the generalized centralities of the complexes in the examples has been discussed.

Eigenvector centrality in simplicial complexes of hypergraphs.

Hypergraph is the model of relations lying in clusters of objects. Identifying vital nodes is a fundamental problem in the analysis of the hypergraph. To reflect the multilayer feature of the hypergraph, in this paper, we deconstruct the hypergraph into a simplicial complex and analyze the homological dual relations of boundary and coboundary between simplices. For clarity, these two relations are summarized into a bidirectional graph, called the simplicial diagram, which provides a global framework for the exploration of the hypergraph. To determine the node importance in the hypergraph, we propose a parameter-free eigenvector centrality for weighted hypergraphs in terms of a simplicial complex, named Simplicial DualRank centrality. For each simplex, we define two indices of importance, the inner centrality and the outer centrality. Inner centrality transmits according to the relation of coboundary, which converts to outer centrality at the hyperlinks; in duality, outer centrality transmits according to the relation of boundary, which converts to inner centrality at the nodes. Therefore, a circuit of centrality is constructed on the simplicial diagram, the steady state of which defines the Simplicial DualRank centrality of all the simplices in the hypergraph. Moreover, we apply the Simplicial DualRank centrality to weighted complex networks, which results in a variant of the classical eigenvector centrality. Finally, experimental results in a science collaboration dataset show that the Simplicial DualRank can identify Nobel laureates from the prize-winning papers in Physics, top scientists should select collaborators more carefully to maintain their research quality, and scholars tend to find relatively effective collaborations in their future research.

Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behavior of phase oscillators on 5-clique simplicial assemblies.

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of four-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise K_{1} and triangle-based triple K_{2}≥0 couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter ν, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength K_{1}∈[-4,2] along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronization. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronized groups that hinder global network synchronization. Remarkably, these groups are differently affected by the size of the shared faces between neighboring 5-cliques and the presence of higher-order interactions. At K_{1}<0, partial synchronization is much higher in the compact community than in the assemblies of cliques sharing single nodes, at least occasionally. These structures also partially desynchronize at a lower triangle-based coupling K_{2} than the compact assembly. Broadening of the internal frequency distribution gradually reduces the synchronization level in the mixed and sparse communities, even at positive pairwise couplings. The order-parameter fluctuations in these partially synchronized states are quasicyclical with higher harmonics, described by multifractal analysis and broad singularity spectra.

Lattices, Garside structures and weakly modular graphs

In this article we study combinatorial non-positive curvature aspects of various simplicial complexes with natural A˜n shaped simplices, including Euclidean buildings of type A˜n and Cayley graphs of Garside groups and their quotients by the Garside elements. All these examples fit into the more general setting of lattices with order-increasing Z-actions and the associated lattice quotients proposed in a previous work by the first named author. We show that both the lattice quotients and the lattices themselves give rise to weakly modular graphs, which is a form of combinatorial non-positive curvature. We also show that several other objects fit into this setting of lattices/lattice quotients, including Artin complexes of Artin-Tits groups of type A˜n, a class of arc complexes and weak Garside groups arising from a categorical Garside structure in the sense of Bessis. Hence our result also implies to these objects and shows that they give weakly modular graphs. Along the way, we also clarify the relationship between categorical Garside structures, lattices with Z action and different classes of complexes studied this article. We use this point of view to describe the first examples of Garside groups with exotic properties, like non-linearity or rigidity results.

Higher cup products on hypercubic lattices: Application to lattice models of topological phases

In this paper, we derive the explicit formula for higher cup products on hypercubic lattices based on the recently developed geometrical interpretation on the simplicial complexes. We illustrate how this formalism can elucidate lattice constructions on hypercubic lattices for various models and derive them from spacetime actions. In particular, we demonstrate explicitly that the (3 + 1)D SPT S=12∫w22+w14 (where w1 and w2 are the first and second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model constructed on the cubic lattice. Other examples include the double-semion model and also the “fermionic” toric code in arbitrary dimensions on hypercubic lattices. In addition, we extend previous constructions of exact boson-fermion dualities and the Gu-Wen Grassmann integral to arbitrary dimensions. Another result that may be of independent interest is a derivation of a cochain-level action for the generalized double-semion model, reproducing a recently derived action on the cohomology level.

Hopf Monoids and Generalized Permutahedra

Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species.Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures.We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences.We highlight some main applications:We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account.We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of one such result for generalized permutahedra.We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday.We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions.We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus.

Large simple d-cycles in simplicial complexes

Large simple d-cycles in simplicial complexes

Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes

We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ) is at least 2−d Πk=0d−1ηk.As an application, we show that for every sufficiently thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings.We complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove• an upper bound of (d + 1)/2d on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite d-dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and• an upper bound of (d + 1)/2d + ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical building associated with {rm{G}}{{rm{L}}_{d + 2}}({mathbb{F}_q}) for any ε > 0 and sufficiently large q. This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes.

Dirac signal processing of higher-order topological signals

Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks. In real scenarios topological signal data might be noisy and an important task is to process these signals by improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as ‘square root’ of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

The homology of random simplicial complexes in the multi-parameter upper model

The homology of random simplicial complexes in the multi-parameter upper model

Purity and Separation for Oriented Matroids

Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in [ n ] [n] has the same cardinality. In this paper, we extend these notions and define M \mathcal {M} -separated collections for any oriented matroid M \mathcal {M} . We show that maximal by size M \mathcal {M} -separated collections are in bijection with fine zonotopal tilings (if M \mathcal {M} is a realizable oriented matroid), or with one-element liftings of M \mathcal {M} in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid M \mathcal {M} is pure if M \mathcal {M} -separated collections form a pure simplicial complex, i.e., any maximal by inclusion M \mathcal {M} -separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank 3 3 , graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank 3 3 is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an n n -gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank 3 3 , graphical, uniform).