D-dimensional Simplicial Complexes Research Articles

Pinning control of simplicial complexes

In this paper, pinning control of simplicial complexes is investigated. The D-dimensional simplicial complex can be successfully pinned to a desired state (i.e., an equilibrium of the intrinsic dynamics) under the proposed control scheme. The local stability of synchronization in the pinned simplicial complex is analyzed and discussed. The relationship between pinning synchronization and the individual dynamics, higher-order structures, coupling strengths, as well as the control strategy is established. Moreover, the effects of control gain and the number of controlled nodes on synchronizability are examined in detail. The obtained results can provide valuable references for designing optimal higher-order networks in practical applications.

Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data - An Algorithm and A Benchmark.

This paper introduces an efficient algorithm for persistence diagram computation, given an input piecewise linear scalar field f defined on a d-dimensional simplicial complex K, with d ≤ 3. Our work revisits the seminal algorithm "PairSimplices" [31], [103] with discrete Morse theory (DMT) [34], [80], which greatly reduces the number of input simplices to consider. Further, we also extend to DMT and accelerate the stratification strategy described in "PairSimplices" [31], [103] for the fast computation of the 0th and (d-1)th diagrams, noted D0(f) and Dd-1(f). Minima-saddle persistence pairs ( D0(f)) and saddle-maximum persistence pairs ( Dd-1(f)) are efficiently computed by processing , with a Union-Find , the unstable sets of 1-saddles and the stable sets of (d-1)-saddles. We provide a detailed description of the (optional) handling of the boundary component of K when processing (d-1)-saddles. This fast pre-computation for the dimensions 0 and (d-1) enables an aggressive specialization of [4] to the 3D case, which results in a drastic reduction of the number of input simplices for the computation of D1(f), the intermediate layer of the sandwich. Finally, we document several performance improvements via shared-memory parallelism. We provide an open-source implementation of our algorithm for reproducibility purposes. We also contribute a reproducible benchmark package, which exploits three-dimensional data from a public repository and compares our algorithm to a variety of publicly available implementations. Extensive experiments indicate that our algorithm improves by two orders of magnitude the time performance of the seminal "PairSimplices" algorithm it extends. Moreover, it also improves memory footprint and time performance over a selection of 14 competing approaches, with a substantial gain over the fastest available approaches, while producing a strictly identical output. We illustrate the utility of our contributions with an application to the fast and robust extraction of persistent 1-dimensional generators on surfaces, volume data and high-dimensional point clouds.

Hamiltonian paths, unit-interval complexes, and determinantal facet ideals

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Pósa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and (tight and weak) Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1)Every unit-interval strongly-connected d-dimensional simplicial complex is traceable.(This extends the well-known result “unit-interval connected graphs are traceable”.)(2)Every unit-interval d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian.(This extends the fact that “unit-interval 2-connected graphs are Hamiltonian”.)(3)Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Gröbner basis for a diagonal term order which is compatible with the traceability of the complex.(This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa–Vandebogert.)(4)Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution.(This extends the result by Kiani and Saeedi-Madani that “only the complete graph has a binomial edge ideal with linear resolution”.)(5)The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.

A Robustness Study for the Extraction of Watertight Volumetric Models from Boundary Representation Data

Geometrically induced topology plays a major role in applications such as simulations, navigation, spatial or spatio-temporal analysis and many more. This article computes geometrically induced topology useful for such applications and extends previous results by presenting the unpublished used algorithms to find inner disjoint (d+1)-dimensional simplicial complexes from a set of intersecting d-dimensional simplicial complexes which partly shape their B-Reps (Boundary Representations). CityGML has been chosen as the input data format for evaluation purposes. In this case, the input data consist of planar segment complexes whose triangulated polygons serve as the set of input triangle complexes for the computation of the tetrahedral model. The creation of the volumetric model and the computation of its geometrically induced topology is partly parallelized by decomposing the input data into smaller pices. A robustness analysis of the implementations is given by varying the angular precision and the positional precision of the epsilon heuristic inaccuracy model. The results are analysed spatially and topologically, summarised and presented. It turns out that one can extract most, but not all, volumes and that the numerical issues of computational geometry produce failures as well as a variety of outcomes.

Local-to-global Urysohn width estimates

Abstract The notion of the Urysohn d-width measures to what extent a metric space can be approximated by a d-dimensional simplicial complex. We investigate how local Urysohn width bounds on a Riemannian manifold affect its global width. We bound the 1-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of n-manifolds of considerable ( n - 1 ) {(n-1)} -width in which all unit balls have arbitrarily small 1-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.

Kirchhoff index of simplicial networks

We introduce a high-dimensional analogue of the Kirchhoff index which is also known as total effective resistance. This analogue, which we call the simplicial Kirchhoff index Kf(X), is defined to be the sum of the simplicial effective resistances of all (d+1)-subsets of the vertex set of a simplicial complex X of dimension d. This definition generalizes the Kirchoff index of a graph G defined as the sum of the effective resistance between all pairs of vertices of G. For a d-dimensional simplicial complex X with n vertices, we give formulas for the simplicial Kirchhoff index in terms of the pseudoinverse of the Laplacian LX in dimension d−1 and its eigenvalues:Kf(X)=n⋅trLX+=n⋅∑λ∈Λ+1λ, where LX+ is the pseudoinverse of LX, and Λ+ is the multi-set of non-zero eigenvalues of LX. Using this formula, we obtain an inequality for a high-dimensional analogue of algebraic connectivity and Kirchhoff index, and propose these quantities as measures of robustness of simplicial complexes. In addition, we derive its integral formula and relate this index to a simplicial dynamical system. We present an open problem for a combinatorial proof of our formula by relating the combinatorial interpretation of Rσ to rooted forests in higher dimensions.

Quantum statistics in Network Geometry with Fractional Flavor

Quantum statistics have been shown to emerge to describe the statistical properties of growing networks when nodes are associated to a fitness value. Recently it has been shown that quantum statistics emerge also in a growing simplicial complex model called Network Geometry with Flavor (NGF) which allows for the description of many-body interactions between the nodes. This model depends on an external parameter called flavor that is responsible for the underlying topology of the simplicial complex. When the flavor takes the value s = −1 the d-dimensional simplicial complex is a manifold in which every -dimensional face can only have an incidence number . In this case the faces of the simplicial complex are naturally described by the Bose–Einstein, Boltzmann and Fermi–Dirac distribution depending on their dimension. In this paper we extend the study of NGF to fractional values of the flavor s = −1/m in which every -dimensional face can only have incidence number . We show that in this case the statistical properties of the faces of the simplicial complex are described by the Bose–Einstein or the Fermi–Dirac distribution only. Finally we comment on the spectral properties of the networks constituting the underlying structure of the considered simplicial complexes.

A sharp threshold for spanning 2‐spheres in random 2‐complexes

A Hamiltonian cycle in a graph is a spanning subgraph that is homeomorphic to a circle. With this in mind, it is natural to define a Hamiltonian d-sphere in a d-dimensional simplicial complex as a spanning subcomplex that is homeomorphic to a d-dimensional sphere. We consider the Linial–Meshulam model for random simplicial complexes, and prove that there is a sharp threshold at p = e / γ n for the appearance of a Hamiltonian 2-sphere in a random 2-complex, where γ = 4 4 / 3 3 .

Extremal hypercuts and shadows of simplicial complexes

Let F be an n-vertex forest. An edge e ∉ F is said to be in F’s shadow if F ∪ {e} contains a cycle. It is easy to see that if F is an “almost tree”, i.e., a forest that contains two components, then its shadow contains at least $$\left\lfloor {\frac{{{{(n - 3)}^2}}}{4}} \right\rfloor $$ edges and this is tight. Equivalently, the largest number of edges in an n-vertex cut is $$\left\lfloor {\frac{{{n^2}}}{4}} \right\rfloor $$ . These notions have natural analogs in higher d-dimensional simplicial complexes which played a key role in several recent studies of random complexes. The higher-dimensional situation differs remarkably from the one-dimensional graph-theoretic case. In particular, the corresponding bounds depend on the underlying field of coefficients. In dimension d = 2 we derive the (tight) analogous theorems. We construct 2-dimensional “ℚ-almost-hypertrees” (defined below) with an empty shadow. We prove that an “ $$\mathbb{F}_2$$ -almost-hypertree” cannot have an empty shadow, and we determine its least possible size. We also construct large hyperforests whose shadow is empty over every field. For d ≥ 4 even, we construct a d-dimensional $$\mathbb{F}_2$$ -almost-hypertree whose shadow has vanishing density. Several intriguing open questions are mentioned as well.

Random Steiner systems and bounded degree coboundary expanders of every dimension

We introduce a new model of random d-dimensional simplicial complexes, for \(d\ge 2\), whose \((d-1)\)-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash’s recent result on designs (The existence of designs; arXiv:1401.3665, 2014), and the proof of the expansion uses techniques developed by Evra and Kaufman in (Bounded degree cosystolic expanders of every dimension; arXiv:1510.00839, 2015). This gives a full solution to a question raised in Dotterrer and Kahle (J Topol Anal 4(4): 499–514, 2012), which was solved in the two-dimensional case by Lubotzky and Meshulam (Adv Math 272: 743–760, 2015).

On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space

We consider d-dimensional simplicial complexes which can be PL embedded in the 2d-dimensional Euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in the $$(2d-1)$$ -dimensional Euclidean space. In addition, we use similar considerations on links of vertices to derive a new asymptotic upper bound on the total number of d-simplices in an (continuously) embeddable complex in 2d-space with n vertices, improving known upper bounds, for all $$d \ge 2$$ . Moreover, we show that the same asymptotic bound also applies to the size of d-complexes linklessly embeddable in the $$(2d+1)$$ -dimensional space.

The Threshold for Integer Homology in Random d-Complexes

Let $$Y \sim Y_d(n,p)$$Y~Yd(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology $$H_{d-1}(Y; \mathbb {Z})$$Hd-1(YźZ) is less than $$40d (d+1) \log n / n$$40d(d+1)logn/n. This bound is tight, up to a constant factor which depends on d.

On the phase transition in random simplicial complexes

It is well-known that the G(n;p) model of random graphs undergoes a dramatic change around p = 1 . It is here that the random graph, almost surely, contains cycles, and here it first acquires a giant (i.e., order ( n)) connected component. Several years ago, Linial and Meshulam have introduced theYd(n;p) model, a probability space of n-vertex d-dimensional simplicial complexes, where Y1(n;p) coincides with G(n;p). Within this model we prove a naturald-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from Yd(n;p). We also compute the real Betti numbers ofYd(n;p) forp = c=n. Finally, we establish the emergence of giant shadow at this threshold. (For d = 1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d 2 the emergence of the giant shadow is a first order phase transition.

Approximation algorithms for Max Morse Matching

In this paper, we prove that the Max Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch [1]. For D-dimensional simplicial complexes, we obtain a (D+1)(D2+D+1)-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. For D≥5, we describe a 2D-factor approximation algorithm for simplicial manifolds by processing the simplices in increasing order of dimension. This algorithm leads to 12-factor approximation for 3-manifolds and 49-factor approximation for 4-manifolds. This algorithm may also be applied to non-manifolds resulting in a 1(D+1)-factor approximation ratio. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.

Uniformly Cohen–Macaulay simplicial complexes and almost Gorenstein* simplicial complexes

In this paper, we study simplicial complexes whose Stanley–Reisner rings are almost Gorenstein and have a-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen–Macaulay simplicial complexes. A d-dimensional simplicial complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the simplicial complex Δ∖{F} is Cohen–Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension ≤2.

K-Decomposable Monomial Ideals

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable.

The threshold for d-collapsibility in random complexes*

In this paper we determine the threshold for d-collapsibility in the probabilistic model Xd(n,p) of d-dimensional simplicial complexes. A lower bound for this threshold was established in (Aronshtam and Linial, Random Struct. Algorithms 46 (2015) 26–35), and here we show that this is indeed the correct threshold. Namely, for every c > ηd, a complex drawn from is asymptotically almost surely not d-collapsible. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 260–269, 2016

Deciding Shellability of Simplicial Complexes with h-Assignments

If a d-dimensional pure simplicial complex C has a shelling, which is a specific total order of all facets of C, C is said to be shellable. We consider the problem of deciding whether C is shellable or not. This problem is solved in linear time of m, the number of all facets of C, if d=1 or C is a pseudomanifold in d=2. Otherwise it is unknown at this point whether the decision of shellability can be solved in polynomial time of m. Thus, for the latter case, we had no choice but to apply a brute force method to the decision problem; namely checking up to the m! ways to see if one can arrange all the m facets of C into a shelling. In this paper, we introduce a new concept, called h-assignment, to C and propose a practical method using h-assignments to decide whether C is shellable or not. Our method can make the decision of shellability of C by smaller sized computation than the brute force method.

Cognitive Behavioral Therapy, Sertraline, and Combination Effective for Childhood Anxiety Disorders

For d≥2, Walkup’s class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d−1)-spheres. Kalai showed that for d≥4, all connected members of K(d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f1≥5f0−152χ, with equality only for X∈K(4). Kühnel observed that this implies f0(f0−11)≥−15χ, with equality only for 2-neighborly members of K(4). Kühnel also asked if there is a triangulated 4-manifold with f0=15, χ=−4 (attaining equality in his lower bound). In this paper, guided by Kalai’s theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S3S1. Because of Kühnel’s inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d+3)-vertex sphere products Sd−1×S1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai’s result.

A Moore bound for simplicial complexes

Let X be a d-dimensional simplicial complex with N faces of dimension d − 1. Suppose that every (d − 1)-face of X is contained in at least k d + 2 faces of X, of dimension d. Extending the classical Moore bound for graphs, it is shown that X must contain a ball B of radius at mostlog k−d Nwhose d-dimensional homology Hd(B) is non-zero. The Ramanujan complexes constructed by Lubotzky, Samuels and Vishne are used to show that this upper bound on the radius of B cannot be improved by more than a multiplicative constant factor. This implies the classical Moore bound. Theorem A. g(G) < 2 logk−1 n +2 . Let dG(u, v) be the distance between the vertices u and v in the graph metric, and let Br(v )= {u ∈ V : dG(u, v) r} denote the ball of radius r around v. Define the acyclicity radius rv(G )o fG at the vertex v to be the maximal r such that the induced graph G(Br(v)) is acyclic. Let r(G) = minv∈V rv(G); then r(G )= � g(G)/2 �− 1. The asymptotic version of Moore's bound is equivalent to the following statement. Theorem A1. If δ(G )= k 3, then for every v ∈ V rv(G) � log k−1 n� . (1.2)